n-categories are sheaves on n-manifoldsn-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

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 ).


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 ).


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.


Dold-Thom


The homotopy groups




X 7→ Conf(X ,C)





Dold-Thom


The homotopy groups




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




Dold-Thom


The homotopy groups




X 7→ Conf(X ,C)





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.


Dold-Thom

Question







Dold-Thom

Question







Dold-Thom

Question







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?



Question




Question




Question




Question




Question




Question




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.



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.



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.




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



Question



Proposition

ConfC(R) ∼= BC.







Question



Proposition

ConfC(R) ∼= BC.







Question



Proposition

ConfC(R) ∼= BC.







Question


Nope. Sheaf⇒ ConfC(−) is determined by its restriction to Emb(R,R).

And

Proposition

ConfC(R) ∼= BC.







Question



Proposition

ConfC(R) ∼= BC.


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),

...



Idea



ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.

Question







...



Idea



ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.

Question







...



Idea



ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.

Question







...



Idea



ConfC(M t S) := (Z , l0, l1) | Z ∩ S = ∅.

Question







...


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).


Weak categories








(S ⊂ M)





Weak categories








(S ⊂ M)





Weak categories








(S ⊂ M)





Weak categories








(S ⊂ M)





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, ...


t-sheaf

Definition




Example

ConfC(−),




t-sheaf

Definition




Example

ConfC(−),




t-sheaf

Definition




Example

ConfC(−),




t-sheaf

Definition




Example

ConfC(−),




t-sheaf

Definition




Example

ConfC(−),




t-sheaf

Definition




Example

ConfC(−),


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, ...


t-sheaf

Definition




Example

ConfC(−),


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, ...


t-sheaf

Definition




Example

ConfC(−),





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).


Statement n = 1





Example

C ↔ ConfC

Cobd ↔ Ψd .





∫S1 C ' HH(C).


Statement n = 1





Example

C ↔ ConfC

Cobd ↔ Ψd .





∫S1 C ' HH(C).


Statement n = 1





Example

C ↔ ConfC

Cobd ↔ Ψd .





∫S1 C ' HH(C).


Statement n = 1





Example

C ↔ ConfC

Cobd ↔ Ψd .





∫S1 C ' HH(C).


Statement n = n





Naturally generalizes in two directions:

1Man nMan (coming soon),

categories n-categories (now).

Theorem (n = n, w/ Rozenblyum)


λ : Weak n-categories t-sheaves on nMan : ρ



Statement n = n













Statement n = n













Statement n = n













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).


Weak n-categories

Definition (Berger)


[p](T1, . . . ,Tp)


Definition (Rezk)





Weak n-categories

Definition (Berger)


[p](T1, . . . ,Tp)


Definition (Rezk)






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.



Definition


Rk prk−−→ Rk−1 prk−1−−−→ . . .pr1−−→ R0.


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)




Definition


Rk prk−−→ Rk−1 prk−1−−−→ . . .pr1−−→ R0.


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)





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).



Definition


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 ′


γ : f (S) S′ ⊂ M ′.


of strata.




Definition


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 ′


γ : f (S) S′ ⊂ M ′.


of strata.




Definition


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 ′


γ : f (S) S′ ⊂ M ′.


of strata.



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.






Definition




Example









Definition




Example









Definition




Example









Definition




Example



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.






Definition




Example









Definition




Example









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.


Iterated HH

Example



Remark



Iterated HH

Example



Remark



Documents

n-categories are sheaves on n-manifoldsn-categories are sheaves on n-manifolds David Ayala (w/ Nick Rozenblyum) Harvard University January 8, 2012 David Ayala (w/ Nick Rozenblyum)