cobordism hypothesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### Plain cobordism hypothesis

What is known as the *Cobordism Hypothesis* is, in its most basic form, the [[conjecture]] that, roughly, the [[(∞,n)-category of cobordisms]] $Bord_n^{fr}$ with [[framed manifold|framing]] is [[equivalence|equivalently]] the  [[free construction|free]] [[symmetric monoidal (∞,n)-category]] [[(∞,n)-category with duals|with duals]] on a single [[object]].

The idea is due to [Baez & Dolan (1995)](#BaezDolan95), who stated it in the further generality of [[k-tuply monoidal n-category|$k$-tuply monoidal $n$-categories]] (i.e. for $k \leq \infty$) under the name *[[Tangle Hypothesis]]* ([p. 23](https://arxiv.org/pdf/q-alg/9503002.pdf#page=23)), before definitions of these [[higher category theory|higher category theoretical]] notions were available. The term "cobordism hypothesis" is used by Baez and Dolan only in passing and without definition ([p. 25](https://arxiv.org/pdf/q-alg/9503002.pdf#page=25)) but was popularized by [Lurie (2009)](#Lurie09) who gave a detailed though still incomplete sketch of a [[proof]] by appeal to the notion of [[n-fold complete Segal space|$n$-fold complete Segal spaces]] as a [[geometric definition of higher categories]], thereby breaking the impasse caused by previous attempts (e.g. [tac:18-10](http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html)) which instead tried [[algebraic definitions of higher categories]].

Since a (framed) fully [[extended topological quantum field theory]] (TQFT) may be identified with a symmetric [[monoidal (∞,n)-functor|monoidal]] [[(∞,n)-functor|$(\infty,n)$-functor]] $Z \colon Bord^{fr}_n \to \mathcal{C}^\otimes$, the cobordism hypothesis implies (and should be implied by, via a higher [[Yoneda lemma]]) that [[fully extended TQFTs]] are entirely determined by their value on the point (the "[[n-vector space]] of [[states]]" of the theory), and conversely that [[fully dualizable objects]] in $\mathcal{C}^{\otimes}$ [[equivalence|equivalently]] encode such fully extended TQFTs. As such the hypothesis is actually called the "*Extended TQFT Hypothesis*" by [Baez & Dolan (1995)](#BaezDolan95) ([p. 28](https://arxiv.org/pdf/q-alg/9503002.pdf#page=28)).

From the cobordism hypothesis for [[framed manifold|framing]] structure it follows fairly readily that the [[infinity-groupoid|$\infty$-groupoid]] of [[fully dualizable objects]] in a symmetric monoidal $(\infty,n)$-category carries an [[group action|action]] of the [[orthogonal group]] $O(n)$ and that cobordisms with [[tangential structure|tangential]] [[G-structure]] for $G \hookrightarrow O(n)$ are similarly classified by the [[homotopy fixed points|homopy fixed loci]] of the [[restricted action|corresponding]] $G$-action [[Lurie (2009, Thm. 2.4.26)](#Lurie09)]. This yields the statement of the hypothesis for common notions of cobordisms, such as with [[orientation]] ([[SO(n)]][[G-structure|-structure]]), [[spin structure]] ([[Spin(n)]][[G-structure|-structure]]), etc.

The idea of the cobordism hypothesis is that:

1. [[cylinders]] count as trivial ([[identity morphism|identity]]) [[cobordisms]] between their [[boundary]] components, corresponding to how local [[spacetime]]-evolution in a [[topological quantum field theory]] is trivial (corresponding to the physics jargon that "their [[Hamiltonian]] vanishes"),

1. so that all the non-trivial information in [[cobordism]] and in [[TQFT]]-propagation is in the structure of [[handle body]]-attachments (in physics known as: "topology change") which obey the [[triangle identity|zig-zag rules]] [known from](adjunction#InTermsOfStringDiagrams) [[adjunctions]], [[2-adjunctions]] and their further higher dimensional analogs,

1. which means that the point -- regarded as a 0-dimensional element of an $n$-dimensional [[cobordism]] -- is equipped with the [[structure]] of, first of all, a [[dualizable object]] under the relevant [[tensor product]] of cobordisms (which is [[disjoint union]]), and next that of a $k$-dualizable object for all $k \leq n$.

### Geometric cobordism hypothesis
 {#GeometricCobordismHypothesis}

More generally, the *geometric* cobordism hypothesis [[Grady & Pavlov (2021)](#GradyPavlov21)] asserts that general non-topological *geometric* [[functorial field theory|extended functorial field theories]] (such as [[conformal field theory]] as well as the usual [[relativistic quantum field theory|relativistic]] [[AQFT on curved spacetimes|QFT on curved spacetimes]]) are similarly enocded by (1.) local data depending only on the notion of geometry and (2.)  global topological data encoded only in higher dualizability structure.

For example, the [[FRS-theorem on rational 2d CFT]] is, at least in spirit, a realization of this idea: It states that [[rational CFT|rational]] [[2d conformal field theories]], when defined on all possible [[worldsheet]] [[surfaces]], are fully encoded (1.) by local geometric data encoded in [[vertex operator algebras]] or [[conformal nets]] or similar and (2.) global purely topological data (a suitable [[Frobenius algebra|Frobenius]] [[algebra object]] [[internalization|internal to]] the [[modular tensor category]] [[representation category|of representations]] of that [[vertex operator algebra]]/[[conformal net]]).

Conversely, the plain (topological) cobordism hypothesis is the special case of the geometric cobordism hypothesis of no geometric structure and in this sense very much captures the idea which in physics is known as "TQFTs are those QFTS without local degrees of freedom".

In general, where the notion of extended (geometric) [[functorial field theory]] is one candidate for a definition of (non-topological) [[non-perturbative quantum field theory]], the (geometric) cobordism hypothesis may be regarded as the first step in establishing the construction of such theories, reducing the problem entirely to the local behaviour of the theory.

Notice that the local behaviour of [[quantum field theory]] is still immensely rich: It keeps busy quantum field theorists most of whom never encounter a topological effect in their life. For example the original statement of the [[Wightman axioms]]/[[Haag-Kastler axioms]] are entirely concerned only with the local behaviour of Lorentzian QFTs (namely on [[Minkowski spacetime]]) and yet the construction of any non-trivial examples in dimensions $d \geq 4$ remains wide open.

In this vein, much in the spirit of [Freyd's dictum](category+theory#WhatIsTriviallyTrivial) on the role of [[categorical algebra]] in general, one may read the (geometric) cobordism hypothesis as trivializing the trivial part of [[non-perturbative quantum field theory]].

A proof of the geometric cobordism hypothesis is thus the important first step towards the [currently wide-open and yet crucial problem](mass+gap#ReferencesMassGapProblem) of mathematically constructing [[non-perturbative quantum field theories]].

## Formalization

In [[Lurie 2009](#Lurie09)] a sketch of an approach to the formalization and proof of the cobordism hypothesis is described.

### For framed cobordisms
 {#ForFramed}

#### Statement

For $\mathcal{C}$ a [[symmetric monoidal (∞,n)-category with duals]] write $Core(\mathcal{C})$ for its [[core]] (the maximal [[∞-groupoid]] in $\mathcal{C}$).

For $\mathcal{C}$, $\mathcal{D}$ two [[symmetric monoidal (∞,n)-categories]], write   $Fun^\otimes(\mathcal{D}, \mathcal{C} )$ for  the [[(∞,n)-category]] of symmetric monoidal [[(∞,n)-functors]] between them.

+-- {: .num_defn #FramedCobordismsNCat}
###### Definition

Write $Bord_n^{fr}$ be the [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[(∞,n)-category of cobordisms]] with [[framed manifold|n-framing]].

=--

+-- {: .num_theorem #CobordismHypothesisFramedVersion}
###### Theorem (cobordism hypothesis, framed version)

Evaluation of any such functor $F$ on the [[point]] ${*}$

$$
  F \mapsto F({*})
$$

induces an [[(∞,n)-functor]]

$$
  pt^* : Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to \mathcal{C} .
$$

such that

* this factors through the [[core]] of $\mathcal{C}$;

* the map

  $$
    pt^* \;\colon\; Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to Core(\mathcal{C})
  $$

  is an [[equivalence in an (∞,1)-category|equivalence]] of [[(∞,n)-categories]].

=--

This is ([Lurie, theorem 2.4.6](#Lurie)).

+-- {: .proof}
###### Proof

The proof is based on

1. the [[Galatius-Madsen-Tillmann-Weiss theorem]],  which characterizes the [[geometric realization]] $|Bord_n^{or}|$ in terms of the [[suspension]] of the [[Thom spectrum]];

1. Igusa's connectivity result which he uses to show that putting "framed [[Morse function]]s" on cobordisms doesn't change their [[homotopy type]] (theorem 3.4.7, page 73)

In fact, the Galatius-Madsen-Weiss theorem is now supposed to be a corollary of Lurie's result.

=--

#### Implications -- The canonical $O(n)$-∞-action
 {#TheCanonicalOnAction}

One of the striking consequences of theorem \ref{CobordismHypothesisFramedVersion} is that it implies that

+-- {: .num_cor #CanonicalOnAction}
###### Corollary

Every [[∞-groupoid]]

$$
  \mathcal{C}^{fd} \hookrightarrow \mathcal{C}
$$

of [[fully dualizable objects]] in a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}$ carries a canonical [[∞-action]] of (the [[∞-group]] structure on the [[homotopy type]] of) the [[orthogonal group]] $O(n)$, induced by the action of $O(n)$ on the [[framed manifold|n-framing]] of the point in $Bord_n^{fr}$.

=--

([Lurie, corollary 2.4.10](#Lurie))

+-- {: .num_example #ExamplesOfTheCanonicalOnActions}
###### Example

The action in corollary \ref{CanonicalOnAction} is

* for $n = 1$: the $O(1) = \mathbb{Z}/2\mathbb{Z}$ action action given by passing to [[dual objects]];

* for $n = 2$ the $O(2)$-action the _[[Serre automorphism]]_.

* for $n = \infty$ the $O(n)$-action on [[n-fold loop spaces]] (see e.g. [Gaudens-Menichi 07, section 5](#GaudensMenichi07)) (see also at _[[orthogonal spectra]]_).

=--

([Lurie, examples 2.4.12, 2.4.14. 2.4.15](#Lurie))

+-- {: .num_prop #CanonicalSOActionOnBnZ}
###### Proposition

For all $n \in \mathbb{N}$, the canonical $SO$-[[∞-action]] on

$$
  B^n \mathbb{Z} \in Ab_\infty(\infty Grpd) \hookrightarrow (\infty,n)CatWithDuals
$$

is trivial.

=--

+-- {: .proof }
###### Proof

The action on a [[connective spectrum]] $\Omega^\infty X$ factors through the [[J-homomorphism]]

$$
  SO \times \Omega^\infty X
  \stackrel{(J,id)}{\longrightarrow}
  \Omega^\infty S^\infty \times \Omega^\infty X
  \stackrel{precomp}{\longrightarrow}
  \Omega^\infty X
  \,.
$$

But on [[homotopy groups]] the [[image of J]] is pure [[torsion]] which means that for $\Omega^\infty X = B^n \mathbb{Z}$ the induced actions on homotopy groups are all trivial. From this and using the [[long exact sequence of homotopy groups]] it follows that the $\infty$-action itself is trivial.

=--

### For cobordisms with extra topological structure
 {#CobsWithExtraTopStructure}

We discuss the cobordism hypothesis for cobordisms that are equipped with the extra [[structure]] of maps into some [[topological space]] equipped with a [[vector bundle]]. This is the case for which an [[extended TQFT]] is (the local refinement of) what has also been called an _[[HQFT]]_.

#### For cobordisms with any structure ("$(X,\xi)$-structure")
 {#ForCobordismsWithXXiStructure}

+-- {: .num_defn #XXiStructure}
###### Definition

Let $X$ be a [[topological space]] and $\xi \to X$ a [[real number|real]] [[vector bundle]] on $X$ of [[rank]] $n$. Let $N$ be a [[smooth manifold]] of [[dimension]] $m \leq n$. An **$(X,\xi)$-structure** on $N$ consists of the following data

* A [[continuous function]] $f : N \to X$;

* An [[isomorphism]] of [[vector bundle]]s

  $$
    T N \oplus \mathbb{R}^{n-m} \simeq f^* \xi
  $$

  between the [[fiber]]wise [[direct sum]] of the [[tangent bundle]] $T N$ with the trivial rank $(n-m)$ bundle and the [[pullback]] of $\xi$ along $f$.

=--

This is ([Lurie, notation 2.4.16](#Lurie)).

The two extreme cases of def. \ref{XXiStructure} are the following

+-- {: .num_example #FramingAsXXiStructur}
###### Example

For $X = \ast$ the point and $\xi = \mathbb{R}^n$, then an $(X,\xi)$-structure is the same as an $n$-[[framing]], hence

$$
  (Bord_n^{(\ast, \mathbb{R}^n)})
  \simeq
  Bord_n^{fr}
$$

reproduces the $(\infty,n)$-category of framed cobordisms of def. \ref{FramedCobordismsNCat}.

=--

+-- {: .num_example #UnorientedAsXXiStructur}
###### Example

For $X = B O(n)$ the [[classifying space]] of real [[vector bundles]] of [[rank]] $n$ (the [[delooping]] of the [[∞-group]] $O(n)$ underlying the [[orthogonal group]]) and for $\xi = E O(n) \underset{O(n)}{\times} \mathbb{R}^n$ the vector bundle [[associated bundle|associated]] to the $O(n)$-[[universal bundle]], then $(X,\xi)$-structure on $n$-dimensional manifolds is essentially _no_-structure (the [[maximal compact subgroup]]-inclusion $O(n)\to GL(n)$ is a [[weak homotopy equivalence]]). Cobordisms with this structure will also be called _unoriented cobordisms_

$$
  Bord_n^{un} \coloneqq Bord_n^{(B O(n), E O(n)\underset{O(n)}{\times} \mathbb{R}^n)}
  \,.
$$

Accordingly, for $X = B SO(n)$ the [[delooping]] of the [[special orthogonal group]], the corresponding $(X,\xi)$-structure makes _oriented manifolds_

$$
  Bord_n^{or} \coloneqq Bord_n^{(B SO(n), E SO(n)\underset{SO(n)}{\times} \mathbb{R}^n)}
  \,.
$$

=--

Generally:

+-- {: .num_example #GStructureAsXXiStructure}
###### Example

For $\chi \colon G \to O(n)$ a [[topological group]] mapping via a [[homomorphism]] to $O(n)$, then $X = B G$ and $\xi = \chi^\ast (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)$, the $(X,\xi)$-structure is [[G-structure]].

This we get to [below](#CobordismsWithGStructure).

=--

+-- {: .num_defn #CatOfCobordismsWithXIStructure}
###### Definition

Let $X$ be a [[topological space]] and $\xi \to X$ an $n$-[[dimensional]] [[vector bundle]]. The [[(∞,n)-category]] $Bord_n^{(X, \xi)}$ is defined analogously to $Bord_n$ but with all manifolds equipped with $(X,\xi)$-structure, def. \ref{XXiStructure}.

=--

This is ([Lurie, def. 2.4.17](#Lurie)).

+-- {: .num_theorem #CobHypTheoremForTargetSpaceAndVectorBundle}
###### Theorem

Let $\mathcal{C}$ be a [[symmetric monoidal (∞,n)-category]] with duals, let $X$ be a [[CW-complex]], let $\xi \to X$ be an $n$-[[dimensional]] [[vector bundle]] over $X$ equipped with an inner product, and let $\tilde X \to X$ be the [[associated bundle|associated]] [[orthogonal group|O(n)]]-[[principal bundle]] of orthonormal [[frames]] in $\xi$.

There is an [[equivalence in an (∞,1)-category|equivalence]] in [[∞Grpd]]

$$
  Fun^\otimes(Bord_n^{(X,\xi)}, \mathcal{C})
  \simeq
  Top_{O(n)}(\tilde X, \tilde \mathcal{C})
  \,,
$$

where on the right we regard $\tilde C$ as a [[topological space]] carrying the canonical $O(n)$-[[action]] discussed [above](#TheCanonicalOnAction).

=--

This is ([Lurie, theorem. 2.4.18](#Lurie)). The following is some aspects of the idea of the proof in ([Lurie, p. 57](#Lurie)).

+-- {: .num_remark #ActionHomomorphismsAsInfinityActionHomomorphisms}
###### Remark

In the language of [[∞-actions]] (as discussed there), the space $Top_{O(n)}(\tilde X, \tilde \mathcal{C})$ is that of horizontal maps fitting into

$$
  \array{
     X && \longrightarrow && \tilde X//O(n)
     \\
     & \searrow && \swarrow
     \\
     && B SO(n)
  }
$$

where the left map is the classifying map for $\xi$ and the right one is the canonical one out of the [[homotopy quotient]].

=--

+-- {: .proof #ProofOf2.4.18}
###### Idea of Proof of theorem \ref{CobHypTheoremForTargetSpaceAndVectorBundle}.

Notice that for each point $x \colon \ast \to X$ there is an induced inclusion

$$
  Bord_n^{fr} \stackrel{x}{\longrightarrow} Bord_n^{(X,\xi)}
$$

of the framed cobordisms, def. \ref{FramedCobordismsNCat}, into those of $(X,\xi)$-structure, def. \ref{CatOfCobordismsWithXIStructure}, including those cobordisms whose map to $X$ is constant on $X$, and observing that for these an $(X,\xi)$-structure is equivalently an $n$-[[framing]]. Moreover, by corollary \ref{CanonicalOnAction} the induced point evaluation is $O(n)$-equivariant, hence yielding a morphism of [[∞-groupoids]]

$$
  \alpha
  \;\colon\;
  Func^\otimes(Bord_n^{(X,\xi)}, \mathcal{C})
  \longrightarrow
  Maps_{O(n)}(\tilde X, \tilde \mathcal{C})
  \,,
$$

where $\tilde X$ denotes the $O(n)$-[[principal bundle]] to which $\xi$ is [[associated bundle|associated]].

More generally, this is true for the pullback structure of $\xi$ along along any map $Y \to X$, yielding

$$
  \alpha_Y
  \;\colon\;
  Func^\otimes(Bord_n^{(Y,\xi|Y)}, \mathcal{C})
  \longrightarrow
  Maps_{O(n)}(\tilde X\underset{X}{\times} Y, \tilde \mathcal{C})
  \,.
$$

By the previous comment, observe that $\alpha_Y$ is an equivalence for $Y = \ast$.

Now the [[codomain]] of this [[natural transformation]] sends [[(∞,1)-colimits]] in $Y$ over $X$ to [[(∞,1)-limits]]. ([Lurie, theorem 3.1.8](#Lurie)) shows that the same is true for the domain. Hence $\alpha_Y$ is an equivalence for all $Y$ that appear as [[(∞,1)-colimits]] of the point. But this is the case for all [[∞-groupoids]] $Y$, by [this proposition](limit+in+a+quasi-category#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself).

=--

We consider now some special cases of the general definition of local structure-topological field theory

#### For framed cobordisms in a topological space
  {#StatementForCobordismsInAManifold}

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-cobordisms (def. \ref{CatOfCobordismsWithXIStructure}) for the case that the vector bundle $\xi$ is the trivial vector bundle $\xi = \mathbb{R}^n \otimes X$.

In this case $\tilde X = O(n) \times X$.  Write

$$
  Bord_n^{fr}(X) := Bord_n^{(X,X \times \mathbb{R}^n)}
  \,.
$$

Write $\Pi(X) \in $ [[∞Grpd]] for the [[fundamental ∞-groupoid]] of $X$.

+-- {: .num_prop }
###### Corollary

There is an [[equivalence in an (∞,1)-category|equivalence]] in [[∞Grpd]]

$$
  Fun^\otimes(Bord^{fr}_n(X), C)
  \simeq
  (\infty,n)Cat(\Pi(X), \tilde C)
  \simeq
   \infty Grpd(\Pi(X), Core(\tilde C))
  \,,
$$

=--

This is a special case of the [above theorem](#CobHypTheoremForTargetSpaceAndVectorBundle).

Notice that one can read this as saying that $Cob_n(X)$ is roughly like the [[free functor|free]] [[symmetric monoidal (∞,n)-category]] on the [[fundamental ∞-groupoid]] of $X$ (relative to $\infty$-categories of fully dualizable objects at least).

#### For cobordisms with $G$-structure
 {#CobordismsWithGStructure}

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-bundles (def. \ref{CatOfCobordismsWithXIStructure}) for the special case of [[G-structure]] (example \ref{GStructureAsXXiStructure}), hence for the case that $X$ is the [[classifying space]] of a [[topological group]].

Let $G$ be a [[topological group]] equipped with a [[homomorphism]] $\chi : G \to O(n)$ to the [[orthogonal group]]. Notice that via the canonical linear [[representation]] $\mathbf{B}O(n) \to$ [[Vect]] of $O(n)$ on $\mathbb{R}^n$, this induces accordingly a representation of $G$ on $\mathbb{R}^n$..

Let then

* $X := B G$ be the [[classifying space]] for $G$;

* $\xi_\chi := \mathbb{R}^n \times_G E G $ be the corresponding [[associated bundle|associated vector bundle]] to the [[universal principal bundle]] $E G \to B G$.

+-- {: .num_defn }
###### Definition

We say

$$
  Bord^G_n := Bord_n^{(B G, \xi_\chi)}
  \,.
$$

is the $(\infty,n)$-category of **cobordisms with $G$-structure**.

=--

See ([Lurie, notation 2.4.21](#Lurie))

+-- {: .num_defn }
###### Definition

We have

* For $G = 1$ the trivial group, a $G$-structure is just a framing and so

  $$
    Bord_n^{(1,\xi)} \simeq Bord_n^{fr}
  $$

  reproduces the $(\infty,n)$-category of [[framed cobordisms]], def. \ref{FramedCobordismsNCat}.

* For $G = SO(n)$ the [[special orthogonal group]] equipped with the canonical embedding $\chi : SO(n) \to O(n)$ a $G$-structure is an [[orientation]]

  $$
    Bord_n^{(SO(n))} \simeq Bord_n^{or}
    \,.
  $$

* For $G = O(n)$ the [[orthogonal group]] itself equipped with the identity map $\chi : O(n) \to O(n)$ a $G$-structure is no structure at all,

  $$
    Bord_n^{O(n)} \simeq Bord_n
    \,.
  $$

=--

See ([Lurie, example 2.4.22](#Lurie)).

Then we have the following version of the cobordism hypothesis for manifolds with $G$-structure.

+-- {: .num_cor #GEquivariantVersion}
###### Corollary

For $G$ an [[∞-group]] equipped with a homomorphism $G \to O(n)$ to the [[orthogonal group]] (regarded as an [[∞-group]] in [[∞Grpd]]), then evaluation on the point induces an equivalence

$$
  Fun^\otimes( Bord_n^{G}, \mathcal{C} )
  \simeq
  (\tilde {\mathcal{C}})^{G}
$$

between extended TQFTs on $n$-dimensional manifolds with [[G-structure]] and the [[∞-groupoid]] [homotopy invariants](invariant#ForInfinityGroupActions) of the [[infinity-action]] of $G$ on $\tilde \mathcal{C}$ (which is induced by the evaluation on the point).

=--

This is ([Lurie, theorem 2.4.26](#Lurie)).

+-- {: .proof #ProofOf2.4.26}
###### Proof

Theorem \ref{CobHypTheoremForTargetSpaceAndVectorBundle} asserts that

$$
  Fun^\otimes( Bord_n^{G}, \mathcal{C} )
  \simeq
  Maps_{G}(E G , \tilde C)
 \,.
$$

Hence it remains to see that the right hand side are equivalently the [[homotopy invariants]] of the $G$-[[∞-action]]. This follows for instance with the discussion at [[∞-action]], by which

$$
  Maps_G(V,W)\simeq \infty Grpd_{/B G}(V/\!/G, W/\!/G)
  \,.
$$

This yields

$$
   Maps_{G}(E G , \tilde C)
   \simeq
   \infty Grpd_{/ B G}( B G, \tilde C /\!/B G )
   \,.
$$

By the discussion at [[dependent product]]

$$
  \infty Grpd_{/ B G}( B G, \tilde C /\!/B G )
  \simeq
  \underset{B G}{\prod} (\tilde C /\!/B G)
$$

which are the [[homotopy invariants]].

=--

#### For (un-)oriented cobordisms
 {#ForUnorientedCobordisms}

The case that $\chi \colon G \longrightarrow O(n)$ is the identity
is at the other extreme of the framed case, and turns out to be similarly fundamental.

For $\mathbf{H}$ an [[(∞,1)-topos]], write $Corr_n(\mathbf{H})^\otimes$ for the [[(∞,n)-category of correspondences]] in $\mathbf{H}$. For $Phases \in DCat_n(\mathbf{H})$ an [[(∞,n)-category with duals]] [[internal (∞,n)-category|internal]] to $\mathbf{H}$, write $Corr_n(\mathbf{H}_/{Phases})^{\otimes_{phases}}$ for the [[(∞,n)-category of correspondences]] over $Phases$ and equipped with the [[phased tensor product]]. There is the [[forgetful functor|forgetful]] [[monoidal (∞,n)-functor]]

$$
  Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}
  \longrightarrow
  Corr_n(\mathbf{H})^\otimes
$$

By the discussion at _[[(∞,n)-category of correspondences]]_ these are [[(∞,n)-categories with duals]] and the canonical $O(n)$-[[∞-action]] on them, corollary \ref{CanonicalOnAction}, is trivial for $Corr_n(\mathbf{H})$. This means that an $O(n)$-[[homotopy fixed point]] in $Corr_n(\mathbf{H})$ is just an object of $\mathbf{H}$ equipped in turn with an $O(n)$-[[∞-action]].  Therefore

+-- {: .num_prop #UnorientedBulkFieldTheories}
###### Proposition

Local unoriented-topological field theory

$$
  Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H})^\otimes
$$

are equivalent to objects $X \in \mathbf{H}$ equipped with an $O(n)$-[[∞-action]].

At least for $\mathbf{H} = $ [[∞Grpd]], then given such, the corresponding field theory $Z_{X/\!/O(n)}$ sends a cobordism $\Sigma$ to the space of maps

$$
  \array{
     \Pi(\Sigma) && \longrightarrow && X//O(n)
     \\
     & {}_{\mathllap{T \Sigma \oplus \mathbb{R}^{n-dim(\Sigma)}}}\searrow
     && \swarrow
     \\
     && B O(n)
  }
$$

hence

$$
  Z_{X//O(n)} \colon \Sigma \mapsto [\Pi(\Sigma),X]^{O(n)}
  \,.
$$

In particular this means that the assignment to the point is again $X$ itself.

=--

This is a slight rephrasing of the paragraph pp 58-59 in ([Lurie](#LurieTFT)).

+-- {: .num_prop #ExchangingFieldsForStructures}
###### Proposition

At least for $\mathbf{H} = $ [[∞Grpd]], with $X \in \mathbf{H}$ an object equipped with an $O(n)$-[[∞-action]], then horizontal lifts in

$$
  \array{
     && Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}
     \\
     & \nearrow & \downarrow
     \\
     Bord_n^\sqcup
     &\underset{X//O(n)}{\longrightarrow}&
     Corr_n(\mathbf{H})^\otimes
  }
$$

are equivalent to

$$
  (Bord_n^{(X//O(n), X \underset{O(n)}{\times}\mathbb{R}^n)})^{\sqcup}
  \longrightarrow
  Phases^\otimes
  \,.
$$

=--

This is ([Lurie, prop. 3.2.8](#Lurie)).

+-- {: .num_remark}
###### Remark

Via the interpretation of local field theories with coefficients in
$Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$ as [[schreiber:Local prequantum field theory]], the statement of prop. \ref{ExchangingFieldsForStructures} translates in quantum field theory jargon to the statement that "All background structures are fields." This is essentially the slogan of [[general covariance]].

=--

+-- {: .num_cor #UnorientedLocalPrequantumFieldTheory}
###### Corollary

Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an [[abelian ∞-group]] object, regarded as a [[(∞,n)-category with duals]] [[internal (∞,n)-category|internal]] to $\mathbf{H}$.

At least if $\mathbf{H} = $ [[∞Grpd]], then local unoriented-topological field theories of the form

$$
  Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}
$$

are equivalent to a choice

1. of $X \in \mathbf{H}$ equipped with an $O(n)$-[[∞-action]]

1. a homomorphism of $O(n)$-[[∞-actions]] $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence (by the discussion at _[[∞-action]]_) to a horizontal morphism in $\mathbf{H}$ fitting into the diagram

$$
  \array{
      X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n)
      \\
      & \searrow &\swArrow_\simeq& \swarrow
      \\
      && B O(n)
  }
  \,.
$$

=--

+-- {: .proof }
###### Proof

By prop. \ref{UnorientedBulkFieldTheories}
the co-restriction

$$
  Bord_n^\sqcup \stackrel{Z}{\longrightarrow} Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}
  \longrightarrow Corr_n(\mathbf{H})^\otimes
$$

is equivalent to an [[∞-action]]

$$
  \array{
     X &\longrightarrow& X//O(n)
     \\
     && \downarrow
     \\
     && B O(n)
  }
$$

Therefore by prop. \ref{ExchangingFieldsForStructures} $Z$ is equivalent to

$$
  (Bord_n^{(X//O(n), X \underset{O(n)}{\times} \mathbb{R}^n)  })^\sqcup
  \longrightarrow
  Phases^\otimes
  \,.
$$

Finally, by theorem \ref{CobHypTheoremForTargetSpaceAndVectorBundle} and in view of remark \ref{ActionHomomorphismsAsInfinityActionHomomorphisms}, this is equivalent to maps of the form

$$
  \array{
      X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n)
      \\
      & \searrow && \swarrow
      \\
      && B O(n)
  }
  \,.
$$

=--

By the discussion at _[[schreiber:Local prequantum field theory]]_, these statements hold also for fields with moduli spaces in more general $(\infty,1)$-toposes $\mathbf{H}$ (one sufficient condition is that $\mathbf{H}$ has an [[(infinity,1)-site]] of definition all whose objects are [[etale homotopy theory|etale contractible]]).

Some examples are discussed at _[[prequantum field theory]]_ in the section _[Higher Chern-Simons field theory -- Levels](prequantum+field+theory#HigherChern-SimonsLocalPrequantumFieldTheoryLevels)_.

#### For HQFTs
  {#ForHQFTs}

If in  def. \ref{CatOfCobordismsWithXIStructure} one chooses $X = B SO(n) \times Y$ for any topological space $Y$, and $\xi$ the [[pullback]] of the canonical vector bundle bundle on $B SO$ to $B SO \times Y$, then an $(\infty,n)$-functor $Bord^{X}_n \to C$ is similar to what Turaev calls an [[HQFT]] over $Y$.

### For cobordisms with singularities (boundaries/branes and defects/domain walls)
 {#ForCobordismsWithSingularities}

There is a vast generalization of the plain $(\infty,n)$-category of cobordisms (with topological structure) considered above given by allowing the [[cobordisms]] to be equipped with various types of [[singularities]] ([Lurie 09, Definition Sketch 4.3.2](#Lurie)).

Each type of singularity in dimension $k$ now corresponds to a new generator [[k-morphisms]], and the (framed) $(\infty,n)$-category of cobordisms with singularities is now no longer the free symmetric monoidal $(\infty,n)$-category freely generated from just a point (a 0-morphisms), but freely generated from these chosen generators. This general version is ([Lurie 09, Theorem 4.3.11](#Lurie)).

For instance if the generator on top of the point $\ast$ is a [[1-morphism]] of the form $\emptyset \to \ast$, then this defines a type of [[codimension]] $(n-1)$-[[boundary]]; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ with all dual are equivalent to choices of morphisms $1 \to A$, where $A \in \mathcal{C}$ is the fully dualizable object assigned to the point, as before, and now equipped with a morphism from the tensor unit into it.
Indeed, this is the usual datum that describes [[branes]] in QFT (see for instance at [[FRS formalism]]).

For more on this see at _[[QFT with defects]]_.

### For noncompact cobordisms
 {#ForNoncompactCobordisms}

One important variant of the category of cobordisms is obtained by discarding all those morphisms which have non-empty incoming (say, dually one could use outgoing) boundary component. Then a representation of this category imposes on its values "cups but no caps", hence only half of the data of a [[dualizable object]] in the given degree.

Accordingly, in this case the cobordism hypothesis says that such a functor is given not quite by a [[fully dualizable object]], but by a weaker structure called a _[[Calabi-Yau object]]_ (see there for more).

([Lurie, section 4.2](#Lurie))

2-dimensional TQFT of this form is known as _[[TCFT]]_, see there for more

### For cobordisms with geometric structure

A non-topological [[quantum field theory]] is a representation of a [[cobordism category]] for cobordisms equipped with extra [[stuff, structure, property]] that is "not just topological", meaning roughly not of the form of def. \ref{CatOfCobordismsWithXIStructure}.

The theory for this more general case is not as far developed yet.

* steps towards classification of quantum field theories with _super-Euclidean structure_ are discussed at

  * [[(1,1)-dimensional Euclidean field theories and K-theory]]

  * [[(2,1)-dimensional Euclidean field theories and tmf]]

* a general definition of a [[cobordism category]] of cobordisms equipped with **geometric structure** given by a morphism into, roughly, a [[smooth infinity-groupoid]] of structure is discussed in  ([Ayala](#Ayala)).

A full-blown geometric cobordism statement is due to [Grady & Pavlov 2021](#GradyPavlov21).

## Remarks ##

### Morphisms of TQFTs

In particular this means that $Fun^\otimes(Bord_n^{fr} , C )$ is itself an $(\infinity,0)$-category, i.e. an [[∞-groupoid]].

When interpreting symmetric monoidal functors from bordisms to $C$ as [[TQFT]]s this means that TQFTs with given codomain $C$ form a [[topological space|space]], an [[∞-groupoid]]. In particular, any two of them are either equivalent or have no morphism between them.

According to [[Chris Schommer-Pries]] interesting morphisms of [[TQFT]]s arise when looking at transformations only on sub-categories on all of $Bord_n$. This is described at [[QFT with defects]] .

### Invariants determined from the point ###

The theorem does say that the invariant attached by an extended TQFT to the point determines all the higher invariants – however it is important to notice that there are strong constraints on what is assigned to the point. For an $n$-dimensional _framed_ theory one needs to assign a [[fully dualizable object]], and the meaning of the term "fully dualizable" depends on $n$, and gets increasingly hard to satisfy as n grows..

For an $n$-dimensional unoriented theory, the object assigned to the point has to be a fixed point for the $O(n)$- action on fully dualizable objects that is obtained from the framed case of the theorem.

In the 1d case, this $O(1)$ action on dualizable objects takes every object to its dual, and an $O(1)$ fixed point is indeed a vector space with a nondegenerate symmetric inner product.

For an _oriented_ theory $n$-dimensional theory need an $SO(n)$-fixed point, which for $n=1$ is nothing but for $n=2$ ends up meaning a [[Calabi-Yau category]] (in the case the target 2-category is that of categories).

In fact something more general is true: if one wants a theory that takes values on manifolds equipped with a $G$-structure, for $G$ any group mapping to $O(n)$ (such as for instance [[orientation]] already discussed or its higher versions [[Spin structure]] or [[String structure]] or [[Fivebrane structure]] or ...) one needs to assign to the point a $G$-fixed point in dualizable objects in your category (with $G$ acting through $O(n)$).

This beautifully includes all the above plus for example [[manifold]]s with maps (up to [[homotopy]]) to some auxiliary (connected) space $X$ – here we take $G$ to be the [[loop space]] $\Omega X$  of $X$ (mapping trivially to $O(n)$), so that a reduction of the structure group of the manifold to $G$ involves a map to the [[delooping]] $\mathcal{B}G \simeq X$.

Such theories are classified by $X$-families of fully dualizable objects.

Notice that there is an important subtlety of Lurie's theorem in the case of manifolds with $G$-structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the $G$-fixed points for the $G$-action on fully dualizable objects, but rather they are the _homotopy_ fixed points. This is very important because a homotopy fixed point is not just a property. It is additional structure. Depending on $G$, this additional structure is often encoded in the higher dimensional portion of the field theory.

 One can see this in the 1 dimensional case: there is no property of vector spaces which automatically endows them with an inner product, but it is [[stuff, structure, property|extra structure]].

## Related concepts

* [[generalized tangle hypothesis]]

* [[conformal cobordism category]]

* [[opetopic type theory]]

[[!include Isbell duality - table]]

## References

### Topological cobordisms

The original hypothesis is formulated in

*  {#BaezDolan95} [[John Baez]], [[James Dolan]], _Higher dimensional algebra and Topological Quantum Field Theory_ J. Math. Phys. **36** (1995) 6073-6105 [[arXiv:q-alg/9503002](http://arxiv.org/abs/q-alg/9503002)]

A sketch of an approach to the formalization and proof of the cobordism hypothesis is described in:

* {#Lurie09} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_, Current Developments in Mathematics **2008** (2009) 129-280 [[arXiv:0905.0465](http://arxiv.org/abs/0905.0465)]

The comparatively simple case of $n = 1$ is spelled out in detail in

* [[Yonatan Harpaz]], _The Cobordism Hypothesis in Dimension 1_ ([arXiv:1210.0229](http://arxiv.org/abs/1210.0229))

and aspects of the case $n = 2$ (see also at _[[2d TQFT]]_) are further discussed in

* {#SchommerPries11} [[Chris Schommer-Pries]], _The Classification of Two-Dimensional Extended Topological Field Theories_ ([arXiv:1112.1000](http://arxiv.org/abs/1112.1000))

Lecture notes and reviews on the topic of the cobordisms hypothesis:

* [[Jacob Lurie]], _TQFT and the Cobordism Hypothesis_ ([video](http://www.ma.utexas.edu/video/dafr/lurie/), [notes](http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html))

* [[Jacob Lurie]], _The Classification of Extended Topological Field Theories_ ([video](https://www.youtube.com/watch?v=svsidHyNk1k))

* [[Julie Bergner]], _[[UC Riverside Seminar on Cobordism and Topological Field Theories]]_  (2009).

* [[Julie Bergner]], _Models for $(\infty,n)$-Categories and the Cobordism Hypothesis_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, AMS 2011

* [[Daniel Freed]], _The cobordism hypothesis_, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, ([arXiv:1210.5100](http://arxiv.org/abs/1210.5100), [doi:10.1090/S0273-0979-2012-01393-9](https://doi.org/10.1090/S0273-0979-2012-01393-9))

* {#SchommerPries13} [[Chris Schommer-Pries]], _Dualizability in Low-Dimensional Higher Category Theory_ ([arXiv:1308.3574](http://arxiv.org/abs/1308.3574))

* {#Teleman14} [[Constantin Teleman]], _Five lectures on topological field theory_, 2014 ([pdf](http://math.berkeley.edu/~teleman/math/barclect.pdf))

An approach to the proof of the cobordism hypothesis via [[factorization homology]] is in

* [[David Ayala]], [[John Francis]], _The cobordism hypothesis_, ([arXiv:1705.02240](https://arxiv.org/abs/1705.02240))

Discussion of the canonical $O(n)$-action on [[n-fold loop spaces]] (which may be thought of as a special case of the cobordism hypothesis) includes

* {#GaudensMenichi07} Gerald Gaudens, [[Luc Menichi]],  section 5 of _Batalin-Vilkovisky algebras and the $J$-homomorphism_, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365–374 ([arXiv:0707.3103](http://arxiv.org/abs/0707.3103))

### Geometric cobordisms
 {#ReferencesGeometricCobordisms}

Higher cobordisms with geometric [[structure]] (i.e. for non-topological [[extended quantum field theory]]) are discussed in

* {#Ayala} [[David Ayala]], _Geometric cobordism categories_ ([arXiv:0811.2280](https://arxiv.org/abs/0811.2280))

* {#GradyPavlov20} [[Daniel Grady]], [[Dmitri Pavlov]], *Extended field theories are local and have classifying spaces* [[arXiv:2011.01208](https://arxiv.org/abs/2011.01208)]

and full proof of the cobordism hypothesis in this geometric generality (hence for non-topological [[extended quantum field theory|extended]] [[FQFT]] such as [[conformal field theory]]) is claimed in:

* {#GradyPavlov21} [[Daniel Grady]], [[Dmitri Pavlov]], *The geometric cobordism hypothesis* [[arXiv:2111.01095](https://arxiv.org/abs/2111.01095)]

based on [Grady & Pavlov 2020](#GradyPavlov20).

Review:

* [[Dmitri Pavlov]], *The geometric cobordism hypothesis*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[pdf](https://dmitripavlov.org/nyuad.pdf), [[Pavlov-GCHatCQTS.pdf:file]], [web](Center+for+Quantum+and+Topological+Systems#PavlovMar2023), video: [YT](https://www.youtube.com/watch?v=n6Eog_z82eA)]

[[!redirects cobordism hypotheses]]

[[!redirects cobordism theorem]]

[[!redirects geometric cobordism hypothesis]]


cobordism theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _[[cobordism]]_ sits at a subtle connection between [[differential topology]]/[[differential geometry]] and [[stable homotopy theory]]/[[higher category theory]], this is the content of what is often called "cobordism theory".

The insight goes back to the seminal thesis ([Thom 54](#Thom54)), which established that the [[Pontryagin-Thom construction]] exhibits the [[cobordism ring]], whose elements are cobordism [[equivalence classes]] of [[manifolds]], as the [[homotopy groups]] of a [[spectrum]], whence called the _[[Thom spectrum]]_.

Via the [[Brown representability theorem]], the full [[Thom spectrum]] represents a [[generalized cohomology theory]], whence called _[[cobordism cohomology theory]]_. For every kind of topological structure carried by manifolds this has a variant, such as notably [[complex cobordism cohomology theory]]. These Thom spectra and their cobordism cohomology theories play a special role in [[stable homotopy theory]], for instance for the concepts of _[[orientation in generalized cohomology]]_ and the concept of _[[genus]]_.

A more recent refinement of this statement is (the proof of)  the [[cobordism hypothesis]] which identified the ([[framed manifold|framed]]) [[(∞,n)-category of cobordisms]] as the free [[symmetric monoidal (∞,n)-category with duals]] on a single object.

For more introduction see at _[[Introduction to Cobordism and Complex Oriented Cohomology]]_.

[[!include cobordism theory -- contents]]

## References

Original articles:

* {#Thom54} [[René Thom]], _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

* {#Atiyah61} [[Michael Atiyah]], _Thom complexes_, Proc. London Math. Soc. (3), 11:291–310, 1961. 10

Textbook accounts:

* [[Robert Stong]], _Notes on cobordism theory_ , Princeton Univ. Press  (1968) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/stongcob.pdf))

* [[Theodor Bröcker]], [[Tammo tom Dieck]], *Kobordismentheorie*, Lecture Notes in Mathematics **178**, Springer (1970) [[ISBN:9783540053415](https://link.springer.com/book/9783540053415)]

* [[Sergei Novikov]], Section 4.3 in: _Topology I -- General survey_, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 ([doi:10.1007/978-3-662-10579-5](https://link.springer.com/book/10.1007/978-3-662-10579-5), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf))

* {#Kochmann96} [[Stanley Kochman]], chapters I and IV of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Rudyak98} [[Yuli Rudyak]], _On Thom spectra, orientability and cobordism_, Springer Monographs in Mathematics, 1998 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf))

Lecture notes include

* [[Tom Weston]], _An introduction to cobordism theory_ ([pdf](http://people.math.umass.edu/~weston/oldpapers/cobord.pdf))

* {#Francis3} [[John Francis]], _Topology of manifolds_ course notes (2010) ([web](http://math.northwestern.edu/~jnkf/classes/mflds/)), Lecture 2 _Cobordisms_ (notes by [[Owen Gwilliam]]) ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf)), Lecture 3 _Thom's theorem_ (notes by A. Smith) ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf))

* {#Malkiewich11} [[Cary Malkiewich]], _Unoriented cobordism and $M O$_, 2011 ([pdf](http://math.uiuc.edu/~cmalkiew/cobordism.pdf))

* {#Ebert12} [[Johannes Ebert]], _A lecture course on Cobordism Theory_, 2012 ([pdf](http://wwwmath.uni-muenster.de/u/jeber_02/skripten/bordism-skript.pdf))

* _[[Introduction to Cobordism and Complex Oriented Cohomology]]_

This one here includes the connection to the [[(infinity,n)-category of cobordisms]]

* [[Dan Freed]], _Bordism: old and new_, 2013 ([pdf](http://www.ma.utexas.edu/users/dafr/bordism.pdf))

For [[complex cobordism cohomology]] see there and see

* {#Adams74} [[Frank Adams]], part I, part II of _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974


cobordism theory -- contents

**[[cobordism theory]]** = [[manifolds and cobordisms - contents|manifolds and cobordisms]] + [[stable homotopy theory]]/[[higher category theory]]

* [[equivariant cobordism theory]]

**Concepts of cobordism theory**

* [[manifold]], [[differentiable manifold]], [[smooth manifold]]

* [[tangential structure]]

* [[cobordism]], [[cobordism class]]

  [[cobordism ring]]

* [[submanifold]],

  [[normal bundle]]

* [[Pontrjagin's theorem]] ([[equivariant Pontrjagin theorem|equivariant]], [[twisted Pontrjagin theorem|twisted]]):

  $\phantom{\leftrightarrow}$ [[Cohomotopy]]

  $\leftrightarrow$ [[cobordism classes]] of [[normally framed submanifolds]]

* [[Thom's theorem]]:

  $\phantom{\leftrightarrow}$ [[homotopy classes]] of maps to [[Thom space]] [[MO]]

  $\leftrightarrow$ [[cobordism classes]] of [[normally oriented submanifolds]]

* [[vector bundle]]

  [[Thom space]]

  [[Thom isomorphism]]

* [[Thom spectrum]]

  [[Pontryagin-Thom collapse construction]]

* [[cobordism cohomology theory]]

  [[complex cobordism cohomology theory]]

  [[orientation in generalized cohomology]]

  [[genus]]

* [[(∞,n)-category of cobordisms]]

  [[cobordism hypothesis]]

[[!include flavours of cobordism cohomology theories -- table]]


cobordism theory determining homology theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[multiplicative cohomology theory]], hence an [[E-∞ ring]] $E$, equipped with a multiplicative universal [[orientation in generalized cohomology|orientation]] for manifolds with [[G-structure]], hence a [[homomorphism]] of $E_\infty$-rings $M G \longrightarrow E$ from the $G$-[[Thom spectrum]] $M G$, then $E$ canonically becomes an $M G$-[[∞-module]] and so one may consider the functor

$$
  X \mapsto M G_\bullet(X) \otimes_{M G_\bullet} E_\bullet
  \,.
$$

If this is first of all a [[generalized homology theory]] itself, hence [[Brown representability theorem|represented]] by a [[spectrum]], then one may ask if this spectrum coincides with the original $E$, hence if there is an [[natural equivalence]]

$$
  M G_\bullet(-) \otimes_{M G_\bullet} E_\bullet \simeq E_\bullet(-)
  \,.
$$

If so one often says that _the cobordism theory determines the homology theory_ (e.g. [Hopkins-Hovey 92](#HopkinsHovey92)).

## Examples

Originally this was shown to be the case by ([Conner-Floyd 66](#ConnerFloyd66), see _[[Conner-Floyd isomorphism]]_) for $E= $[[KU]] with its canonical [[complex oriented cohomology theory|complex orientation]] $MU \to KU$ (they also showed the case for [[KO]] with [[MSp]] $\to$[[KO]]). Later the [[Landweber exact functor theorem]] ([Landweber 76](#Landweber76)) generalized this to all [[complex oriented cohomology theories]] [[MU]]$\to E$.

The generalization to the actual [[Atiyah-Bott-Shapiro orientations]] of [[topological K-theory]], namely [[MSpin^c|MSpinc]] $\to$ [[KU]] and [[MSpin]] $\to$ [[KO]] is due to ([Hopkins-Hovey 92](#HopkinsHovey92)).

For [[elliptic cohomology]] with the [[SO orientation of elliptic cohomology]] $M SO \to Ell$ the statement is due to ([Landweber-Ravenel-Stong 93](#LandweberRavenelStong93)). For the refinement to the [[spin orientation of elliptic cohomology]] of ([Kreck-Stolz 93](#KreckStolz93)) a statement is due to ([Hovey 95](#Hovey95)).

## Refereces

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#Landweber76} [[Peter Landweber]], _Homological properties of comodules over $MU^\ast(MU)$ and $BP^\ast(BP)$_, American Journal of Mathematics (1976): 591-610.

* {#HopkinsHovey92} [[Michael Hopkins]], [[Mark Hovey]], _Spin cobordism determines real K-theory_, Mathematische Zeitschrift 210.1 (1992): 181-196. ([[HopkinsHoveyCobordismK.pdf:file]])

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et. al. (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

* {#KreckStolz93} [[Matthias Kreck]], [[Stefan Stolz]], _$HP^2$-bundles and elliptic homology_, Acta Math, 171 (1993) 231-261 ([[KreckStolzElliptic.pdf:file]])

* {#Hovey95} [[Mark Hovey]], _Spin Bordism and Elliptic Homology_, Mathematische Zeitschrift 219, 163-170 1995 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.3277))


cobordisms as presheaves > history

< [[cobordisms as presheaves]]


coboundary

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[cochain complex]] $(V^\bullet,d)$ a **coboundary** is an element in the image of the [[differential]].

More generally, in the context of the [[cohomology|intrinsic cohomology]] of an [[(∞,1)-topos]] $\mathbf{H}$, for $X$ and $A$ two objects, a [[cocycle]] on $X$ with coefficients in $A$ is an object in $\mathbf{H}(X,A)$ and a coboundary between cocycles is a [[morphism]] in there.

## Related concepts

[[!include chains and cochains - table]]

* [[homotopy]], [[concordance]]

[[!redirects coboundaries]]


cocartesian closed category

A **cocartesian closed category** is a [[cocartesian monoidal category]] which is a [[closed monoidal category]].  The notion is not very interesting, however, because of the following:

+--{: .un_theorem}
###### Theorem
Any cocartesian closed category is equivalent to the [[terminal category]].
=--
+--{: .proof}
###### Proof
Let $C$ be cocartesian closed.  Since it has an initial object, it is inhabited; thus it suffices to show that between any two objects there is a unique morphism.  But in any closed monoidal category, for any objects $x$ and $y$ the set $C(x,y)$ is in bijection with $C(I,[x,y])$ where $I$ is the unit object and $[-,-]$ the [[internal-hom]].  But since $I$ is an initial object, $C(I,[x,y])$ is a [[singleton]], hence so is $C(x,y)$.
=--

Note that the proof actually applies to any *semi*-cocartesian closed monoidal category, i.e. any closed monoidal category whose unit is an initial object.

## See also

* [[cartesian closed category]]

* [[cocartesian coclosed category]]


cocartesian coclosed category

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The notion of a *cocartesian coclosed category* is [[abstract duality|dual]] to that of a *[[cartesian closed category]]*.

## Definition

A **cocartesian coclosed category** is a [[locally cocartesian coclosed category]] $C$ with an [[initial object]]. Equivalently, it is a category $C$ with [[finite coproducts]] and [[coexponential objects]].

## Properties

Any category which is both cartesian closed and cocartesian coclosed is a [[thin category]], though it may not be the terminal category (e.g., any [[Boolean algebra]] is such a category).

+--{: .proof}
###### Proof
Let a category be given which is both cartesian closed and cocartesian coclosed. Cartesian closure tells us that the [[product]] of any object with the initial object $0$ will be itself initial (as [[left adjoints]] preserve [[colimits]]). Furthermore, given any [[morphism]] from an object $A$ to $0$, we can pair this with the [[identity morphism]] on $A$ to obtain a morphism from $A$ into $A \times 0$ with a [[left inverse]] given by projection, thus identifying $A$ as a [[retract]] of an initial object, and therefore as an initial object itself. It follows immediately that any two [[parallel morphisms]] to $0$ are equal; equivalently, by the dual reasoning, any two parallel morphisms out of $1$ are equal. But this means any two parallel morphisms in general are equal, as maps from $X$ to $Y$ can be identified with maps from $1$ to $Y^X$; accordingly, the category must be a thin category.
=--

## Examples

The [[Kleisli category]] of a [[continuation monad]] $K_R = R^{R^{(-)}}$ on a cartesian closed category with coproduct $\mathbf{C}$ is cocartesian coclosed.
In fact $(-) \times R^A \dashv A + (-)$, which we can prove using the cartesian closure of $\mathbf{C}$:
$$
\mathbf{Kl}(K_R)(B \times R^A, C)\cong
\mathbf{C}(B \times R^A, R^{R^C}) \cong
\mathbf{C}(B \times R^A \times R^C, R) \cong
$$
$$
\mathbf{C}(B \times R^{A + C}, R)\cong
\mathbf{C}(B, R^{R^{A+C}}) \cong
\mathbf{Kl}(K_R)(B, A+C)
$$

## See also

* [[coclosed category]]

* [[coclosed monoidal category]]

* [[cocartesian monoidal category]]

* [[locally cocartesian coclosed category]]

* [[cartesian closed category]]

* [[cocartesian closed category]]

[[!redirects cocartesian coclosed categories]]


coCartesian fibration of (∞,1)-operads

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#### Higher algebra
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#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #coCartesianFibrationOfInfinityOperads}
###### Definition

Let $\mathcal{O}^\otimes$ be an [[(∞,1)-operad]]. A **coCartesian fibration of (∞,1)-operads** is

1. a [[coCartesian fibration]] $p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ of the underlying [[quasi-categories]];

1. such that the composite

   $$
     \mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to FinSet^{*/} = Comm^\otimes
   $$

   exhibits $\mathcal{C}^\otimes$ as an [[(∞,1)-operad]].

In this case we say that the underlying [[(∞,1)-category]]

$$
  \mathcal{C}  = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}
$$

is equipped by $p$ with the [[structure]] of an **$\mathcal{O}$-[[monoidal (∞,1)-category]]** (see remark \ref{EquivalenceToOMonoidsInInfinityCat} below).

=--

This is ([Lurie, def. 2.1.2.13](#Lurie)).

+-- {: .num_remark #EquivalenceToOMonoidsInInfinityCat}
###### Remark

For $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ a coCartesian fibration of $(\infty.1)$-operads by def. \ref{coCartesianFibrationOfInfinityOperads}, the underlying map

$$
  \mathcal{C} \to \mathcal{O}
$$

is a [[coCartesian fibration]] of [[(∞,1)-categories]]. Therefore by the [[(∞,1)-Grothendieck construction]] it is classified by an [[(∞,1)-functor]]

$$
  \chi \colon \mathcal{O} \to (\infty,1)Cat
  \,.
$$

This inherits monoidal structure (with respect to the [[cartesian monoidal (∞,1)-category]] structure on [[(∞,1)Cat]]) and hence exhibits an $\mathcal{O}$-algebra in [[(∞,1)Cat]]. This way coCartesian fibrations of $(\infty,1)$-operads over some $\mathcal{O}^\otimes$ are equivalently $\mathcal{O}$-algebras in [[(∞,1)Cat]]. Therefore their identification with $\mathcal{O}$-[[monoidal (∞,1)-categories]].

=--

([Lurie, remark 2.1.2.17, 2.4.2.6](#Lurie))

## Related concepts

* [[monoidal (∞,1)-category]]

* [[morphism of (∞,1)-operads]]

  * [[fibration of (∞,1)-operads]]

  * **coCartesian fibration of (∞,1)-operads**

## References

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects coCartesian fibration of (infinity,1)-operads]]

[[!redirects coCartesian fibrations of (∞,1)-operads]]
[[!redirects coCartesian fibrations of (infinity,1)-operads]]


cocartesian monoidal (infinity,1)-category

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###Context###
#### $(\infty,1)$-Category theory
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#### Monoidal categories
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[[!include monoidal categories - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _cocartesian monoidal (∞,1)-category_ is a [[symmetric monoidal (∞,1)-category]] whose [[tensor product]] is given by the categorical [[coproduct]].
This is dual to the notion of [[cartesian monoidal (∞,1)-category]].

## Definition

(...)

([Lurie, def. 2.4.0.1](#Lurie))

(...)

## Properties

+-- {: .un_lemma}
###### Lemma
If $C$ is an [[(infinity,1)-category]] admitting finite coproducts, then the cocartesian symmetric monoidal structure on $C$ is equivalent to the symmetric monoidal structure obtained as the opposite of the [[cartesian monoidal (infinity,1)-category|cartesian monoidal structure]] on the [[opposite (infinity,1)-category]] $C^{op}$.
=--

+-- {: .un_theorem}
###### Theorem
If $C$ is an [[(infinity,1)-category]] with finite coproducts, then the cocartesian model structure on $C$ is universal among [[symmetric monoidal (infinity,1)-categories]] $D$ for which there exists a functor $C \to CAlg(D)$, to the [[symmetric monoidal (infinity,1)-category]] of [[commutative monoid in an (infinity,1)-category|commutative algebra objects]] in $D$.
=--

See ([Lurie, Theorme 2.4.3.18](#Lurie)).

## Examples

* If $C$ is a [[symmetric monoidal (infinity,1)-category]], then the canonical symmetric monoidal structure on $CAlg(C)$, the [[(infinity,1)-category]] of [[commutative monoid in a symmetric monoidal (infinity,1)-category|commutative algebra objects]] in $C$, is cocartesian, by ([Lurie, Proposition 3.2.4.7](#Lurie)).

## References

Section 2.4 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects cocartesian monoidal (∞,1)-category]]
[[!redirects cocartesian monoidal (∞,1)-categories]]
[[!redirects cocartesian monoidal (infinity,1)-categories]]

[[!redirects coCartesian monoidal (∞,1)-category]]
[[!redirects coCartesian monoidal (∞,1)-categories]]
[[!redirects coCartesian monoidal (infinity,1)-categories]]


cocartesian monoidal category

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#Contents#
* table of contents
{:toc}

## Idea

A __cocartesian monoidal category__ is a [[monoidal category]] $\big(\mathcal{C}, 0, \sqcup \big)$ whose monoidal product functor $\sqcup \,\colon\, \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is given by the [[coproduct]] (and so whose [[tensor unit]] is an [[initial object]] "$0$").

This is the [[opposite category|dual]] notion of that of a *[[cartesian monoidal categories|cartesian monoidal category]]*.

Sometime we refer to a category as __cocartesian monoidal__ just to indicate that it *has* all [[finite coproducts]].

## Related concepts

* [[cocartesian coclosed category]]

* [[cartesian monoidal category]]

* [[cocartesian monoidal dagger category]]

* [[cocartesian monoidal preordered object]]

## References

The terminology appears for instance in:

* [[Harley Eades III]], Gianluigi Bellin, Def. 16 of: *A Cointuitionistic Adjoint Logic* [[arXiv:1708.05896](https://arxiv.org/abs/1708.05896)]

[[!redirects cocartesian monoidal category]]
[[!redirects cocartesian monoidal categories]]
[[!redirects coCartesian monoidal category]]
[[!redirects coCartesian monoidal categories]]
[[!redirects co-Cartesian monoidal category]]
[[!redirects co-Cartesian monoidal categories]]


cocartesian monoidal dagger category

[[!redirects cocartesian monoidal dagger categories]]

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#Contents#
* table of contents
{:toc}

## Definition ##
A **cocartesian monoidal dagger category** is a [[monoidal dagger category]] $(C, +, 0)$ with

* a morphism $i_A \in Hom_C(A,A + B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
* a morphism $i_B \in Hom_C(B,A + B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
* a morphism $d_{A + B} \in Hom_C(A + B,D)$ for an object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$
* a morphism $0_A \in Hom_C(0,A)$ for every object $A \in C$

such that

* for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$, $d_{A + B} \circ i_A = d_A$
* for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$, $d_{A + B} \circ i_B = d_B$
* for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $f \in Hom_C(A,B)$, $f \circ 0_A = 0_B$.

In a cocartesian monoidal dagger category, the tensor product is called a **coproduct** and the tensor unit is called an **initial object**.

## Examples ##

* [[semiadditive dagger category]]

## See also ##

* [[dagger category]]
* [[monoidal dagger category]]


cocartesian monoidal preordered object

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### Context
#### Relations
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[[!include relations - contents]]
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#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

In a [[finitely complete category]] $C$, a **cocartesian monoidal preordered object** $X$ is a [[preordered object]] with internal preorder $R\stackrel{(s,t)}\hookrightarrow X \times X$ and a [[monoid object]] in $C$ with multiplication $(-)\vee(-):X \times X \to X$ and a [[global element|global]] unit $\bot:* \to X$, where $*$ is the [[terminal object]] in $C$, with

* a function $\beta:(* \to X) \to (* \to R)$ such that for all [[global elements]] $a:* \to X$, $s \circ \beta(a) = \bot$ and $t \circ \beta(a) = a$.

* functions
$$\kappa_l:((* \to X) \times (* \to X)) \to (* \to R)$$
$$\kappa_r:((* \to X) \times (* \to X)) \to (* \to R)$$
such that for all [[global elements]] $a:* \to X$ and $b:* \to X$, $s \circ \kappa_l(a,b) = a$, $t \circ \kappa_l(a,b) = a \vee b$, $s \circ \kappa_r(a,b) = b$, and $t \circ \kappa_r(a,b) = a \vee b$.

## See also

* [[preordered object]]

* [[monoid object]]

* [[cartesian monoidal preordered object]]

* [[bicartesian preordered object]]

* [[join-semilattice object]]

* [[cocartesian monoidal category]]

[[!redirects cocartesian monoidal preordered objects]]


cocategory

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
A (small) _cocategory_ is a [[category]] [[internal category|internal to]] $Set^{op}$, the [[opposite category]] of [[Set]].

More generally, if $C$ is [[finitely complete category|finitely cocomplete]], there is a notion of cocategory internal to $C$, namely a [[comonad]] in the [[bicategory]] of [[span|cospans]] in $C$.

If $c_0 \to c_1 \leftarrow c_0$ is a cocategory object in $C$, then by homming out of $c_\bullet$, one obtains a limit-preserving functor $C \to Cat$. Under reasonable conditions, the [[adjoint functor theorem]] conversely implies that all limit-preserving functors $C \to Cat$ are obtained in this way.

## Examples

 - Many [[interval object|interval objects]] are cocategory objects. For example, the arrow category is a cocategory object in $Cat$.

 - Any [[coalgebra]] object is a cocategory object. This includes [[coring|corings]], [[Hopf algebroid|Hopf algebroids]], [[cogroupoid|cogroupoids]], etc.

 - In $Set$, every cocategory object is a coproduct of copies of the trivial cocategory object $1 \to 1 \leftarrow 1$ and the cocategory object $1 \to 2 \leftarrow 1$ where the two points are distinct. These represent the [[discrete category]] functor and the [[codiscrete category]] functors $Set \to Cat$, respectively.

## Related entries

* [[Trimble n-category]]


cochain

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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[duality|dual]] to a [[chain]].

## Related concepts

[[!include chains and cochains - table]]

[[!redirects cochains]]


cochain algebra > history

< [[cochain algebra]]
[[!redirects cochain algebra -- history]]


cochain complex

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###Context###
#### Homological algebra
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#Contents#
* table of contents
{:toc}

## Idea

A **cochain complex** $V^\bullet$ in an [[Ab-enriched category]] $C$ may be regarded as a [[chain complex]] (see there for more) in $C$, but with the sign of the $\mathbb{Z}$-grading reversed.

This means that for a cochain complex $(V^\bullet,d)$ the [[differential]] (or *coboundary* operator) $d$ _increases_ the degree

$$
  d \colon V^{k} \to V^{k+1}
  \,.
$$

## Examples

* An archetypical example is the deRham cochain complex of [[differential form]]s on a [[manifold]] $X$, or more generally of [[differential forms in synthetic differential geometry]].

* The dual (in some suitable sense) of a [[chain complex]] $(V_\bullet, \partial_k : V_k \to V_{k-1})$ is canonically a cochain complex with $V^k := (V_k)^*$ and $d_k := (\partial_k)^*$.

* A [[monoid]] in chain complexes as well as cochain complexes is a [[differential graded algebra]].

  * In their dual formulation [[L-infinity-algebra]]s and more generall [[Lie infinity-algebroid]]s are encoded by such monoids in cochain complexes.

* By the dual [[Dold-Kan correspondence]] cochain complexes in non-negative degree are equivalent to [[simplicial object|cosimplicial]] abelian groups.

[[!redirects cochain complexes]]


cochain on a simplicial set

> This entry may need to be merged with _[[simplicial cochain]]_.

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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The collection $[S^\bullet,R]$ of $R$-valued functions on a [[simplicial set]] $S^\bullet$ is a commutative [[cosimplicial algebra]]. Under the [[monoidal Dold–Kan correspondence]] it maps to its [[Moore complex|Moore cochain complex]] $C^\bullet([S^\bullet,R])$ which is a [[dg-algebra]] under the [[cup product]]: this is the **cochain complex of the simplicial set**.

Notably, this cochain complex is an [[E-∞ algebra]] (an [[algebra over an operad|algebra]] over the [[E-∞ operad]]). In [[chain homology and cohomology|cohomology]] it becomes a [[graded algebra|graded-commutative algebra]].

## Definition

Let $R$ be commutative [[ring]].

For $S$ a set, write
$$[S,R] = R^S$$
for the $R$-valued functions on $S$: the set of maps from $S$ to $R$ (using either [[internal hom]] notation or [[exponential object]] notation).

This is in particular naturally

* a  [[group]] (using the addition in $R$);

* and even an $R$-[[module]]

* and even an $R$-[[algebra]].

* and even a commutative $R$ algebra (since $R$ is assumed to be commutative ring).

Similarly, for $S = (S_\bullet) : \Delta^{op} \to Set$ a [[simplicial set]] write $[S_\bullet,R] $
for the [[cosimplicial algebra]] obtained by taking $R$-valued functions in each degree. This is naturally

* a [[simplicial group|cosimplicial group]]

* and even a cosimplicial $R$-module

* and even a [[cosimplicial algebra]] over $R$ .

Equivalently, if we write $R [S_\bullet]$ for the [[simplicial object|simplicial]] $R$-module which is in degree $n$ the free $R$-module on the set $S_n$, we have a canonical [[isomorphism]]

$$
  [S_\bullet,R]
    \simeq
  Hom_{R Mod}(R[S_\bullet], R)
  \,.
$$

This latter point of view is often preferred in the literature when $R[S_\bullet]$ is regarded as the collection of _chains_ on $S$ and $[S_\bullet,R]$ as that of _cochains_ .

More precisely, we should speak of chains and cochains after applying the [[Moore complex]] functor. Write

$$
  C^\bullet(S,R) := C^\bullet([S_\bullet,R])
$$

for the [[Moore complex|Moore cochain complex]] obtained from the [[simplicial group|cosimplicial group]] $[S_\bullet,R]$. This is the **cochain complex** of the simplicial set $S$. Using the [[cup product]], this is even a [[dg-algebra]].

## Properties

+-- {: .un_prop}
###### Proposition

The functor
$$
  [-,R] : SSet \to [\Delta^\op,R Mod]
$$
is a [[symmetric monoidal functor|symmetric]] [[lax monoidal functor]].

=--

+-- {: .proof}
###### Proof

For instance Prop 3.8 in ([May03](#May03)) .

=--

...

### Homotopy commutativity {#homotopycommutativity}

The [[dg-algebra]] of cochains $C^\bullet(S,R)$ is not, in general, (graded) commutative. But it is homotopy commutative in that it is an [[algebra over an operad]] for an [[E-infinity operad|E-∞ operad]].

+-- {: .un_theorem}
###### Theorem

The cochain functor
$$
  C^\bullet[-,R] : SSet \to dgAlg
$$
naturally factors through [[algebra over an operad|algebras over]] an [[E-infinity operad|E-∞ operad]], notably the [[EilenbergZilber operad]] as well as the [[Barratt-Eccles operad]].

In both these cases the complex of binary operations in these operads has a 0-cycle whose action $C^\bullet(S,R) \otimes C^\bullet(S,R) \to C^\bullet(S,R)$ is the usual [[cup product]].

=--

+-- {: .proof}
###### Proof

The statement for the Eilenberg--Zilber operad goes back to [HinSch87](HinSch87) . A good review is in ([May03](#May03)) . The statement for the Barrat--Eccles operad is in ([BerFre01](#BerFre01)) .

=--

## Examples

* For $X$ a [[topological space]] and $\Delta_{Top} : \Delta \to Top$ the canonical topological simplices, the simplicial set $X^{\Delta^\bullet_{Top}}$ is the [[singular simplicial complex]] of $X$. It cochain dg-algebra is the one that computes the [[singular cohomology]] of $X$.

## Related concepts

* [[chain homology]]

## References {#References}

Basics are for instance in _Application 1.1.3_ of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

An explicit description of the cochains that express the homotopy symmetry of the cup product is given from page 30 on of the old

* [[Norman Steenrod]], _[[SteenrodOnCohomologyOperations.pdf:file]]_ , Colloquium lectures (1957)

The modern [[operad]]-theoretic statement that for $S \in$ [[SSet]] a [[simplicial set]], the cochain complex  $C^\bullet([S,R])$ is an [[E-∞ algebra]] apparently goes back to

* [[V. Hinich]] and [[V. Schechtman]], _On homotopy limits of homotopy algebras_, in _K-theory, arithmetic and geometry_, Lecture notes Vol. 1289, Berlin 1987 pp. 240--264
{#HinSch87}

A particularly clear exposition is in

* [[Michael Mandell]], _Cochain multiplication_, Amer. J. Math.  124 (2002)

This in turn is nicely reviewed and spelled out in section 3 of

* [[Peter May]], _Operads and sheaf cohomology_ (2003).
[PDF](https://www.math.uchicago.edu/~may/PAPERS/Esheaf.pdf).
(unpublished private notes -- but maybe we get permission to upload them here?)
{#May03}

These describe actions of the [[Eilenberg-Zilber operad]] on $C^\bullet([S^\bullet,R])$.

An action of instead the [[Barratt-Eccles operad]] is described in

* [[Clemens Berger]], [[Benoit Fresse]] _Combinatorial operad actions on cochains_ ([arXiv:0109158](http://arxiv.org/abs/math/0109158))
{#BerFre01}

An action of a [[sequence operad]], which is isomorphic to the [[surjection operad]] of Berger and Fresse was constructed by

* [[James E. McClure]], [[Jeffrey H. Smith]], _Multivariable cochains operations and little $n$-cubes__ ([arXiv:math/0106024](https://arxiv.org/abs/math/0106024))

[[!redirects cochains on simplicial sets]]


CocommCoalg

# Contents #
* table of contents
{:toc}

This page describes some nice properties of the [[category]] $R Cocomm Coalg$ of [[cocommutative coalgebra|cocommutative]] [[coalgebras]] over a [[ground ring]] $R$, in particular details of the proof that it is a [[complete category|complete]], [[cocomplete category|cocomplete]], [[lextensive category|lextensive]], [[cartesian closed category]] with a [[generating set]].

In the sequel, we will simply say "coalgebra", although we really mean *cocommutative* (coassociative, counital) coalgebra.

## Colimits

Colimits in $R Cocomm Coalg$ are created (reflected) by the forgetful functor $U: R Cocomm Coalg \to R Mod$. As the codomain is cocomplete, so is the domain. Thus for example, the coproduct of coalgebras $C, D$ is the direct sum module $C \oplus D$ equipped with the evident comultiplication $C \oplus D \to (C \otimes C) \oplus (D \otimes D) \hookrightarrow (C \oplus D) \otimes (C \oplus D)$.

It is perhaps well to point out explicitly that if $f: C \to D$ is a coalgebra map, then the [[image]] $f(C)$ in $R Mod$ inherits a coalgebra structure from $D$ and provides the image in $R Cocomm Coalg$. This is an easy consequence of the fact that $f(C \otimes C) = f(C) \otimes f(C)$ as submodules of $D \otimes D$.

## Local presentability

Most of the nice properties can be derived as consequences of the [[locally presentable category|local presentability]] of $R Cocomm Coalg$. For this we refer to the paper by [Porst](#Porst).

In the case where $R$ is a [[field]] $k$, the category $k Cocomm Coalg$ is locally finitely presentable. The finitely presentable objects of $k Cocomm Coalg$ are those coalgebras that are [[finite-dimensional vector space|finite-dimensional]] as [[vector spaces]]. The first step towards establishing locally finite presentability is the fundamental theorem of coalgebras, which guarantees that every coalgebra is a filtered colimit of finite-dimensional coalgebras:

+-- {: .num_theorem}
###### Theorem
Every coalgebra $C$ is the union of its finite-dimensional subcoalgebras, i.e., is the [[directed colimit]] of the system of finite-dimensional subcoalgebras of $C$ and inclusion maps between them.
=--

(For the nonce, we define "subcoalgebra" of $C$ to mean vector subspace $i: V \hookrightarrow C$ such that the restricted comultiplication $\delta_C \circ i$ is contained in the subspace $V \otimes V$ of $C \otimes C$. Later we will see that subcoalgebras are actually the same thing as [[subobjects]] in $k Cocomm Coalg$ in the sense of equivalence classes of [[monomorphisms]].)

+-- {: .num_prop}
###### Proposition
Every finite-dimensional coalgebra $C$ is finitely presentable, i.e., $\hom(C, -): k Cocomm Coalg \to Set$ preserves filtered colimits.
=--

+-- {: .proof}
###### Proof
If $D = colim_j D_j$ is a directed colimit, then the image of a coalgebra map $f: C \to D$ is a finite-dimensional subcoalgebra inclusion $f(C) \to D$ which, as a finitely presentable vector space, is included in one of the components $D_j \to D$ of the colimit cone; this inclusion is a subcoalgebra inclusion.
=--

It follows easily from these results and cocompleteness of $k Cocomm Coalg$ that $k Cocomm Coalg$ is locally finitely presentable. As a result we have a [[Gabriel-Ulmer duality|Gabriel-Ulmer]] equivalence

$$k Cocomm Coalg \simeq Lex(k Cocomm Coalg_{fd}^{op}, Set)$$

where the category of finite-dimensional coalgebras is dual to the category of finite-dimensional algebras, so that also

$$k Cocomm Coalg \simeq Lex(Comm Alg_{fd}, Set).$$

Naturally, choosing a representative of each isomorphism class of finite-dimensional coalgebras, we obtain a generating set of $k Cocomm Coalg$.

## Totality and completeness

As is the case for any locally presentable category, $R Cocomm Coalg$ is not only cocomplete but is a [[total category|totally cocomplete category]] and is (therefore) [[complete category|complete]] as well.

The construction of [[limits]] can be described explicitly. The [[equalizer]] of two coalgebra maps $f, g: C \stackrel{\to}{\to} D$ is the largest subcoalgebra contained in the $R Mod$ equalizer (the latter coincides with the equalizer as computed in [[Set]]). This can be described even more explicitly in [[Sweedler notation]]: the equalizer of $f, g$ is the set

$$E = \{c \in C: c_{(1)} \otimes f(c_{(2)}) \otimes c_{(3)} = c_{(1)} \otimes g(c_{(2)}) \otimes c_{(3)}\}$$

with the unique structure of coalgebra that makes it a subcoalgebra of $C$.

Assuming the construction of [[cofree coalgebra|cofree cocommutative coalgebra]]s, viz. the right adjoint $K: R Mod \to R Cocomm Coalg$ to the forgetful functor $U: R Cocomm Coalg \to R Mod$ (which we also touch on below), the [[product]] of a family of coalgebras $C_i$ can be described as follows. Consider the product $\prod_i C_i$ taken in $R Mod$, and let $p: U K(\prod_i C_i) \to \prod_i C_i$ be the component of the counit of the adjunction $U \dashv K$ at that product. Then the product $C$ of the $C_i$ taken in $R Cocomm Coalg$ is the largest subcoalgebra $j: C \hookrightarrow K(\prod_i C_i)$ such that each composite $\pi_i p j: C \to C_i$ is a coalgebra map. For a proof, see the [article](#Agore) by Agore.

## Cartesian closure

The [[cartesian product]] of two coalgebras $C, D$ is given by $C \otimes D$ with the evident coalgebra structure. The functor $C \otimes -: R Cocomm Coalg \to R Cocomm Coalg$ is a [[cocontinuous functor]], since colimits are reflected from $R Mod$ and $C \otimes -: R Mod \to R Mod$ is cocontinuous there.

For locally presentable or more generally total categories $\mathbf{C}$, cocontinuity of a functor $F: \mathbf{C} \to \mathbf{C}$ is enough to guarantee that $F$ has a right adjoint. It follows that $R Cocomm Coalg$ is cartesian closed.

## Comonadicity over $R Mod$

Again, since $U: R Cocomm Coalg \to R Mod$ is cocontinuous and $R Cocomm Coalg$ is locally presentable, $U$ has a right adjoint $K$. This is described more explicitly at [[cofree coalgebra|cofree cocommutative coalgebra]].

The comonadicity of $U$ is proven in the [article](#Barr) by Barr, section 4.

## Extensivity

We will show that $k Cocomm Coalg$ is a [[lextensive category]].

+-- {: .num_prop}
###### Proposition
Coproducts of coalgebras are disjoint.
=--

+-- {: .proof}
###### Proof
The coproduct of coalgebras $C, D$ is $C \oplus D$, and the pullback of the two coalgebra inclusions $i_C: C \to C \oplus D, i_D: D \to C \oplus D$ is the coalgebra equalizer of the two maps

$$\array{
 & & C & &  \\
 & _\mathllap{\pi_C} \nearrow & & \searrow _\mathrlap{i_C} \\
C \otimes D & & & & C \oplus D \\
 & _\mathllap{\pi_D} \searrow & & \nearrow _\mathrlap{i_D} \\
 & & D & &
}$$

This coalgebra equalizer is constructed as the largest subcoalgebra of $C \otimes D$ contained in the $k Vect$ equalizer. But the $k Vect$ equalizer is easily seen to be $0$ (cf. the fact that $k Vect$ coproducts are themselves disjoint), and so the coalgebra equalizer must be $0$ as well, concluding the proof.
=--

+-- {: .num_prop}
###### Proposition
Coproducts of coalgebras are stable under pullback.
=--

+-- {: .proof}
###### Proof
Consider the following diagram:

$$\array{
& & F & & \\
& & \downarrow j & & \\
& & X & & \\
& & \downarrow f & & \\
C & \underset{i_C}{\to} & C \oplus D & \underset{i_D}{\leftarrow} & D
}$$

where $f$ is an arbitrary coalgebra map and $j: F \hookrightarrow X$ is a finite-dimensional subcoalgebra inclusion. We will show that the pullback of the coproduct diagram along $f \circ j$ is a coproduct decomposition of $F$.
=--

## References

* {#Barr} [[Michael Barr]], *Coalgebras over a commutative ring*, Journal of Algebra **32** 3 (1974) 600-610 [doi:10.1016/0021-8693(74)90161-6, [pdf](https://www.math.mcgill.ca/barr/ftp/pdffiles/coalgebra.pdf), [[Barr-Coalgebras.pdf:file]]]

* {#Porst} [[Hans-Eberhard Porst]], *On corings and comodules*, Archivum Mathematicum **42** 4 (2006) 419-425 [[dmlcz:108017](https://dml.cz/handle/10338.dmlcz/108017), [pdf](http://dml.cz/bitstream/handle/10338.dmlcz/108017/ArchMathRetro_042-2006-4_7.pdf)]

* Ana L. Agore, _Limits of Coalgebras, Bialgebras and Hopf Algebras_, arxiv.org/pdf/1003.0318, 2010. ([pdf](http://arxiv.org/pdf/1003.0318.pdf))
 {#Agore}

[[!redirects Cocomm Coalg]]
[[!redirects CommCoalg]]
[[!redirects Comm Coalg]]
[[!redirects category of cocommutative coalgebras]]
[[!redirects category of commutative coalgebras]]

category: category


cocommutative coalgebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _cocommutative [[coalgebra]]_ is the formal [[duality|dual]] of [[commutative algebra]].

Cocommutative coalgebras form the [[category]] [[CocommCoalg]].

## Definition

For $(C,\otimes, I, \sigma)$ a [[symmetric monoidal category]], a [[comonoid object]] in $C$ is an [[object]] $A$ equipped with a [[morphism]]

$$
  \Delta : A \to A \otimes A
$$

("[[comultiplication]]") and a morphism

$$
  \varepsilon: A \to I
$$

("[[counit]]") that satisfy the [[coassociativity]] and [[counitality]] equations:

$$
  \array{
     A &\stackrel{\Delta}{\to}& A \otimes A
     \\
     ^\mathllap{\Delta} \downarrow && \downarrow^{\mathrlap{Id \otimes \Delta}}
     \\
     A \otimes A &\underset{\Delta \otimes Id}{\to}&
     A \otimes A \otimes A
  }
\qquad \qquad \qquad
  \array{
     & & A & &
     \\
     & ^\mathllap{Id} \swarrow & \downarrow^\mathrlap{\Delta} & \searrow^\mathrlap{Id} &
     \\
     A & \underset{Id \otimes \varepsilon}{\leftarrow} & A \otimes A & \underset{\varepsilon \otimes Id}{\to} & A
   }
  \,.
$$

This is **co-commutative** if we have a [[commuting diagram]]

$$
  \array{
    && A \otimes A
    \\
    & {}^{\mathllap{\Delta}}\nearrow && \searrow^\mathrlap{\sigma_{A,A}}
    \\
    A  &&\underset{\Delta}{\to}&& A \otimes A
  }
  \,,
$$

where $\sigma_{A,A} : A \otimes A \to A \otimes A$ is the [[braiding]] [[isomorphism]] of $C$.

In the case of the symmetric monoidal category of [[module|modules]] over a [[commutative ring]] $R$, such an object is called a **cocommutative coalgebra** over $R$. In the case of the symmetric monoidal category of [[chain complex|chain complexes]] (or differential graded spaces), such an object is called a **DG coalgebra**.

Sometimes "cocommutative coalgebra" in a symmetric monoidal category is used as a synonym for cocommutative comonoid object. We will follow that convention below. If $M$ is symmetric monoidal, then $CocommCoalg(M)$ denotes the category of cocommutative algebras and coalgebra maps in $M$.

## Relation to cartesian monoidal categories

Cocommutative coalgebras form a bridge between the [[doctrine|doctrines]] of symmetric monoidal categories and [[cartesian monoidal category|cartesian monoidal]] categories. Notice that every object in a cartesian monoidal category carries a unique cocommutative comonoid structure (the counit is the unique map to the terminal object, and the counicity equations then force the comultiplication map to be the diagonal), and every morphism in a cartesian monoidal category is thereby a coalgebra map.

If $C$ and $D$ are two cocommutative coalgebras, then $C \otimes D$ becomes a cocommutative coalgebra with comultiplication

$$C \otimes D \stackrel{\Delta_C \otimes \Delta_D}{\to} C \otimes C \otimes D \otimes D \stackrel{Id \otimes \sigma \otimes Id}{\to} C \otimes D \otimes C \otimes D$$

and counit

$$C \otimes D \stackrel{\varepsilon_C \otimes \varepsilon_D}{\to} I \otimes I \cong I.$$

+-- {: .un_prop}
###### Proposition
$C \otimes D$ is the cartesian product of $C$ and $D$ in $CocommCoalg(M)$.
=--

+-- {: .un_theorem}
###### Theorem
The forgetful 2-functor

$$U: CartMonCat \to SymMonCat,$$

from cartesian monoidal categories and product-preserving functors to symmetric monoidal categories and strong symmetric monoidal functors, is coreflective. That is to say, it has a right [[biadjunction|bi-adjoint]] which sends $M$ to $CocommCoalg(M)$, and the unit of the 2-adjunction is an equivalence.
=--

+-- {: .un_cor}
###### Corollary
The 2-category of cartesian monoidal categories is comonadic (in the bicategorical sense) over the 2-category of symmetric monoidal categories.
=--

## Related concepts

* [[comonoid object]]

* [[monoid]], [[commutative monoid]], [[category of monoids]]

* [[gs-monoidal category]]

[[!redirects cocommutative]]
[[!redirects cocommutativity]]
[[!redirects co-commutative]]
[[!redirects co-commutativity]]

[[!redirects cocommutative coalgebras]]
[[!redirects co-commutative coalgebra]]
[[!redirects co-commutative coalgebras]]

[[!redirects cocommutative comonoid]]
[[!redirects cocommutative comonoids]]


cocomplete category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

## Definition

A [[category]] $C$ is **cocomplete** if it has all [[small diagram|small]] [[colimit|colimits]]: that is, if every [[small diagram]]

$$ F: D \to C$$

where $D$ is a [[small category]] has a [[colimit]] in $C$. Equivalently, a category $C$ is **cocomplete** if it has all small [[wide pushouts]] and an [[initial object]].

The most natural [[morphisms]] between cocomplete categories are the [[cocontinuous functor|cocontinuous functors]].

## Remarks

* Dually, a category with all small limits is a [[complete category]].
* A category $D$ is cocomplete if and only if $D^{op}$ is complete, so the abstract properties of cocompleteness mimic those of completeness.
* If a category has not all small colimits but all _finite_ colimits, then it is a [[finitely cocomplete category]].

## Examples

Many familiar categories of mathematical structures are cocomplete: to name just a few examples, [[Set]], [[Grp]], [[Ab]], [[Vect]] and [[Top]] are cocomplete.

For a small category $C$, the presheaf category $[C^{op},Set]$ is cocomplete, and the [[Yoneda embedding]] exhibits it as the [[free cocompletion]] of $C$.

## The 2-category of cocomplete categories

### Properties

- The 2-category of cocomplete categories is [[2-monadic]] over $Cat$, and [[complete]] and [[cocomplete]] as a 2-category. (See [this MathOverflow answer](https://mathoverflow.net/a/65999/152679) by [[Mike Shulman]].)
- The [[copower]] $C \cdot A$ where $C$ is a category and $A$ is a cocomplete category, is given by $[C^{op}, A]$. The [[power]] $C \pitchfork A$ is given by $[C, A]$ (see [Pitts 1985](#Pitts)).

## Related concepts

* [[complete category]]

* [[bicomplete category]]

* [[M-complete category]]

## References

* {#Pitts} [[Andrew M. Pitts]], _On product and change of base for toposes_ , Cah. Top. Géom. Diff. Cat. **XXVI** no.1 (1985) pp.43-61.

[[!redirects cocomplete categories]]
[[!redirects cocomplete]]
[[!redirects mplete category]]


cocomplete well-pointed topos

# The (co)complete, locally small, well-pointed topos
* table of contents
{: toc}

## Idea

The [[Set|category of sets]] is, among other things, a [[well-pointed topos]] with a [[natural numbers object]] satisfying the [[axiom of choice]]---i.e. a model of [[ETCS]].  However, $Set$ cannot be characterized *uniquely* by this or any other elementary property, i.e. by a property that doesn't refer, explicitly or implicitly, to the notion of set.  This is a version of Gödel's [[incompleteness theorem]].

However, $Set$ can be characterized uniquely by the addition of non-elementary categorial properties, the most notable and obvious of which are [[complete category|completeness]] and [[cocomplete category|cocompleteness]].  On the other hand, it is also true that any [[topos]] is complete and cocomplete relative to itself, in the sense that its [[self-indexing]] is complete and cocomplete as an [[indexed category]].  Thus, in order to characterize $Set$ in this way we have to mean "external" completeness and cocompleteness (which is the intuitive notion of completeness, in the context of some notion of set).

## Properties

+-- {: .un_theorem}
###### Theorem
Let $S$ be a [[locally small category|locally small]], well-pointed topos (such as a locally small model of ETCS); the following are equivalent.

1. $S$ is equivalent either to $Set$ or to the category of sets in some [[Grothendieck universe]].
2. $S$ admits all coproducts indexed by its own [[homsets]].
3. $S$ admits all products indexed by its own [[homsets]].
=--
+-- {: .proof}
###### Proof
The implications from 1 to 2 and 1 to 3 are easy exercises, following from the "second-order replacement axiom" of a Grothendieck universe.  And the implication from 3 to 2 follows because any [[topos]] which has some type of [[limit]] automatically also has the same sort of [[colimit]], because $S^{op}$ is [[monadic adjunction|monadic]] over $S$ (via the contravariant powerset functor).  Thus, it remains to prove that 2 implies 1.  So suppose that $S$ admits coproducts indexed by its homsets.  In particular, for any $X\in S$ we can form the coproduct $\coprod_{x\colon 1\to X} 1$, which of course comes with a canonical map to $X$.  This map is a bijection on [[global elements]], so since $S$ is well-pointed, it is an isomorphism.

Consider the "global sections" functor $\Gamma = S(1,-) \colon S\to Set$.  Well-pointedness implies that this functor is [[faithful functor|faithful]] and [[conservative functor|conservative]].  However, since by the above we have $X\cong \coprod_{\Gamma(X)} 1$, it follows that $\Gamma$ is [[full functor|full]] as well, so that $S$ is equivalent to a [[full subcategory]] of $Set$.  From now on we identify it with its [[essential image]].

Let $X\in S$ and suppose that $Y\subset X$ is a subset of $X$.  By assumption on $S$, it admits the coproduct $\coprod_{x\colon 1\to X} Z_x$ where $Z_x$ is $1$ if $x\in Y$ and $\emptyset$ otherwise.  But this coproduct maps to $Y$ via a bijection on global elements as before, so it is an isomorphism, and thus $Y\in S$ as well.  Hence $S$ is closed in $Set$ under subsets.  It follows that the full inclusion $S\hookrightarrow Set$ is a bijection on [[subobject]] lattices, and thus a [[logical functor]] (preserves [[power objects]]).  Finally, the hypothesis on coproducts implies that the union of an $S$-set of $S$-sets is again an $S$-set.

If $S$ is essentially small, then these properties imply that it must be a Grothendieck universe.  If $S$ is large, then it contains sets of arbitrarily large cardinality, hence (by closure under subsets) it contains sets of all cardinalities---thus it is all of $Set$.

(We have proven the axioms of a universe in a "structural" form.  We can alternately directly construct an [[inaccessible cardinal]] such that $S$ is the category of sets in $V_\kappa$.  We argue as before that $S$ is a full subcategory of $Set$, and let $\kappa$ be the smallest cardinal number not the cardinality of a set in $S$; it follows that $S$ consists precisely of the sets of cardinality $\lt\kappa$.  Since $S$ is a topos with a NNO, $\kappa$ must be an uncountable strong [[limit cardinal|limit]].  Finally, if $X\in S$ and $Y_x \in S$ for each $x\in X$, then $\coprod_{x\colon 1\to X} Y_x\in S$, showing that $\kappa$ is [[regular cardinal|regular]], and hence inaccessible.)
=--

+-- {: .un_cor}
###### Corollary
$Set$ is, up to equivalence, the unique locally small and cocomplete well-pointed topos (hence the unique locally small and cocomplete model of ETCS), and the unique locally small and complete well-pointed topos (hence the unique locally small and complete model of ETCS).
=--
+-- {: .proof}
###### Proof
Since the category of sets in some Grothendieck universe does not admit all small coproducts (for instance, not those of the size of the universe itself), if any of the equivalent statements the theorem hold for some cocomplete topos, that topos must be all of Set.
=--

A more direct proof of the corollary is possible.

+-- {: .proof}
###### Proof
We show as in the theorem that $\Gamma$ is a full inclusion.  Cocompleteness of $S$ then shows that it has a left adjoint given by $X\mapsto \coprod_X 1$, so it suffices to show that the unit $X\to S(1,\coprod_X 1)$ of this adjunction is a bijection.

For injectivity, we need to show that if $j_x=j_y$, then $x=y$.  If $x\neq y$, then since coproducts in a topos are [[disjoint coproduct|disjoint]], the pullback of $j_x$ and $j_y$ is the initial object $0$.  But since $j_x=j_y$, their pullback is also $1$.  By well-pointedness, $0$ is not isomorphic to $1$, so by contradiction, $x=y$.

For surjectivity, we need to show that any map $k\colon 1\to \coprod_X 1$ is equal to $j_x$ for some $x$.  Now for each $x$, the pullback of $k$ and $j_x$ is a [[subterminal object|subobject of 1]], call it $U_x$.  Since a well-pointed topos is [[two-valued topos|two-valued]], every $U_x$ must be $0$ unless some $U_x$ is $1$.  Since coproducts in a topos are [[pullback stability|stable under pullback]], $1 = \coprod_X U_x$.  But then if $U_x=0$ for every $x$, we would have $1 = \coprod_X 0 = 0$, contradicting well-pointedness.  Thus there must be an $x$ with $U_x=1$, which implies that $k=j_x$.
=--

## In Constructive/Intuitionistic Mathematics

In [[constructive mathematics]], $Set$ is not a model of $ETCS$, but it is a model of the [[IETCS|intuitionistic version]] thereof -- that is, it is a (constructively) well-pointed topos with a NNO.  It is also locally small, complete and cocomplete.

However, the above theorem and corollary both fail intuitionistically, at least as stated above.  The proof of the theorem goes through up to the point where we identify $S$ with a full subcategory of $Set$.  The next paragraph, however, proves only that $S$ is closed under indexed unions and [[detachable subsets]] (more precisely, under subsets whose [[characteristic function]] lands in the set of truth values belonging to $S$ --- whereas for arbitrary $S$, the only truth values we can be sure belong to $S$ are "true" and "false").

Depending on how we choose to define "Grothendieck universe" intuitionistically, this may imply that if $S$ is essentially small then it is a Grothendieck universe.  However, it definitely does *not* follow intuitionistically that if $S$ is large, it must be all of $Set$; the argument for this in the classical case implicitly uses the [[axiom of choice]] in the guise of the [[well-ordering]] (or at least [[total ordering]]) of [[cardinal numbers]].

In particular, the corollary fails to hold intuitionistically.  A concrete [[weak counterexample]] (which can be made into a strong counterexample by working internally to some topos) is given by the following.

Let $\mathcal{U}$ be a truth value (identified with the subset $\{\star \;|\; \mathcal{U}\} \subseteq \{\star\} = 1$) such that $\not\not\mathcal{U}$ is valid and $\mathcal{U}$ is [[indecomposable object|indecomposable]] and [[projective object|projective]], and let $S$ be the topos $Set/\mathcal{U}$.  Then $S$ is a [[Grothendieck topos]], hence cocomplete and locally small, and the assumptions on $\mathcal{U}$ ensure that it is well-pointed, but in an intuitionistic theory they don't imply that $\mathcal{U} = \top$.

Note that in this example, the "global sections" functor $S\to Set$ is *not* the forgetful functor $Set/\mathcal{U} \to Set$ (which doesn't even preserve the terminal object), but the [[exponential]] functor $\Pi_U = Hom(U,-)$.  This is the [[direct image functor]] in the [[geometric morphism]] $Set/\mathcal{U} \to Set$, whereas the obvious forgetful functor is the left adjoint to the inverse image functor that exhibits $S$ as a [[locally connected topos]].