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![Page 1: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/1.jpg)
Finitary 2-categories and their2-representations
Volodymyr MazorĚuk(Uppsala University)
Workshop “NWDR 17”July 12, 2013, Wuppertal, Germany
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 1/23
![Page 2: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/2.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 3: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/3.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 4: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/4.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 5: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/5.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 6: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/6.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 7: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/7.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 8: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/8.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 9: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/9.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 10: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/10.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 11: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/11.jpg)
Finite dimensional algebras and k-linear categories
k — algebraically closed field
A — finite dimensional algebra over k
mod-A — the category of right finitely generated A-modules
P1,P2, . . . ,Pk — indecomposable projectives in mod-A up to iso.
P — the full subcategory of mod-A with objects P1,P2, . . . ,Pk
P — k-linear category (i.e. enriched over k-mod)
P-mod — the category of k-linear functors from P to k-mod
Theorem. P-mod ∼= mod-A.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 2/23
![Page 12: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/12.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 13: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/13.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 14: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/14.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 15: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/15.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 16: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/16.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 17: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/17.jpg)
Finitary k-linear categories
P is finitary in the following sense:
I It has finitely many objects.I Morphism spaces are finite dimensional over k.
Additionally: Each idPi is primitive
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 3/23
![Page 18: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/18.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 19: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/19.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 20: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/20.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 21: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/21.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 22: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/22.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 23: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/23.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 24: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/24.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 25: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/25.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 26: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/26.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 27: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/27.jpg)
2-categories: definition
Definition. A 2-category is a category enriched over the monoidalcategory Cat of small categories (in the latter the monoidal structure isinduced by the cartesian product).
This means that a 2-category C is given by the following data:
I objects of C ;I small categories C(i, j) of morphisms;I bifunctorial composition C(j, k)× C(i, j)→ C(i, k);I identity objects 1j;
which are subject to the obvious set of (strict) axioms.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 4/23
![Page 28: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/28.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 29: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/29.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 30: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/30.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 31: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/31.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 32: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/32.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 33: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/33.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 34: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/34.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 35: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/35.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 36: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/36.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 37: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/37.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 38: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/38.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 39: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/39.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 40: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/40.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 41: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/41.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 42: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/42.jpg)
2-categories: terminology and the principal example
Terminology.
I An object in C(i, j) is called a 1-morphism of C .I A morphism in C(i, j) is called a 2-morphism of C .I Composition in C(i, j) is called vertical and denoted ◦1.I Composition in C is called horizontal and denoted ◦0.
Principal example. The category Cat is a 2-category.
I Objects of Cat are small categories.I 1-morphisms in Cat are functors.I 2-morphisms in Cat are natural transformations.I Composition is the usual composition.I Identity 1-morphisms are the identity functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 5/23
![Page 43: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/43.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 44: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/44.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 45: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/45.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 46: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/46.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 47: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/47.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 48: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/48.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 49: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/49.jpg)
Finitary 2-categories
Definition: A 2-category C is called finitary (over k) provided that
I C has finitely many objects;I each C(i, j) is additive, k-linear, idempotent split, with finitely
many indecomposables (up to isomorphism);I all spaces of 2-morphisms are finite dimensional (over k);I the identity 1-morphisms are indecomposable.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 6/23
![Page 50: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/50.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 51: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/51.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 52: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/52.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 53: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/53.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 54: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/54.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 55: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/55.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 56: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/56.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 57: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/57.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 58: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/58.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
![Page 59: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/59.jpg)
Example 1: projective functors on A-mod
A — finite dimensional connected associative k-algebra
A-mod — the category of finitely generated left A-modules
Definition: F : A-mod→ A-mod is projective is it is isomorphic totensoring with a projective bimodule.
Definition: [M.-Miemietz] The 2-category PA is defined as follows:
I PA has one object ♣ (which is identified with A-mod);I 1-morphisms in PA(♣,♣) are functors isomorphic to direct sums of
the identity and projective functors;I 2-morphisms in PA(♣,♣) are natural transformations of functors.
Remark: PA is a “simple” finitary 2-category (∼ Artin-Wedderburn)
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 7/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: projection functors
Γ — finite acyclic quiver
kΓ — the path algebra of Γ
For i ∈ Γ let Fi : kΓ-mod→ kΓ-mod be the i-th projection functor“factor out the trace of the i-th simple module”
Fact: Fi : kΓ-inj→ kΓ-inj.
Gi : kΓ-mod→ kΓ-mod — the unique (up to iso) left exact functor suchthat Gi|kΓ-inj ∼= Fi|kΓ-inj
Fact: Gi is exact.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 8/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 2: the 2-category generated by projection functors
Definition: [Grensing-M.] The 2-category GΓ is defined as follows:
I GΓ has one object ♣ (which is identified with kΓ-mod);I 1-morphisms in GΓ(♣,♣) are functors isomorphic to direct
summands of sums of compositions of projection functors;I 2-morphisms in GΓ(♣,♣) are natural transformations of functors.
Fact: GΓ is finitary.
To check: GΓ has only finitely many 1-morphisms up to iso.
Problem: Classify indecomposable 1-morphisms in GΓ.
Inspired by: A.-L. Grensing’s PhD Thesis.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 9/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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Example 3: Soergel bimodules
(W , S) — finite Coxeter system
C — the coinvariant algebra of a fixed geometric realization of (W , S)
Bw — the Soergel C-C-bimodule corresponding to w ∈W
Definition: [M.-Miemietz] The 2-category S (W ,S) is defined as follows:
I S (W ,S) has one object ♣ (which is identified with C-mod);I 1-morphisms in S (W ,S)(♣,♣) are functors isomorphic to tensoring
with directs sums of Soergel bimodules;I 2-morphisms in S (W ,S)(♣,♣) are natural transformations of
functors.
Inspired by: W. Soergel’s combinatorial description of projectivefunctors acting on the regular block of the BGG category O.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 10/23
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2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
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2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 90: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/90.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 91: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/91.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 92: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/92.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
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2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 94: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/94.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 95: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/95.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 96: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/96.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 97: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/97.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 98: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/98.jpg)
2-functors and 2-representations
A and C — two 2-categories
Definition. A 2-functor F : A → C is a functor which sends1-morphisms to 1-morphisms and 2-morphisms to 2-morphisms in a waythat is coordinated with all the categorical structures (domains,codomains, identities and compositions).
Definition. A 2-representation of a 2-category C is a 2-functor from Cto some “classical” 2-category.
Example. Categories PA, GΓ and S (W ,S) were defined using thecorresponding defining 2-representation
“Classical” 2-representations:
I in Cat;I in the 2-category Add of additive categories and additive functors;I in the 2-subcategory add of Add consisting of all fully additive
categories with finitely many isoclasses of indecomposable objects;I a the 2-category ab of abelian categories and exact functors.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 11/23
![Page 99: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/99.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
![Page 100: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/100.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
![Page 101: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/101.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
![Page 102: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/102.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
![Page 103: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/103.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
![Page 104: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/104.jpg)
Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
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Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
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Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
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Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
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Decategorification
A — additive category
Definition. The split Gorthendieck group [A]⊕ of A is the quotient ofthe free abelian group generated by [M], M ∈ A, modulo the relations[X ]− [Y ]− [Z ] whenever X ∼= Y ⊕ Z in A.
A — finitary 2-category
Definition. The decategorification [A ] of A is a category with sameobjects as A , with morphisms defined via [A ](i, j) := [A(i, j))]⊕ andwith multiplication and identities induced from A .
Examples.
I [S (W ,S)](♣,♣) ∼= ZW .I for Γ = • // • // . . . // • (with n − 1 vertices),
[GΓ](♣,♣) is isomorphic to the integral semigroup algebra of then-th Catalan monoid Cn, that is the monoid of order-preserving andorder-decreasing transformations of 1, 2, . . . , n.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 12/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
![Page 110: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/110.jpg)
Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
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Decategorification of 2-representations
A — finitary 2-category
A 2-representation of A can often be understood as a functorial action ofA on a collection of certain categories.
If A acts on M , then [A ] acts on [M ](⊕).
Hence: A 2-representation of A decategorifies to a usual representationof [A ].
Examples:
I The functorial action of S (W ,S) on O0 decategorifies to the regularrepresentation of ZW [Bernstein–S. Gelfand].
I for Γ = • // • // . . . // • (with n − 1 vertices), thedefining representation of GΓ decategorifies to the (linearization ofthe) defining representation of Cn.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 13/23
![Page 119: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/119.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C-afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
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Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C-afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 121: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/121.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C-afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 122: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/122.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C-afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 123: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/123.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 124: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/124.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 125: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/125.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
![Page 126: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/126.jpg)
Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
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Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
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Principal 2-representations
C — finitary 2-category
Definition. For i ∈ C the corresponding principal 2-representation Pi ofC is defined as the 2-functor C(i,−).
We have:
I j 7→ C(i, j);I F : j→ k 7→ F ◦ − : C(i, j)→ C(i, k);I α : F→ G 7→ α ◦0 id− : F ◦ − → G ◦ −.
C -afmod — 2-category of additive finitary 2-representations of C(morphisms – additive strong transformations)
Fact. Pi is projective in C-afmod in the sense that it preserves all smallcolimits.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 14/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C -proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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2-Morita Theory
C — finitary 2-category
C -proj — 2-category of projective 2-representations of C
Definition. A progenerator is a projective 2-representation P of C suchthat any other projective 2-representation is a retract of some2-representation from add(P).
Example.⊕i∈C
Pi is a progenerator.
Theorem: [M.-Miemietz] A , C — finitary 2-categories. Then TFAE:
I A-afmod and C-afmod are biequivalent.I A-proj and C-proj are biequivalent.I There is a progenerator P ∈ C -proj such that the endomorphism
2-category of P is biequivalent to Aop.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 15/23
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Multisemigroup of a finitary category
F ,G are composable indecomposable 1-morphisms in C , then
F ◦ G ∼=∑
H indec.
H⊕mHF,G .
C — finitary 2-category
Definition. The multisemigroup (S(C), �) of C is defined as follows:S(C) is the set of isomorphism classes of 1-morphisms in C (including 0),
[F ] � [G ] =
{{[H] : mH
F ,G 6= 0}, F ◦ G defined and 6= 0;
0, else.
Sometimes S(C)′ := S(C) \ {0} is closed with respect to �.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 16/23
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Multisemigroup of a finitary category
F ,G are composable indecomposable 1-morphisms in C , then
F ◦ G ∼=∑
H indec.
H⊕mHF,G .
C — finitary 2-category
Definition. The multisemigroup (S(C), �) of C is defined as follows:S(C) is the set of isomorphism classes of 1-morphisms in C (including 0),
[F ] � [G ] =
{{[H] : mH
F ,G 6= 0}, F ◦ G defined and 6= 0;
0, else.
Sometimes S(C)′ := S(C) \ {0} is closed with respect to �.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 16/23
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Multisemigroup of a finitary category
F ,G are composable indecomposable 1-morphisms in C , then
F ◦ G ∼=∑
H indec.
H⊕mHF,G .
C — finitary 2-category
Definition. The multisemigroup (S(C), �) of C is defined as follows:S(C) is the set of isomorphism classes of 1-morphisms in C (including 0),
[F ] � [G ] =
{{[H] : mH
F ,G 6= 0}, F ◦ G defined and 6= 0;
0, else.
Sometimes S(C)′ := S(C) \ {0} is closed with respect to �.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 16/23
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Multisemigroup of a finitary category
F ,G are composable indecomposable 1-morphisms in C , then
F ◦ G ∼=∑
H indec.
H⊕mHF,G .
C — finitary 2-category
Definition. The multisemigroup (S(C), �) of C is defined as follows:S(C) is the set of isomorphism classes of 1-morphisms in C (including 0),
[F ] � [G ] =
{{[H] : mH
F ,G 6= 0}, F ◦ G defined and 6= 0;
0, else.
Sometimes S(C)′ := S(C) \ {0} is closed with respect to �.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 16/23
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Multisemigroup of a finitary category
F ,G are composable indecomposable 1-morphisms in C , then
F ◦ G ∼=∑
H indec.
H⊕mHF,G .
C — finitary 2-category
Definition. The multisemigroup (S(C), �) of C is defined as follows:S(C) is the set of isomorphism classes of 1-morphisms in C (including 0),
[F ] � [G ] =
{{[H] : mH
F ,G 6= 0}, F ◦ G defined and 6= 0;
0, else.
Sometimes S(C)′ := S(C) \ {0} is closed with respect to �.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 16/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
![Page 152: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/152.jpg)
Cells
C — finitary 2-category
(S(C), �) — the multisemigroup of C
Definition. [F ] ∼L [G ] if S(C) � [F ] = S(C) � [G ]
Definition. Equivalence classes of ∼L are called left cells.
Similarly: ∼R (right cells) and ∼J (two-sided cells)
Examples:
I if (S(C), �) is a semigroup, we get Green’s relationsI for S (W ,S) we get Kazhdan-Lusztig cells
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 17/23
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Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
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Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
![Page 155: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/155.jpg)
Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
![Page 156: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/156.jpg)
Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
![Page 157: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/157.jpg)
Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
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Abelianization
Definition. The abelianization 2-functor · is defined as follows:
given M ∈ C-afmod and i ∈ C the category M(i) has objects
Xα // Y , X ,Y ∈M(i), α : X → Y ;
and morphisms
Xα //
β
��
Y
γ
��X ′
α′// Y ′
modulo Xα //
β
��
Y
α′ξ
��
ξ
~~||||
||||
X ′α′
// Y ′
the 2-action of C is defined component-wise
extends to a 2-functor component-wise
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 18/23
![Page 159: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/159.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 160: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/160.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 161: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/161.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 162: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/162.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 163: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/163.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 164: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/164.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 165: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/165.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 166: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/166.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 167: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/167.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 168: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/168.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 169: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/169.jpg)
Fiat 2-categories
C — finitary 2-category
Definition. C is fiat (finitary - involution - adjunction - two category)provided that it has
I a weak object preserving involution ?;
I adjunction morphisms F ◦ F ? → 1i and 1j → F ? ◦ F .
Examples.
I S (W ,S) is fiat;
I PA is fiat iff A is self-injective and weakly symmetric;
I GΓ is not fiat.
Fact. If C is fiat, then each left cell of C has a unique Duflo involution
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 19/23
![Page 170: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/170.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
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Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 172: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/172.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 173: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/173.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 174: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/174.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 175: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/175.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 176: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/176.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 177: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/177.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 178: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/178.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 179: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/179.jpg)
Cell 2-representation
C — fiat category;
L — left cell of C ;
i ∈ C — the source object for 1-morphisms in L;
Pi — principal 2-representation
Pi — its abelianization
GL – Duflo involution
LGL – the corresponding simple module in Pi(i)
Theorem. X := add{F LGL : F ∈ L} is closed under the action of C
Definition. The cell 2-representation CL of C corresponding to L is thefinitary 2-representation obtained by restricting the action of C to X .
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 20/23
![Page 180: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/180.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 181: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/181.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 182: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/182.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 183: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/183.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 184: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/184.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 185: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/185.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 186: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/186.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 187: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/187.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 188: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/188.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 189: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/189.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 190: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/190.jpg)
Properties of cell 2-representation
Assume:
I J be a 2-sided cell of C ;
I different left cells inside J are not comparable w.r.t. the left order;
I for any L,R ⊂ J we have |L ∩ R| = 1;
I the function F 7→ mF , where F ∗ ◦ F = mFH is constant on rightcells of J .
Theorem. [M.-Miemietz]
I For any two left cells L and L′ of J the corresponding cell2-representations CL and CL′ are equivalent.
I EndC CL ∼= k-mod.
I If C admits a positive grading, then the last technical assumption isredundant.
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 21/23
![Page 191: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/191.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 192: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/192.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 193: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/193.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 194: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/194.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 195: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/195.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 196: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/196.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 197: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/197.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 198: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/198.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 199: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/199.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 200: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/200.jpg)
Some challenges
I find most suitable setup;
I what are simple 2-representations?
I any Jordan-Hölder theory?
I Morita theory for abelian representations;
I general categorification algorithms;
I any homological methods for 2-representations?
I understand combinatorics of available examples;
I many more...
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 22/23
![Page 201: Finitary 2-categories and their 2-representations › ~mazor › PREPRINTS › DOKLADY › wuppertal.pdf · 2013-07-12 · 2-functorsand2-representations A andC —two2-categories](https://reader033.vdocuments.mx/reader033/viewer/2022060207/5f03d04b7e708231d40ae65e/html5/thumbnails/201.jpg)
THANK YOU!!!
Volodymyr Mazorchuk Finitary 2-categories and their 2-representations 23/23