n-categories are sheaves on n-manifoldsn-categories are sheaves on n-manifolds david ayala (w/ nick...
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n-categories are sheaves on n-manifolds
David Ayala(w/ Nick Rozenblyum)
Harvard University
January 8, 2012
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom theory
• X a (locally compact Hausdorff) topological space.• C a (discrete) commutative group.
Consider the set of configurations in X with labels in C
ConfC(X ) = (Z ,Z l−→ C) | Z ⊂ X is finite,
equipped with a topology so that• (Multiplication) points can collide and their labels add.• (Units) points labeled by 0 ∈ C can disappear.• (Non-compact) points can disappear at “∞”.
Base point ∅ ∈ ConfC(X ).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom theory
• X a (locally compact Hausdorff) topological space.• C a (discrete) commutative group.
Consider the set of configurations in X with labels in C
ConfC(X ) = (Z ,Z l−→ C) | Z ⊂ X is finite,
equipped with a topology so that• (Multiplication) points can collide and their labels add.• (Units) points labeled by 0 ∈ C can disappear.• (Non-compact) points can disappear at “∞”.
Base point ∅ ∈ ConfC(X ).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Theorem ((non-compact) Dold-Thom)
The homotopy groups
π∗ConfC(X ) ∼= H∗(X∗; C)
agree with the reduced homology groups of the one-point compactification ofX with coefficients in C.
Ingredients: Fix C. The assignment
X 7→ Conf(X ,C)
is co-variantly functorial among closed inclusions and contra-variantlyfunctorial among open embeddings.Moreover it is:• (Continuous) ConfC(−) is Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excisive) For Z → X a (nice) closed embedding, the sequence
ConfC(Z )→ ConfC(X )→ ConfC(X \ Z )
is a homotopy fibration sequence.This is the hard part.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Theorem ((non-compact) Dold-Thom)
The homotopy groups
π∗ConfC(X ) ∼= H∗(X∗; C)
agree with the reduced homology groups of the one-point compactification ofX with coefficients in C.
Ingredients: Fix C. The assignment
X 7→ Conf(X ,C)
is co-variantly functorial among closed inclusions and contra-variantlyfunctorial among open embeddings.Moreover it is:• (Continuous) ConfC(−) is Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excisive) For Z → X a (nice) closed embedding, the sequence
ConfC(Z )→ ConfC(X )→ ConfC(X \ Z )
is a homotopy fibration sequence.This is the hard part.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Theorem ((non-compact) Dold-Thom)
The homotopy groups
π∗ConfC(X ) ∼= H∗(X∗; C)
agree with the reduced homology groups of the one-point compactification ofX with coefficients in C.
Ingredients: Fix C. The assignment
X 7→ Conf(X ,C)
is co-variantly functorial among closed inclusions and contra-variantlyfunctorial among open embeddings.
Moreover it is:• (Continuous) ConfC(−) is Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excisive) For Z → X a (nice) closed embedding, the sequence
ConfC(Z )→ ConfC(X )→ ConfC(X \ Z )
is a homotopy fibration sequence.This is the hard part.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Theorem ((non-compact) Dold-Thom)
The homotopy groups
π∗ConfC(X ) ∼= H∗(X∗; C)
agree with the reduced homology groups of the one-point compactification ofX with coefficients in C.
Ingredients: Fix C. The assignment
X 7→ Conf(X ,C)
is co-variantly functorial among closed inclusions and contra-variantlyfunctorial among open embeddings.Moreover it is:• (Continuous) ConfC(−) is Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excisive) For Z → X a (nice) closed embedding, the sequence
ConfC(Z )→ ConfC(X )→ ConfC(X \ Z )
is a homotopy fibration sequence.This is the hard part.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Question
Do these axioms characterize commutative groups?
Nope - but in a weak sense, yes.Namely, they characterize group-like E∞-algebras (in spaces) (Brownrepresentability).
For instance, Dold-Thom⇒ ConfC(Rk ) ' Bk C. Together with the adjoint of
R∗ ∧ ConfC(Rk )→ ConfC(Rk+1)
being an equivalence (scanning), we get an Ω∞-space as claimed.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Question
Do these axioms characterize commutative groups?
Nope - but in a weak sense, yes.Namely, they characterize group-like E∞-algebras (in spaces) (Brownrepresentability).
For instance, Dold-Thom⇒ ConfC(Rk ) ' Bk C. Together with the adjoint of
R∗ ∧ ConfC(Rk )→ ConfC(Rk+1)
being an equivalence (scanning), we get an Ω∞-space as claimed.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Question
Do these axioms characterize commutative groups?
Nope - but in a weak sense, yes.Namely, they characterize group-like E∞-algebras (in spaces) (Brownrepresentability).
For instance, Dold-Thom⇒ ConfC(Rk ) ' Bk C. Together with the adjoint of
R∗ ∧ ConfC(Rk )→ ConfC(Rk+1)
being an equivalence (scanning), we get an Ω∞-space as claimed.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Dold-Thom
Question
Do these axioms characterize commutative groups?
Nope - but in a weak sense, yes.Namely, they characterize group-like E∞-algebras (in spaces) (Brownrepresentability).
For instance, Dold-Thom⇒ ConfC(Rk ) ' Bk C. Together with the adjoint of
R∗ ∧ ConfC(Rk )→ ConfC(Rk+1)
being an equivalence (scanning), we get an Ω∞-space as claimed.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Question
What if C = A is associative but not commutative?
Cannot define a (reasonable) topology on ConfA(X ) for general X .
But for X = M a framed 1-manifold, the set ConfA(M) can be topologized justas before.
Question
What if A, regarded as a category with a single object, is replaced by anarbitrary (small) category C?
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Question
What if C = A is associative but not commutative?
Cannot define a (reasonable) topology on ConfA(X ) for general X .
But for X = M a framed 1-manifold, the set ConfA(M) can be topologized justas before.
Question
What if A, regarded as a category with a single object, is replaced by anarbitrary (small) category C?
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Question
What if C = A is associative but not commutative?
Cannot define a (reasonable) topology on ConfA(X ) for general X .
But for X = M a framed 1-manifold, the set ConfA(M) can be topologized justas before.
Question
What if A, regarded as a category with a single object, is replaced by anarbitrary (small) category C?
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Question
What if C = A is associative but not commutative?
Cannot define a (reasonable) topology on ConfA(X ) for general X .
But for X = M a framed 1-manifold, the set ConfA(M) can be topologized justas before.
Question
What if A, regarded as a category with a single object, is replaced by anarbitrary (small) category C?
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Fix C small. (For example, take C = A an associative monoid).
Let M be a framed smooth 1-manifold. Consider the set
ConfC(M) := (Z , l0, l1)
where• Z ⊂ M is finite,• l0 : M \ Z → ob C,• l1 : Z → mor C, such that “source-target”,
equipped with a topology so that• (Multiplication) points can collide and their labels compose,• (Units) points labeled by identities 1c can disappear,• (Non-compact) points can disappear at “∞”.
There is no canonical base point.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
Fix C small. (For example, take C = A an associative monoid).
Let M be a framed smooth 1-manifold. Consider the set
ConfC(M) := (Z , l0, l1)
where• Z ⊂ M is finite,• l0 : M \ Z → ob C,• l1 : Z → mor C, such that “source-target”,
equipped with a topology so that• (Multiplication) points can collide and their labels compose,• (Units) points labeled by identities 1c can disappear,• (Non-compact) points can disappear at “∞”.
There is no canonical base point.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.
Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R). And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R). And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R). And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R). And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R).
And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Commutative to categorical
The assignmentM 7→ ConfC(M)
is contra-variantly functorial among framed open smooth embeddings.Moreover, it is:• (Continuous) Top-enriched.• (Sheaf) ... in particular t 7→ × and ∅ 7→ ∗.• (Excision - No!) For Z ⊂ M a closed inclusion, the sequence
ConfC(Z )→ ConfC(M)→ ConfC(M \ Z )
just doesn’t work.
Question
Does this continuous sheaf remember the category C?
Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R). And
Proposition
ConfC(R) ∼= BC.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Transversality for excision
Idea
The sheaf ConfC has a notion of transversality which can take the place ofexcision.
Let S ⊂ M be a finite subset. Consider the subspace
ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.
Question
Does the continuous sheaf ConfC equipped with the additional structure oftransversality remember the category C?
Nope - but in a weak sense, yes.
ConfC(R t 0) '−→ ob C,
ConfC(R t 0, 1) '−→ mor C,
ConfC(R t 0, 1, 2) '−→ mor ×ob mor ,
(Conf(t 0, 1, 2)→ Conf(t 0, 2))'−→ (mor ×ob mor −→ mor),
...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Transversality for excision
Idea
The sheaf ConfC has a notion of transversality which can take the place ofexcision.
Let S ⊂ M be a finite subset. Consider the subspace
ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.
Question
Does the continuous sheaf ConfC equipped with the additional structure oftransversality remember the category C?
Nope - but in a weak sense, yes.
ConfC(R t 0) '−→ ob C,
ConfC(R t 0, 1) '−→ mor C,
ConfC(R t 0, 1, 2) '−→ mor ×ob mor ,
(Conf(t 0, 1, 2)→ Conf(t 0, 2))'−→ (mor ×ob mor −→ mor),
...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Transversality for excision
Idea
The sheaf ConfC has a notion of transversality which can take the place ofexcision.
Let S ⊂ M be a finite subset. Consider the subspace
ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.
Question
Does the continuous sheaf ConfC equipped with the additional structure oftransversality remember the category C?
Nope - but in a weak sense, yes.
ConfC(R t 0) '−→ ob C,
ConfC(R t 0, 1) '−→ mor C,
ConfC(R t 0, 1, 2) '−→ mor ×ob mor ,
(Conf(t 0, 1, 2)→ Conf(t 0, 2))'−→ (mor ×ob mor −→ mor),
...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Transversality for excision
Idea
The sheaf ConfC has a notion of transversality which can take the place ofexcision.
Let S ⊂ M be a finite subset. Consider the subspace
ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.
Question
Does the continuous sheaf ConfC equipped with the additional structure oftransversality remember the category C?
Nope - but in a weak sense, yes.
ConfC(R t 0) '−→ ob C,
ConfC(R t 0, 1) '−→ mor C,
ConfC(R t 0, 1, 2) '−→ mor ×ob mor ,
(Conf(t 0, 1, 2)→ Conf(t 0, 2))'−→ (mor ×ob mor −→ mor),
...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Transversality for excision
Idea
The sheaf ConfC has a notion of transversality which can take the place ofexcision.
Let S ⊂ M be a finite subset. Consider the subspace
ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.
Question
Does the continuous sheaf ConfC equipped with the additional structure oftransversality remember the category C?
Nope - but in a weak sense, yes.
ConfC(R t 0) '−→ ob C,
ConfC(R t 0, 1) '−→ mor C,
ConfC(R t 0, 1, 2) '−→ mor ×ob mor ,
(Conf(t 0, 1, 2)→ Conf(t 0, 2))'−→ (mor ×ob mor −→ mor),
...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak categories
Definition (following Rezk)
A weak category (complete Segal space) is a map of quasi-categories
C : ∆op → Spaces
which is local with respect to ∆[p]→ ∆[r ]⊔
∆[0] ∆[s] for each[p] = [r + s] ∈ ∆ and ∆[0]→ N[1]Gpd .
Definition (Towards “sheaf on 1Man with transversality”)
An object of the quasi-category 1Mant is a pair
(S ⊂ M)
for which M admits an atlas (by Euclideans) for which each elementintersects S non-empty; and an edge (M,S)→ (M ′,S′) is a pair (f , γ) wheref : M → M ′ is a framed embedding and γ : f (S) S′ ⊂ M ′ is a path of finitesubsets.
Subcategory 1Mant0 ⊂ 1Mant - γ increases cardinality.
1Mant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak categories
Definition (following Rezk)
A weak category (complete Segal space) is a map of quasi-categories
C : ∆op → Spaces
which is local with respect to ∆[p]→ ∆[r ]⊔
∆[0] ∆[s] for each[p] = [r + s] ∈ ∆ and ∆[0]→ N[1]Gpd .
Definition (Towards “sheaf on 1Man with transversality”)
An object of the quasi-category 1Mant is a pair
(S ⊂ M)
for which M admits an atlas (by Euclideans) for which each elementintersects S non-empty; and an edge (M,S)→ (M ′,S′) is a pair (f , γ) wheref : M → M ′ is a framed embedding and γ : f (S) S′ ⊂ M ′ is a path of finitesubsets.
Subcategory 1Mant0 ⊂ 1Mant - γ increases cardinality.
1Mant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak categories
Definition (following Rezk)
A weak category (complete Segal space) is a map of quasi-categories
C : ∆op → Spaces
which is local with respect to ∆[p]→ ∆[r ]⊔
∆[0] ∆[s] for each[p] = [r + s] ∈ ∆ and ∆[0]→ N[1]Gpd .
Definition (Towards “sheaf on 1Man with transversality”)
An object of the quasi-category 1Mant is a pair
(S ⊂ M)
for which M admits an atlas (by Euclideans) for which each elementintersects S non-empty; and an edge (M,S)→ (M ′,S′) is a pair (f , γ) wheref : M → M ′ is a framed embedding and γ : f (S) S′ ⊂ M ′ is a path of finitesubsets.
Subcategory 1Mant0 ⊂ 1Mant - γ increases cardinality.
1Mant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak categories
Definition (following Rezk)
A weak category (complete Segal space) is a map of quasi-categories
C : ∆op → Spaces
which is local with respect to ∆[p]→ ∆[r ]⊔
∆[0] ∆[s] for each[p] = [r + s] ∈ ∆ and ∆[0]→ N[1]Gpd .
Definition (Towards “sheaf on 1Man with transversality”)
An object of the quasi-category 1Mant is a pair
(S ⊂ M)
for which M admits an atlas (by Euclideans) for which each elementintersects S non-empty; and an edge (M,S)→ (M ′,S′) is a pair (f , γ) wheref : M → M ′ is a framed embedding and γ : f (S) S′ ⊂ M ′ is a path of finitesubsets.
Subcategory 1Mant0 ⊂ 1Mant - γ increases cardinality.
1Mant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak categories
Definition (following Rezk)
A weak category (complete Segal space) is a map of quasi-categories
C : ∆op → Spaces
which is local with respect to ∆[p]→ ∆[r ]⊔
∆[0] ∆[s] for each[p] = [r + s] ∈ ∆ and ∆[0]→ N[1]Gpd .
Definition (Towards “sheaf on 1Man with transversality”)
An object of the quasi-category 1Mant is a pair
(S ⊂ M)
for which M admits an atlas (by Euclideans) for which each elementintersects S non-empty; and an edge (M,S)→ (M ′,S′) is a pair (f , γ) wheref : M → M ′ is a framed embedding and γ : f (S) S′ ⊂ M ′ is a path of finitesubsets.
Subcategory 1Mant0 ⊂ 1Mant - γ increases cardinality.
1Mant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,
All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,
All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
t-sheaf
Definition
A t-sheaf on 1Man is a map of quasi-categories
Ψ: 1Mant → Spaces
which restricts to a sheaf on 1Mant0 .
Example
ConfC(−),
Ψd (M,S) = W ⊂ M × R∞ | W pr−→ M is proper and t S,
Morse(M,S) = M f−→ R Morse | S ∩ Crit(f ) = ∅,All micro-flexible sheaves sheaves on (framed) 1-manifolds which have anatural notion of transversality (subspaces of mapping spaces satisfyingelliptic PDE’s).For instance, maps with prescribed singularities, contact structures, ...
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = 1
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Example
C ↔ ConfC
Cobd ↔ Ψd .
A t-sheaf Ψ determines a functor Ψ: 1Manop → Spaces, given by
Ψ(M) = colimS⊂M Ψ(M t S)
which is NOT a sheaf. The composition∫: Weak − categories → Fun(1Manop,Spaces)
is a (non-compact) version of chiral/factorization homology defined forcategories, not merely E1-algebras.For instance
∫S1 C ' HH(C).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = 1
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Example
C ↔ ConfC
Cobd ↔ Ψd .
A t-sheaf Ψ determines a functor Ψ: 1Manop → Spaces, given by
Ψ(M) = colimS⊂M Ψ(M t S)
which is NOT a sheaf. The composition∫: Weak − categories → Fun(1Manop,Spaces)
is a (non-compact) version of chiral/factorization homology defined forcategories, not merely E1-algebras.For instance
∫S1 C ' HH(C).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = 1
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Example
C ↔ ConfC
Cobd ↔ Ψd .
A t-sheaf Ψ determines a functor Ψ: 1Manop → Spaces, given by
Ψ(M) = colimS⊂M Ψ(M t S)
which is NOT a sheaf. The composition∫: Weak − categories → Fun(1Manop,Spaces)
is a (non-compact) version of chiral/factorization homology defined forcategories, not merely E1-algebras.For instance
∫S1 C ' HH(C).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = 1
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Example
C ↔ ConfC
Cobd ↔ Ψd .
A t-sheaf Ψ determines a functor Ψ: 1Manop → Spaces, given by
Ψ(M) = colimS⊂M Ψ(M t S)
which is NOT a sheaf. The composition∫: Weak − categories → Fun(1Manop,Spaces)
is a (non-compact) version of chiral/factorization homology defined forcategories, not merely E1-algebras.For instance
∫S1 C ' HH(C).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = 1
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Example
C ↔ ConfC
Cobd ↔ Ψd .
A t-sheaf Ψ determines a functor Ψ: 1Manop → Spaces, given by
Ψ(M) = colimS⊂M Ψ(M t S)
which is NOT a sheaf. The composition∫: Weak − categories → Fun(1Manop,Spaces)
is a (non-compact) version of chiral/factorization homology defined forcategories, not merely E1-algebras.For instance
∫S1 C ' HH(C).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = n
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Naturally generalizes in two directions:
1Man nMan (coming soon),
categories n-categories (now).
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = n
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Naturally generalizes in two directions:
1Man nMan (coming soon),
categories n-categories (now).
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = n
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Naturally generalizes in two directions:
1Man nMan (coming soon),
categories n-categories (now).
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Statement n = n
Theorem (n = 1, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak categories t-sheaves on 1Man : ρ
implementing an equivalence.
Naturally generalizes in two directions:
1Man nMan (coming soon),
categories n-categories (now).
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak n-categories
Definition (Berger)
An object of the category Θn is of the form
[p](T1, . . . ,Tp)
where [p] ∈ ∆ and Ti ∈ Θn−1, with Θ0 = ?. Morphisms are simple enough ...
Definition (Rezk)
A weak n-category is a map of quasi-categories
C : Θopn → Spaces
which is local with respect to a specified collection of morphisms (Segal andcomplete).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak n-categories
Definition (Berger)
An object of the category Θn is of the form
[p](T1, . . . ,Tp)
where [p] ∈ ∆ and Ti ∈ Θn−1, with Θ0 = ?. Morphisms are simple enough ...
Definition (Rezk)
A weak n-category is a map of quasi-categories
C : Θopn → Spaces
which is local with respect to a specified collection of morphisms (Segal andcomplete).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Weak n-categories
Definition (Berger)
An object of the category Θn is of the form
[p](T1, . . . ,Tp)
where [p] ∈ ∆ and Ti ∈ Θn−1, with Θ0 = ?. Morphisms are simple enough ...
Definition (Rezk)
A weak n-category is a map of quasi-categories
C : Θopn → Spaces
which is local with respect to a specified collection of morphisms (Segal andcomplete).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
A basic iterated submersion is the diagram of projections for some k ≥ 0
Rk prk−−→ Rk−1 prk−1−−−→ . . .pr1−−→ R0.
An iterated submersion (of dimension ≤ n) is a sequence of submersions offramed smooth manifolds
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
which is locally isomorphism to a basic iterated submersion.
A subcomplex of an iterated submersion is a diagram
S = (Sn pn−→ Sn−1 pn−1−−−→ . . .p1−→ S0)
∩
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
where each Sk is a (certain) compact singular k -manifold and each pk is asubmersion of such.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
A basic iterated submersion is the diagram of projections for some k ≥ 0
Rk prk−−→ Rk−1 prk−1−−−→ . . .pr1−−→ R0.
An iterated submersion (of dimension ≤ n) is a sequence of submersions offramed smooth manifolds
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
which is locally isomorphism to a basic iterated submersion.
A subcomplex of an iterated submersion is a diagram
S = (Sn pn−→ Sn−1 pn−1−−−→ . . .p1−→ S0)
∩
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
where each Sk is a (certain) compact singular k -manifold and each pk is asubmersion of such.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
A basic iterated submersion is the diagram of projections for some k ≥ 0
Rk prk−−→ Rk−1 prk−1−−−→ . . .pr1−−→ R0.
An iterated submersion (of dimension ≤ n) is a sequence of submersions offramed smooth manifolds
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
which is locally isomorphism to a basic iterated submersion.A subcomplex of an iterated submersion is a diagram
S = (Sn pn−→ Sn−1 pn−1−−−→ . . .p1−→ S0)
∩
M = (Mn sn−→ Mn−1 sn−1−−−→ . . .s1−→ M0)
where each Sk is a (certain) compact singular k -manifold and each pk is asubmersion of such.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
Define the quasi-category nMant with objects
S ⊂ M
for which M admits an atlas (by basics) each of whose elements intersect Snon-empty;
and a morphism (M,S)→ (M ′,S′) is a map of diagrams
f : M → M ′
level-wise a framed open smooth embedding,together with a path of subcomplexes
γ : f (S) S′ ⊂ M ′.
Subcategory nMant0 ⊂ nMant - paths γ increase the number of components
of strata.
nMant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
Define the quasi-category nMant with objects
S ⊂ M
for which M admits an atlas (by basics) each of whose elements intersect Snon-empty;
and a morphism (M,S)→ (M ′,S′) is a map of diagrams
f : M → M ′
level-wise a framed open smooth embedding,together with a path of subcomplexes
γ : f (S) S′ ⊂ M ′.
Subcategory nMant0 ⊂ nMant - paths γ increase the number of components
of strata.
nMant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
Define the quasi-category nMant with objects
S ⊂ M
for which M admits an atlas (by basics) each of whose elements intersect Snon-empty;and a morphism (M,S)→ (M ′,S′) is a map of diagrams
f : M → M ′
level-wise a framed open smooth embedding,together with a path of subcomplexes
γ : f (S) S′ ⊂ M ′.
Subcategory nMant0 ⊂ nMant - paths γ increase the number of components
of strata.
nMant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated submersions
Definition
Define the quasi-category nMant with objects
S ⊂ M
for which M admits an atlas (by basics) each of whose elements intersect Snon-empty;and a morphism (M,S)→ (M ′,S′) is a map of diagrams
f : M → M ′
level-wise a framed open smooth embedding,together with a path of subcomplexes
γ : f (S) S′ ⊂ M ′.
Subcategory nMant0 ⊂ nMant - paths γ increase the number of components
of strata.
nMant0 has a natural notion of open cover (Grothendieck site).
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,
All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Definition
A t-sheaf on nMan is a map of quasi-categories
Ψ: (nMant)op → Spaces
which restricts to a sheaf on nMant0 .
Example
For C a strict n-category, ConfC ,
Defects - a topological version for n = 2,
Ψd (M,S) = W ⊂ Mn × R∞ | W pr−→ Mn proper and t Sn,All micro-flexible sheaves on framed n-manifolds with a geometric notionof transversality.
Theorem (n = n, w/ Rozenblyum)
There is an adjunction of quasi-categories
λ : Weak n-categories t-sheaves on nMan : ρ
implementing an equivalence.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated HH
Example
The torus (S1)×n, together with its projections and framing, is an iteratedsubmersion T n.So given a weak n-category C, there is an iterated Hochschild homology
nHH(C) := colimS⊂T n (T n t S)
Remark
Much of the Yoga of higher categories is quite tractable in this geometricsetting:• Delooping monoidal structures,• n-categories enriched over (n − 1)-categories,• Categories of correspondences,• Maximal subgroupoids and groupoidifications.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated HH
Example
The torus (S1)×n, together with its projections and framing, is an iteratedsubmersion T n.So given a weak n-category C, there is an iterated Hochschild homology
nHH(C) := colimS⊂T n (T n t S)
Remark
Much of the Yoga of higher categories is quite tractable in this geometricsetting:• Delooping monoidal structures,• n-categories enriched over (n − 1)-categories,• Categories of correspondences,• Maximal subgroupoids and groupoidifications.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds
Iterated HH
Example
The torus (S1)×n, together with its projections and framing, is an iteratedsubmersion T n.So given a weak n-category C, there is an iterated Hochschild homology
nHH(C) := colimS⊂T n (T n t S)
Remark
Much of the Yoga of higher categories is quite tractable in this geometricsetting:• Delooping monoidal structures,• n-categories enriched over (n − 1)-categories,• Categories of correspondences,• Maximal subgroupoids and groupoidifications.
David Ayala (w/ Nick Rozenblyum) n-categories are sheaves on n-manifolds