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Accessible aspects of 2-category theory John Bourke Department of Mathematics and Statistics Masaryk University CT2019, Edinburgh John Bourke Accessible aspects of 2-category theory

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Page 1: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Accessible aspects of 2-category theory

John Bourke

Department of Mathematics and StatisticsMasaryk University

CT2019, Edinburgh

John Bourke Accessible aspects of 2-category theory

Page 2: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Plan

1. Locally presentable categories and accessible categories.

2. Two dimensional universal algebra.

3. A general approach to accessibility of weak/cofibrantcategorical structures.

4. Quasicategories and related structures (w’ Lack/Vokrınek).

John Bourke Accessible aspects of 2-category theory

Page 3: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Plan

1. Locally presentable categories and accessible categories.

2. Two dimensional universal algebra.

3. A general approach to accessibility of weak/cofibrantcategorical structures.

4. Quasicategories and related structures (w’ Lack/Vokrınek).

John Bourke Accessible aspects of 2-category theory

Page 4: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Plan

1. Locally presentable categories and accessible categories.

2. Two dimensional universal algebra.

3. A general approach to accessibility of weak/cofibrantcategorical structures.

4. Quasicategories and related structures (w’ Lack/Vokrınek).

John Bourke Accessible aspects of 2-category theory

Page 5: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Plan

1. Locally presentable categories and accessible categories.

2. Two dimensional universal algebra.

3. A general approach to accessibility of weak/cofibrantcategorical structures.

4. Quasicategories and related structures (w’ Lack/Vokrınek).

John Bourke Accessible aspects of 2-category theory

Page 6: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Locally presentable and accessible categories

I C is λ-accessible if it has a set of λ-presentable objects ofwhich every object is a λ-filtered colimit. Accessible ifλ-accessible for some λ. (Book of Makkai-Pare 1989)

I Locally presentable = accessible + complete/cocomplete.(GU 1971)

I Capture “algebraic” categories.

I Very nice: easy to construct adjoint functors between assolution set condition easy to verify. Stable under lots of limitconstructions.

I Interested in the world in between accessible and locallypresentable! E.g. weakly locally λ-presentable: λ-accessibleand products/weak colimits. (AR1990s)

John Bourke Accessible aspects of 2-category theory

Page 7: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Locally presentable and accessible categories

I C is λ-accessible if it has a set of λ-presentable objects ofwhich every object is a λ-filtered colimit. Accessible ifλ-accessible for some λ. (Book of Makkai-Pare 1989)

I Locally presentable = accessible + complete/cocomplete.(GU 1971)

I Capture “algebraic” categories.

I Very nice: easy to construct adjoint functors between assolution set condition easy to verify. Stable under lots of limitconstructions.

I Interested in the world in between accessible and locallypresentable! E.g. weakly locally λ-presentable: λ-accessibleand products/weak colimits. (AR1990s)

John Bourke Accessible aspects of 2-category theory

Page 8: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Locally presentable and accessible categories

I C is λ-accessible if it has a set of λ-presentable objects ofwhich every object is a λ-filtered colimit. Accessible ifλ-accessible for some λ. (Book of Makkai-Pare 1989)

I Locally presentable = accessible + complete/cocomplete.(GU 1971)

I Capture “algebraic” categories.

I Very nice: easy to construct adjoint functors between assolution set condition easy to verify. Stable under lots of limitconstructions.

I Interested in the world in between accessible and locallypresentable! E.g. weakly locally λ-presentable: λ-accessibleand products/weak colimits. (AR1990s)

John Bourke Accessible aspects of 2-category theory

Page 9: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Locally presentable and accessible categories

I C is λ-accessible if it has a set of λ-presentable objects ofwhich every object is a λ-filtered colimit. Accessible ifλ-accessible for some λ. (Book of Makkai-Pare 1989)

I Locally presentable = accessible + complete/cocomplete.(GU 1971)

I Capture “algebraic” categories.

I Very nice: easy to construct adjoint functors between assolution set condition easy to verify. Stable under lots of limitconstructions.

I Interested in the world in between accessible and locallypresentable! E.g. weakly locally λ-presentable: λ-accessibleand products/weak colimits. (AR1990s)

John Bourke Accessible aspects of 2-category theory

Page 10: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Locally presentable and accessible categories

I C is λ-accessible if it has a set of λ-presentable objects ofwhich every object is a λ-filtered colimit. Accessible ifλ-accessible for some λ. (Book of Makkai-Pare 1989)

I Locally presentable = accessible + complete/cocomplete.(GU 1971)

I Capture “algebraic” categories.

I Very nice: easy to construct adjoint functors between assolution set condition easy to verify. Stable under lots of limitconstructions.

I Interested in the world in between accessible and locallypresentable! E.g. weakly locally λ-presentable: λ-accessibleand products/weak colimits. (AR1990s)

John Bourke Accessible aspects of 2-category theory

Page 11: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i .

Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 12: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 13: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 14: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 15: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 16: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 17: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Two dimensional universal algebra – Sydney 1980s

I Two-dimensional universal algebra: e.g. 2-category MonCatpof monoidal categories and strong monoidal functors:f (a⊗ b) ∼= fa⊗ fb and f (i) ∼= i . Also SMonCatp,Lex , Reg .

I What properties do such 2-categories of pseudomaps have?

I Not all limits (e.g. equalisers/pullbacks) so not locallypresentable.

I BKP89: pie limits – those nice 2-d limits like products,comma objects, pseudolimits whose defining cone does notimpose any equations between arrows.

I BKPS89: 2-categories of weak structures (e.g. algebras for aflexible – a.k.a cofibrant – 2-monad) also admit splittings ofidempotents (in summary, flexible/cofibrant weighted limits).

I Today, we’ll see such 2-cats are moreover accessible.

John Bourke Accessible aspects of 2-category theory

Page 18: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 19: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 20: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 21: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 22: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 23: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai and generalised sketches 1

I After Phd in Sydney, was postdoc in Brno where Makkai was.

I Makkai interested in developing theory of locally presentable2-categories/bicategories involving filtered bicolimits etc.

I Some years later, I read his paper “Generalised sketches . . . ”in which he described structures defined by universalproperties and their pseudomaps as cats of injectives – itfollows such categories of weak maps are genuinely accessible!

I Lack and Rosicky also observed cat NHom of bicategories andnormal pseudofunctors is accessible, by identifying bicategorieswith their 2-nerves – certain injectives. [LR2012]

I Visited Makkai in Budapest 2015 and chatted about all ofthis.

I Will describe general approach to accessibility of weak objectsand weak maps. Some parts worked out by Makkai and someby me.

John Bourke Accessible aspects of 2-category theory

Page 24: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 25: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 26: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 27: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 28: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}

(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 29: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}

(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 30: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}

(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

Page 31: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}

I Then XT is set of all terminal objects, so TObp ↪→ Sk is thefull subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

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Makkai’s generalised sketches 2 – terminal objects

I Consider cat Sk of 3-truncated simplicial sets X equippedwith set XT ⊂ X [0] of marked 0-simplices, and simplicialmaps preserving these. The cat Sk is l.p.

I Will describe cat TObp of small cats with terminal object andpseudomaps as injectivity class in category Sk.

I Fully faithful functor TObp → Sk sending C to truncatednerve C with CT the set of all terminal objects.

(1) Add in inner horns (and codiagonals) with trivial markings tocapture categories with a distinguished set of objects asinjectives in Sk .

(2) Non-emptiness of XT : ∅→ {•}(3) Objects in XT are terminal 1: {0 1} → {0→ 1}(4) Objects in XT are terminal 2: {0⇒ 1} → {0→ 1}(5) Repleteness of XT : {0 ∼= 1} → {0 ∼= 1}I Then XT is set of all terminal objects, so TObp ↪→ Sk is the

full subcat of injectives, so accessible.

John Bourke Accessible aspects of 2-category theory

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Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 34: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 35: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 36: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 37: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 38: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Properties of 2-categories of weak objects and pseudomaps

A locally small 2-category C belongs to K if:

I C has flexible limits;

I its underlying category is accessible with filtered colimits;

I finite flexible limits (those generated by finite products,inserters and equifiers and splittings of idempotents) commutewith filtered colimits in C.

Morphisms of K are 2-functors preserving flexible limits and filteredcolimits; 2-cells are 2-natural transformations.

I For C ∈ K we say that C ∈ K+ if the full subcategoryRE (C)→ Arr(C) of retract equivalences in C is accessible andaccessibly embedded in the arrow category of C.

Proposition

K+ is closed in 2-Cat under bilimits – in particular, pullbacks ofisofibrations.

John Bourke Accessible aspects of 2-category theory

Page 39: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Cellular 2-categories

I Let J = {ji : δDi → Di : i = 0, 1, 2, 3} be the generatingcofibrations in 2-Cat.

I δD0 → D0: ∅→ (•).

I δD1 → D1: (0 1)→→ (0→ 1).

I δD2 → D2: (0 1)%%99 (0 1)//

%%99��

I δD3 → D3: (0 1)%%99�� �� (0 1)//

%%99��

John Bourke Accessible aspects of 2-category theory

Page 40: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Cellular 2-categories

I Let J = {ji : δDi → Di : i = 0, 1, 2, 3} be the generatingcofibrations in 2-Cat.

I δD0 → D0: ∅→ (•).

I δD1 → D1: (0 1)→→ (0→ 1).

I δD2 → D2: (0 1)%%99 (0 1)//

%%99��

I δD3 → D3: (0 1)%%99�� �� (0 1)//

%%99��

John Bourke Accessible aspects of 2-category theory

Page 41: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 42: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.

Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 43: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.

Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 44: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.

A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 45: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.

Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 46: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

The core result

Let Ps(A, C) denote the 2-category of 2-functors andpseudonatural transformations.

TheoremLet C ∈ K+. Then Ps(Di , C)→ Ps(δDi , C) ∈ K+ for i = 0, 1, 2, 3and each such 2-category has flexible limits and filtered colimitspointwise.

Proof.Tricky bit to prove that Ps(D1, C) is accessible – i.e. the cat ofpseudocommutative squares.Taking the pseudolimit of f : A→ B in C gives span A← Pf → B,and pseudocommuting squares correspond to strict maps of theassociated spans.A span A← R → B is of this form iff R → A is a retractequivalence and R → A× B is a discrete isofibration.Using accessibility of these notions, we deduce result.

John Bourke Accessible aspects of 2-category theory

Page 47: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+. Bottom leg preserves limits andfiltered colimits, and so belongs to K+.K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 48: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+. Bottom leg preserves limits andfiltered colimits, and so belongs to K+.K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 49: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+. Bottom leg preserves limits andfiltered colimits, and so belongs to K+.K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 50: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+.

Bottom leg preserves limits andfiltered colimits, and so belongs to K+.K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 51: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+. Bottom leg preserves limits andfiltered colimits, and so belongs to K+.

K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 52: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Examples – monoidal cats

I Goal: construction MonCatp as cocellular object – iteratedpullbacks of the maps Ps(Di ,Cat)→ Ps(δDi ,Cat).

I For magma structure form pullback in 2-Cat:

T -Alg1

U1

��

// Ps(D1,Cat)

Ps(j1,Cat)

��Cat

C 7→(C2,C)// Ps(δD1,Cat)

X 2 Y 2

X Y

f 2 //

mX

��

mY

��

f//

∼=f

I Pseudomorphisms as above right.

I Right leg isofibration in K+. Bottom leg preserves limits andfiltered colimits, and so belongs to K+.K+ closed in 2-Catunder pullbacks of isofibrations – hence T -Alg1 → Cat ∈ K+.

John Bourke Accessible aspects of 2-category theory

Page 53: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Monoidal cats 2

I Add associators by forming a pullback

T -Alg2

��

// Ps(D2,Cat)

Ps(K1,Cat)

��T -Alg1

R // Ps(δD1,Cat)

X 3 X 2

X X 2

m×1 //

1×m��

m

��m

//

α +3

Here R sends (C ,m) to the two paths from C 3 to C as on theright above.

Now K1 : P2 → I2 is the inclusion of theboundary of the free invertible 2-cell – thus an associator isobtained in the pullback. Arguing as before, T -Alg2 ∈ K+.

I Add pentagon equation and so on by considering δD3 → D2.

I Conclude that MonCatp belongs to K+.

John Bourke Accessible aspects of 2-category theory

Page 54: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Monoidal cats 2

I Add associators by forming a pullback

T -Alg2

��

// Ps(D2,Cat)

Ps(K1,Cat)

��T -Alg1

R // Ps(δD1,Cat)

X 3 X 2

X X 2

m×1 //

1×m��

m

��m

//

α +3

Here R sends (C ,m) to the two paths from C 3 to C as on theright above. Now K1 : P2 → I2 is the inclusion of theboundary of the free invertible 2-cell – thus an associator isobtained in the pullback.

Arguing as before, T -Alg2 ∈ K+.

I Add pentagon equation and so on by considering δD3 → D2.

I Conclude that MonCatp belongs to K+.

John Bourke Accessible aspects of 2-category theory

Page 55: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Monoidal cats 2

I Add associators by forming a pullback

T -Alg2

��

// Ps(D2,Cat)

Ps(K1,Cat)

��T -Alg1

R // Ps(δD1,Cat)

X 3 X 2

X X 2

m×1 //

1×m��

m

��m

//

α +3

Here R sends (C ,m) to the two paths from C 3 to C as on theright above. Now K1 : P2 → I2 is the inclusion of theboundary of the free invertible 2-cell – thus an associator isobtained in the pullback. Arguing as before, T -Alg2 ∈ K+.

I Add pentagon equation and so on by considering δD3 → D2.

I Conclude that MonCatp belongs to K+.

John Bourke Accessible aspects of 2-category theory

Page 56: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Monoidal cats 2

I Add associators by forming a pullback

T -Alg2

��

// Ps(D2,Cat)

Ps(K1,Cat)

��T -Alg1

R // Ps(δD1,Cat)

X 3 X 2

X X 2

m×1 //

1×m��

m

��m

//

α +3

Here R sends (C ,m) to the two paths from C 3 to C as on theright above. Now K1 : P2 → I2 is the inclusion of theboundary of the free invertible 2-cell – thus an associator isobtained in the pullback. Arguing as before, T -Alg2 ∈ K+.

I Add pentagon equation and so on by considering δD3 → D2.

I Conclude that MonCatp belongs to K+.

John Bourke Accessible aspects of 2-category theory

Page 57: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Monoidal cats 2

I Add associators by forming a pullback

T -Alg2

��

// Ps(D2,Cat)

Ps(K1,Cat)

��T -Alg1

R // Ps(δD1,Cat)

X 3 X 2

X X 2

m×1 //

1×m��

m

��m

//

α +3

Here R sends (C ,m) to the two paths from C 3 to C as on theright above. Now K1 : P2 → I2 is the inclusion of theboundary of the free invertible 2-cell – thus an associator isobtained in the pullback. Arguing as before, T -Alg2 ∈ K+.

I Add pentagon equation and so on by considering δD3 → D2.

I Conclude that MonCatp belongs to K+.

John Bourke Accessible aspects of 2-category theory

Page 58: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

More examples and results using cocellularity

I Likewise symmetric monoidal categories, finitely completecategories, regular categories, exact categories, bicategories. . . and their respective pseudomorphisms can be constructedas co-cellular objects in K+, and so belong to K+.

I Or internal versions of these . . .

I Arguing in a similar fashion, if T is a finitary 2-monad onC ∈ K+ then the 2-categories Lax-T-Algp,Ps-T-Algp andColax-T-Algp belongs to K+.

I If T , as above, has the property that each pseudoalgebra isisomorphic to a strict T -algebra (e.g. if T isflexible/cofibrant) then T -Algp belongs to K+ – this includesa broad class of examples, including many of the above.

I Also more general results for finite limit 2-theories. . . .

John Bourke Accessible aspects of 2-category theory

Page 59: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

More examples and results using cocellularity

I Likewise symmetric monoidal categories, finitely completecategories, regular categories, exact categories, bicategories. . . and their respective pseudomorphisms can be constructedas co-cellular objects in K+, and so belong to K+.

I Or internal versions of these . . .

I Arguing in a similar fashion, if T is a finitary 2-monad onC ∈ K+ then the 2-categories Lax-T-Algp,Ps-T-Algp andColax-T-Algp belongs to K+.

I If T , as above, has the property that each pseudoalgebra isisomorphic to a strict T -algebra (e.g. if T isflexible/cofibrant) then T -Algp belongs to K+ – this includesa broad class of examples, including many of the above.

I Also more general results for finite limit 2-theories. . . .

John Bourke Accessible aspects of 2-category theory

Page 60: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

More examples and results using cocellularity

I Likewise symmetric monoidal categories, finitely completecategories, regular categories, exact categories, bicategories. . . and their respective pseudomorphisms can be constructedas co-cellular objects in K+, and so belong to K+.

I Or internal versions of these . . .

I Arguing in a similar fashion, if T is a finitary 2-monad onC ∈ K+ then the 2-categories Lax-T-Algp,Ps-T-Algp andColax-T-Algp belongs to K+.

I If T , as above, has the property that each pseudoalgebra isisomorphic to a strict T -algebra (e.g. if T isflexible/cofibrant) then T -Algp belongs to K+ – this includesa broad class of examples, including many of the above.

I Also more general results for finite limit 2-theories. . . .

John Bourke Accessible aspects of 2-category theory

Page 61: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

More examples and results using cocellularity

I Likewise symmetric monoidal categories, finitely completecategories, regular categories, exact categories, bicategories. . . and their respective pseudomorphisms can be constructedas co-cellular objects in K+, and so belong to K+.

I Or internal versions of these . . .

I Arguing in a similar fashion, if T is a finitary 2-monad onC ∈ K+ then the 2-categories Lax-T-Algp,Ps-T-Algp andColax-T-Algp belongs to K+.

I If T , as above, has the property that each pseudoalgebra isisomorphic to a strict T -algebra (e.g. if T isflexible/cofibrant) then T -Algp belongs to K+ – this includesa broad class of examples, including many of the above.

I Also more general results for finite limit 2-theories. . . .

John Bourke Accessible aspects of 2-category theory

Page 62: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

More examples and results using cocellularity

I Likewise symmetric monoidal categories, finitely completecategories, regular categories, exact categories, bicategories. . . and their respective pseudomorphisms can be constructedas co-cellular objects in K+, and so belong to K+.

I Or internal versions of these . . .

I Arguing in a similar fashion, if T is a finitary 2-monad onC ∈ K+ then the 2-categories Lax-T-Algp,Ps-T-Algp andColax-T-Algp belongs to K+.

I If T , as above, has the property that each pseudoalgebra isisomorphic to a strict T -algebra (e.g. if T isflexible/cofibrant) then T -Algp belongs to K+ – this includesa broad class of examples, including many of the above.

I Also more general results for finite limit 2-theories. . . .

John Bourke Accessible aspects of 2-category theory

Page 63: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures. Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 64: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures. Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 65: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures. Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 66: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures.

Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 67: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures. Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 68: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Quasicategories with limits etc

I Moral of the story: weak objects and weak maps formaccessible categories.

I So if we consider only weak structures (as in weak highercategory theory) most stuff should be accessible!

I Ongoing (w. Lack-Vokrınek): extend some of these resultsfrom 2-categories to ∞-cosmoi (Riehl-Verity), which arecertain simplicial categories admitting flexible limits.

I Our first results: we have shown that QCatt , the infinitycosmos of quasicategories with a terminal object and functorspreserving terminal objects is accessible. Proof uses firstapproach in spirit of Makkai’s generalised sketches. Plan toextend this to other quasicategorical structures. Would like aproof internal to ∞ cosmos too.

I Open problem: understand accessiblity of weak objects andweak maps in more contexts. E.g. when is the Kleisli categoryfor a comonad accessible?

John Bourke Accessible aspects of 2-category theory

Page 69: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Final thoughts!

I Paper “Accessible aspects of 2-category theory” in the comingmonths, if you are interested.

I Thanks for listening!

John Bourke Accessible aspects of 2-category theory

Page 70: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

Final thoughts!

I Paper “Accessible aspects of 2-category theory” in the comingmonths, if you are interested.

I Thanks for listening!

John Bourke Accessible aspects of 2-category theory

Page 71: Accessible aspects of 2-category theoryconferences.inf.ed.ac.uk/ct2019/slides/48.pdf · Locally presentable and accessible categories ICis -accessibleif it has asetof -presentable

References

AR Weakly locally presentable categories, Adamek and Rosicky,1994.

BKP : Two-dimensional monad theory, Blackwell, Kelly and Power,1989.

BKPS : Flexible limits . . . , Bird, Kelly, Power and Street, 1989.

GU : Lokal Prsentierbare Kategorien, Gabriel and Ulmer, 1971.

MP : Accessible categories: the foundations of categorical modeltheory, Makkai and Pare, 1989.

M : Generalized sketches . . . , Makkai, 1997.

LR : Enriched weakness, Lack and Rosicky, 2012.

RV : Elements of infinity category theory, Riehl and Verity, inpreparation.

John Bourke Accessible aspects of 2-category theory