co-Heyting negation

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### Context
#### Category theory
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#### Topos Theory
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[[!include topos theory - contents]]
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# Contents
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[[!redirects coHeyting negation]]
[[!redirects Co-Heyting negation]]
[[!redirects supplement]]
[[!redirects bi-Heyting negation]]
[[!redirects Brouwerian negation]]

## Idea

In a [[co-Heyting algebra]] it is possible to define a [[negation|negation operator]] $\sim$ which validates the _law of excluded middle_ but invalidates the _law of non-contradiction_ dual to the negation in a [[Heyting algebra]].

## Definition

Let $a$ be an element of a [[co-Heyting algebra]] $L$ with subtraction $\backslash$. The **co-Heyting negation** of $a$ , called _non a_ (Lawvere 1991) or the _supplement_ of $a$, is defined as $\sim a:=1\backslash a$.

## Remark

A [[co-Heyting algebra|bi-Heyting algebra]] is naturally equipped with two negation operators: the Heyting negation $\neg$ and the co-Heyting supplement $\sim$. Both coincide in a Boolean algebra considered as a bi-Heyting algebra.

In applications, such co-occuring pairs of negation operators $\neg$ and $\sim$ are the most interesting cases as their combination give rise to [[adjoint modalities]] e.g. in mereology they yield _thickened boundaries_ (Stell&Worboys 1997). Beside [[mereology]] they have found applications in [[linguistics]], [[intuitionistic logic]] and [[physics]].

## Properties

*  $\sim a$ minimally supplements $a$ to truth in the sense that $\sim a$ is the least $x$ with $a\vee x=1$. This follows from the adjointness of $\backslash$ which unwinds as $\sim a\leq x$ iff $1=a\vee x$.

* $\sim 1=0$ and $\sim 0= 1$.

* $\sim$ can be used to define the [[co-Heyting boundary|co-Heyting boundary operator]] $\partial :L\to L$ by $\partial a:=a\wedge\sim a$. That $\partial a$ is not necessary trivial is dual to the non-validity of the _tertium non datur_ for general Heyting algebras and already points to the utility of the co-Heyting negation for [[paraconsistent logic]].

* The co-Heyting supplement satisfies the dual **de Morgan rule** $\sim (a\wedge b)=(\sim a)\vee(\sim b)$, but not $\sim (a\vee b) = (\sim a)\wedge (\sim b)$ in general.

* For $a\in L$ define its **core** as $\sim\sim a$. Then $a=\partial a\vee(\sim\sim a)$. Call $a$ with $a=\sim\sim a$ **regular**. Lawvere (1986) proposes in the vein of classical mereology e.g. Tarski 1927 on regions as regular open sets, to consider only regular subbodies as bodies in the full sense.

* Suppose the element $a$ of a co-Heyting algebra has a [[complement]] $x$ i.e. $a\vee x = 1$ and $a\wedge x = 0$, then the complement $x$ coincides with $\sim a$. Because from $a\vee x=1$ follows $\sim a\leq x$ since $\sim a$ is the least element with this property; conversely, $\sim a=\sim a\vee (a\wedge x)=(\sim a\vee a)\wedge (\sim a\vee x)=\sim a\vee x$ whence $x\leq \sim a$. {#co-Heyting_complement}

* A complemented element is obviously regular. The converse is not true. Complemented elements and regular elements of a co-Heyting algebra both form Boolean algebras, but in general regular elements do not inherit the meet operation while complemented elements do. See [[Heyting algebra#ToBooleanAlgebras|relations between Heyting and Boolean algebras]].

* In a bi-Heyting algebra: $\neg a=\neg a\wedge 1=\neg a\wedge (a\vee\sim a)=(\neg a\wedge a)\vee (\neg a\wedge \sim a)= 0\vee (\neg a\wedge\sim a)=\neg a\wedge\sim a$ hence $\neg a\leq\sim a$ and we see that $\neg$ is more strongly negative than $\sim$.

## Related entries

* [[co-Heyting boundary]]
* [[co-Heyting algebra]]
* [[negation]]
* [[Heyting algebra]]
* [[Heyting category]]
* [[mereology]]
* [[adjoint modality]]
* [[paraconsistent logic]]

## References

* [[Andreas Döring|A. Döring]], _Topos-based Logic for Quantum Systems and Bi-Heyting Algebras_ , arXiv:1202.2750 (2013). ([pdf](http://arxiv.org/pdf/1202.2750.pdf))

* Y. Gauthier, _A Theory of Local Negation: The Model and some Applications_ , Arch. Math. Logik **25** (1985) pp.127-143. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002045923))

* {#Law86} [[William Lawvere|F. W. Lawvere]], _Introduction_ , pp.1-16 in Lawvere, Schanuel (eds.), _Categories in Continuum Physics_ , LNM **1174** Springer Heidelberg 1986.

* {#Law91a} F. W. Lawvere, _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in Springer LNM **1488** (1991).

* {#RRZ94} M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _The non-Boolean logic of natural language negation_ , Phil. Math. **2** no.1 (1994) pp.45-68.

* M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Models for non-Boolean negation in natural languages based on aspect analysis_ , pp.241-260 in Gabbay, Wansing (eds.), _What is Negation?_, Kluwer Dordrecht 1999.

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* [[Matías Menni|M. Menni]], C. Smith, _Modes of Adjointness_ , J. Philos. Logic **43** no.3-4 (2014) pp.365-391.

* {#Mor13} T. Mormann, _Heyting Mereology as a Framework for Spatial Reasoning_ , Axiomathes **23** no.1 (2013) pp.237-264. ([draft](http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf))

* C. Rauszer, _Semi-Boolean algebras and their applications to intuitionistic logic with dual operations_, Fund. Math. **83** no.3 (1974) pp.219-249. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83120.pdf))

* [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari,  _Bi-Heyting Algebras, Toposes and Modalities_ , J. Phi. Logic **25** (1996) pp.25-43.

* J.G. Stell, M.F. Worboys, _The algebraic structure of sets of regions_ , pp.163-174 in Hirtle, Frank (eds.), _Spatial Information Theory_, Springer LNCS **1329** (1997).

* H. Wansing, _Negation_ , pp.415-436 in Goble (ed.), _The Blackwell Guide to Philosophical Logic_ , Blackwell Oxford 2001.


co-unitality

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## Idea

The [[formal dual]] of *[[unitality]]*:

An object in a [[monoidal category]] which is equipped with a [[comultiplication]] map which satisfies both co-unitality and [[co-associativity]] is called a *[[co-monoid]]*.

## Definition

Given a [[monoidal category]] $(\mathcal{C}, \otimes)$ and an [[object]] $A$ in $\mathcal{C}$ equipped with a morphism ("[[co-multiplication]]") $\Delta \colon A \longrightarrow A \otimes A$, and a morphism $\epsilon \colon A \longrightarrow 1$, $\epsilon$ is called a *counit* if the following [[commuting diagram|diagrams commute]]

\begin{tikzcd}[sep=20pt]
  &
  A \otimes A
  \ar[
    dr,
    "{ \epsilon \otimes \mathrm{id} }"
  ]
  \\
  A
  \ar[
    ur,
    "{ \Delta }"
  ]
  \ar[
    rr,
    equals
  ]
  &&
  A
  \mathrlap{\,,}
\end{tikzcd}
\begin{tikzcd}[sep=20pt]
  &
  A \otimes A
  \ar[
    dr,
    "{ \mathrm{id} \otimes \epsilon }"
  ]
  \\
  A
  \ar[
    ur,
    "{ \Delta }"
  ]
  \ar[
    rr,
    equals
  ]
  &&
  A
  \mathrlap{\,.}
\end{tikzcd}

## Related concepts

* [[associativity]], [[co-associativity]]

[[!redirects counitality]]


co-well-pointed cotopos

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# Contents
* table of contents
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## Idea

The notion of *co-well-pointed cotopos* is [[abstract duality|dual]] to that of *[[well-pointed topos]]*.

## Definition

A [[cotopos]] $C$ is **co-well-pointed** if the [[initial object]] is a [[cogenerator]] and the initial object is not a [[zero object]].

## See also

* [[cotopos]]

* [[well-pointed topos]]


co-Yoneda lemma

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#Contents#
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## Idea

What is sometimes called the _co-Yoneda lemma_ or _density formula_ is a basic fact about [[presheaves]] (a basic fact of [[topos theory]]): it says that every [[presheaf]] is a [[colimit]] of [[representable functor|representables]] and more precisely that it is the "colimit over itself of all the representables contained in it".

One might think of this as related by [[duality]] to the [[Yoneda lemma]], hence the name.

## Every presheaf is a colimit of representables
 {#EveryPresheafIsAColimitOfRepresentables}

Throughout, let

* $V$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] (for instance $V = $ [[Set]] with its [[Cartesian product]]);

* $\mathcal{C}$ a [[small category|small]] $V$-[[enriched category]].

For $c,d \in Obj(\mathcal{V})$ two objects, we write $\mathcal{C}(c,d) \in V$ for the [[hom-object]].

+-- {: .num_remark}
###### Remark
**([[Yoneda reduction]])**

Recall that the ([[enriched Yoneda lemma|enriched]]) [[Yoneda lemma]] says that for $F \colon \mathcal{C}^{op} \to V$ a $V$-[[enriched functor]] out of the [[opposite category]] of $\mathcal{C}$, hence a $V$-valued [[presheaf]] on $\mathcal{C}$ and $c \in \mathcal{C}$ an [[object]] of $\mathcal{C}$, there is a [[natural isomorphism]] in $V$

$$
  [\mathcal{C}^{op},V]
  \big(
    \mathcal{C}(-,c)
    ,\,
    F
  \big)
   \;\simeq\;
  F(c)
  \,,
$$

where on the left we have the [[hom-object]] in the [[enriched functor category]] between the [[representable functor|functor represented]] by $c$ and the given functor $F$.

Using the expression of these hom-objects on the left as _[[ends]]_, this reads

$$
  \int_{c' \in \mathcal{C}}
  V\big(
    \mathcal{C}(c',c)
    ,\, F(c')
  \big)
    \;\simeq\;
  F(c)
  \,.
$$

In this form the Yoneda lemma is also referred to as _[[Yoneda reduction]]_.

=--

Under [[duality|abstract duality]] an [[end]] turns into a [[coend]], so a co-Yoneda lemma ought to be a similarly fundamental expression for $F(c)$ in terms of a [[coend]].

The natural candidate is the statement that:

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

Every [[presheaf]] $F$ is a [[colimit]] of [[representable functor|representables]], in that

$$
  F(c) \simeq \int^{c' \in C} \mathcal{C}(c,c')\otimes F(c')
$$

hence

$$
  F(-) \simeq \int^{c' \in \mathcal{C}} Y(c')\otimes F(c')
  \,,
$$

where $Y$ denotes the [[Yoneda embedding]]. In [[module]]-language, using the [[tensor product of functors]], this reads

$$
  F(c) \simeq \mathcal{C}(c,-)\otimes_{\mathcal{C}} F
  \,.
$$

Yet another way to state this is as a [[colimit]] over the [[comma category]] $(Y,F)$, for $Y$ the [[Yoneda embedding]]:

$$
  F \simeq colim_{(U \to F) \in (Y,F)} Y(U)
  \,.
$$

=--

This statement we call the **co-Yoneda lemma**.

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

To show that a presheaf $F \colon \mathcal{C}^{op} \to V$ is canonically presented as a colimit of representables, we exhibit a natural isomorphism

$$
  \int^{c} F(c) \otimes \mathcal{C}(-, c)
   \;\cong\;
  F
$$

By the definition of the [[coend]], maps

$$
  \int^c F(c) \times \mathcal{C}(-, c) \to G(-)
$$

are in [[natural bijection]] with families of maps

$$
  F(c) \otimes \mathcal{C}(d, c) \to G(d)
$$

[[extranatural transformation|extranatural]] in $c$ and [[natural transformation|natural]] in $d$. Those are in natural bijection with families of maps

$$
  F(c) \to V(\mathcal{C}(d, c), G(d))
$$

natural in $c$ and extranatural in $d$. These are in natural bijection with families of maps

$$
  F(c) \to Nat(\mathcal{C}(-, c), G) \cong G(c)
$$

(natural in $c$), where the isomorphism is by the Yoneda lemma. Thus we have exhibited a natural isomorphism

$$
  Nat(\int^c F(c) \times \mathcal{C}(-, c), G) \cong Nat(F, G)
$$

(natural in $G$). By Yoneda again, this gives

$$
  \int^c F(c) \times \mathcal{C}(-, c) \cong F
  \,.
$$

=--

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

The statement of the [[co-Yoneda lemma]] in prop. \ref{coYonedaLemma} is a kind of [[categorification]] of the following statement in [[analysis]] (whence the notation with the integral signs):

For $X$ a [[topological space]], $f \colon X \to\mathbb{R}$ a [[continuous function]] and $\delta(-,x_0)$ denoting the [[Dirac distribution]], then

$$
  \int_{x \in X} \delta(x,x_0) f(x)
  =
  f(x_0)
  \,.
$$

=--

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

If one follows the Yoneda-lemma argument at the end of the proof of prop. \ref{coYonedaLemma}, one arrives at the explicit isomorphism

$$
  \int^c F(c) \times \mathcal{C}(-, c) \to F
  \,.
$$

Namely, it corresponds to the family of maps

$$
  F(c) \times \mathcal{C}(d, c) \to F(d)
$$

(extranatural in $c$ and natural in $d$) which in turn corresponds to the natural family

$$
  \mathcal{C}(d, c) \to \hom(F(c), F(d))
$$

associated with the structure of the functor $F \colon \mathcal{C}^{op} \to V$.

=--

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

Let $V = $ [[Set]], and recall the definition of the [[coend]] as a [[coequalizer]]

$$
  \underset{c,d \in \mathcal{C})}{\coprod}
   \mathcal{C}(c,d) \times \mathcal{C}(c_0,c) \times F(d)
   \underoverset
     {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}}
     {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}}
     {\phantom{AAAAAAAA}}
  \underset{c \in \mathcal{C}}{\coprod} \mathcal{C}(c_0,c) \times F(c)
   \overset{coeq}{\longrightarrow}
  \overset{c\in \mathcal{C}}{\int} \mathcal{C}(c_0,c) \times F(c)
  \,.
$$

This says that the coend is the set of [[equivalence classes]] of [[pairs]]

$$
  ( c_0 \overset{}{\to} c,\; x \in F(c) )
  \,,
$$

where two such pairs

$$
  ( c_0 \overset{f}{\to} c,\; x \in F(c) )
  \,,\;\;\;\;
  ( c_0 \overset{g}{\to} d,\; y \in F(d) )
$$

are regarded as equivalent if there exists

$$
  c \overset{\phi}{\to} d
$$

such that

$$
  g = \phi \circ f
  \,,
  \;\;\;\;\;and\;\;\;\;\;
  x = \phi^\ast(y)
  \,.
$$

(Because then the two pairs are the two images of the pair $(f,y)$ under the two morphisms being coequalized.)

But now considering the case that $c = c_0$ and $f = id_{c_0}$, so that $g= \phi$ shows that any pair

$$
  ( c_0 \overset{\phi}{\to} d, \; y \in F(d))
$$

is identified, in the coequalizer, with the pair

$$
  (id_{c_0},\; \phi^\ast(y) \in F(c_0))
  \,,
$$

hence with $\phi^\ast(y)\in F(c_0)$, and that this coequalizing operation is the action

$$
  Hom(c_0,d)\times F(d)\longrightarrow F(c_0)
$$

of morphisms on elements of the presheaf by pullback.

=--

## MacLane's co-Yoneda lemma
 {#MacLanesCoYonedaLemma}

In a brief uncommented exercise on [MacLane, p. 62](#MacLane) the following statement, which is attributed to Kan, is called the **co-Yoneda lemma**.

For $D$ a [[category]], [[Set]] the [[category]] of [[set]]s, $K : D \to Set$ a [[functor]], let $(* \darr K)$ be the [[comma category]] of elements $x \in K d$, let $\Pi: (* \darr K) \to D$ be the projection $(x \in K d) \mapsto d$ and let for each $a \in D$  the functor $\Delta_a: (* \darr K) \to D$ be the diagonal functor sending everything to the constant value $a$.

The **co-Yoneda lemma in the sense of Kan/MacLane** is the statement that there is a [[natural isomorphism]] of [[functor category|functor categories]]

$$
[D,Set](K, D(a, -)) \cong [(*\darr K), D](\Delta_a, \Pi).
$$

Here is an outline of an explicit proof:

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

A natural transformation $\phi: K \to D(a, -)$ assigns to each element $x \in K c$ an element $\phi_c(x) \in D(a, c)$, i.e., an arrow $\phi_c(x): a \to c$. We define a corresponding transformation $\psi: \Delta_a \to \Pi$ which assigns to each object $(c, x \in K c)$ in $(*\darr K)$ the morphism $\phi_c(x): a \to c = \Pi(c, x)$. It is easy to check that the naturality condition on $\phi$ corresponds to the naturality condition on $\psi$, and that the correspondence is bijective.

=--

Here is a more conceptual proof in terms of [[comma categories]]:

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

[[Set]] classifies discrete fibrations, in the sense that a functor $G : D \to Set$ classifies the discrete [[fibration]]

$$
  Q : \Pi_G : El(G) \to D
$$

and natural transformations $\alpha : G \to F$ correspond to maps of fibrations

$$
  El(G) \to El(F)
$$

i.e. functor which commute on the nose with the projections $\Pi_G$, $\Pi_F$ to the base category $D$).

This applies in particular to $F = hom(a,-)$. Notice the [[category of elements]] $El(hom(a,-))$ is the co-slice $(a \downarrow D)$, with its usual projection $\Pi$ to $D$.

However, the [[comma category]] $(a \downarrow D)$ is the "lax pullback" (really, the [[comma object]], see discussion at [[2-limit]]) appearing in

$$
  \array{
    (a \downarrow D) &\stackrel{\Pi}{\to}& D
    \\
    \downarrow &\Uparrow& \downarrow^{Id}
    \\
    * &\stackrel{a}{\to}& D
  }
$$

and so a fibration map $El(G) \to (a \downarrow D)$ corresponds exactly to a lax square

$$
  \array{
    El(G) &\stackrel{\Pi_G}{\to}& D
    \\
    \downarrow &\Uparrow& \downarrow^{Id}
    \\
    * &\stackrel{a}{\to}& D
  }
   \,.
$$

This yields the co-Yoneda lemma in the sense of MacLane's exercise.

=--

## References

For enrichment over $V = Top^{\ast/}_{cg}$  ([[pointed topological spaces|pointed]] [[compactly generated topological spaces]]) the co-Yoneda lemma in the sense of [every presheaf is a colimit of representables](#EveryPresheafIsAColimitOfRepresentables) appears for instance as

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], lemma 1.6 of _[[Model categories of diagram spectra]]_, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [publisher](http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc))

The [co-Yoneda lemma in the sense of MacLane](#MacLanesCoYonedaLemma) appears as a brief uncommented exercise on p. 63 of

* {#MacLane} [[Saunders MacLane]], _[[Categories Work|Categories for the Working Mathematician]]_,

where it is attributed to [[Daniel Kan]].

See also:

* [[Tom Leinster]], [MO:a/20451](https://mathoverflow.net/a/20451/381)

* {#Lor21} [[Fosco Loregian]], Prop. 2.2.1 in: _Coend calculus_, Cambridge University Press 2021 ([arXiv:1501.02503](http://arxiv.org/abs/1501.02503), [doi:10.1017/9781108778657]( https://doi.org/10.1017/9781108778657), ISBN:9781108778657).

[[!redirects Coyoneda lemma]]
[[!redirects coYoneda lemma]]
[[!redirects co-yoneda lemma]]
[[!redirects coyoneda lemma]]
[[!redirects density formula]]
[[!redirects ninja Yoneda lemma]]

[[!redirects enriched co-Yoneda lemma]]


coadjoint action

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## Definition

For $G$ a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$, the **coadjoint action** or **coadjoint representation** of $G$ is the [[action]]/[[representation]] on the [[dual vector space]] $\mathfrak{g}^*$ which is dual to the [[adjoint action]] of $G$ on $\mathfrak{g}$.

An [[orbit]] of the coadjoint action is accordingly called a _[[coadjoint orbit]]_.

## Related concepts

* [[orbit method]]

* [[Borel-Weil-Bott theorem]]

[[!redirects coadjoint actions]]

[[!redirects coadjoint representation]]
[[!redirects coadjoint representations]]


coadjoint orbit

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## Definition

Given a [[Lie group]] $G$, it [[action|acts]] smoothly on the [[dual vector space|dual]] $g^*$ of its [[Lie algebra]] $g$ by the _[[coadjoint action]]_.  The [[orbit|orbits]] of that [[action]] are called __coadjoint orbits__.

Coadjoint orbits are especially important in the [[orbit method]] of [[representation theory]] or, more generally, [[geometric quantization]].

Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of [[Virasoro algebra]]).

## Properties

### As symplectic leaves of the Lie-Poisson structure

The dual $g^*$ of a (say finite-dimensional real) [[Lie algebra]] has a structure of a [[Poisson manifold]] with the Poisson structure due to A. Kirillov and Souriau, called the _[[Lie-Poisson structure]]_, namely for any $a\in g^*$,

$$
  \{ f, g\}(a)
  \;\coloneqq\;
  \langle [d f_a, d g_a],a\rangle
$$

The coadjoint orbits are the [[symplectic leaves]] of that structure; hence each orbit is a [[symplectic manifold]].

## Examples

* [Wilson loops and 1d Chern-Simons σ-models with target the coadjoint orbit](orbit+method#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit)

## Related concepts

* [[orbit method]]

* [[flag variety]], [[coset space]]

* [[Schubert calculus]]

## References

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], pp. 66 in: _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* Bradley N. Currey, _The Structure of the Space of Coadjoint Orbits of an Exponential Solvable Lie Group_, ransactions of the American Mathematical Society
Vol. 332, No. 1 (Jul., 1992), pp. 241-269, ([JSTOR](http://www.jstor.org/stable/2154031))

[[!redirects coadjoint orbits]]


coalgebra

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#### Algebra
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# Coalgebra
* table of contents
{: toc}

## Definition

In the [most familiar sense](http://en.wikipedia.org/wiki/Coalgebra), a _coalgebra_ is just like an
[[associative algebra]], but with all the structure maps 'turned around' (a "[[co-monoid]]").   More precisely, fix a [[ground field]] $k$.   Then an **[[associative algebra]]** $A$ over $k$ is a [[vector space]] equipped with a multiplication

$$
  m \colon A \otimes A \to A
$$

and a unit

$$
  i \colon k \to A
$$

satisfying the [[associativity]] law and left/right [[unit laws]], which can be drawn as [[commutative diagrams]].

[[formal duality|Dually]], a **coalgebra** $C$ is a vector space equipped with a **[[comultiplication]]**

$$
  \Delta \colon A \to A \otimes A
$$

and a **counit**

$$
  e \colon A \to k
$$

satisfying *[[co-associativity]]* and left/right *[[co-unitality]]* laws.

The [[commutative diagrams]] for these laws are obtained by taking the diagrams for the associative and left/right unit laws and turning all the arrows around.

![axioms](http://presheaf.com/cache/d1n601e593w4j122e1nrx3j1d3v68b.png)

We can express this idea much more efficiently using the concept of the [[opposite category|opposite]] of a [[category]], together with [[internalization]].  Namely: a **coalgebra** is a [[monoid]] [[internalization|in]] the $Vect^{op}$, just as an algebra is a monoid in $Vect$.

Coalgebras of this sort are an important ingredient in more sophisticated structures such as [[bialgebra|bialgebras]], [[Hopf algebra|Hopf algebras]] and [[Frobenius algebra|Frobenius algebras]].

More generally:

* a [[coalgebra for an endofunctor]] $F : C \to C$ on a [[category]] $C$ -- an _$F$-coalgebra_ --  is

    * an [[object]] $A$ of $C$;

    * and a [[morphism]] $\alpha : A \to F(A)$;

* a general **coassociative coalgebra** is a [[coalgebra over a comonad]], dual to the concept of an [[algebra over a monad]].

## Examples

### Special cases

* For $R$ a commutative ring, if the endofunction $F : C \to C$ is $F : R Mod \to R Mod$ given by $F : N \mapsto N \otimes N$, then $F$-coalgebras are precisely non-coassociative coalgebras in the specific sense of non-associative monoids in $R Mod^{op}$. (See [[Tom Leinster]]'s comment [here](https://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html#c020741)).

* $L_\infty$-[[L-infinity-algebra|algebra]]s are [[CoDGCA|cocommutative comonoids]] in the category of chain complexes.

### Classes with extra properties and structure

* [[Banach coalgebra]], [[C-star coalgebra]]

* [[measure coalgebra]]

### Other

* [[dyadic interval coalgebra]]
* [[decimal interval coalgebra]]
* [[rational interval coalgebra]]

## Differential graded coalgebras

These are explored briefly in the lexicon style entry [[differential graded coalgebra]].  (At present this is 'bare bones' with little or no motivation or discussion.)

## Cocommutative coassociative coalgebras

These, in most cases, form a complete cocomplete Cartesian Closed Category, $Coalg$ over which the category, $Alg$, of commutative associative algebras is enriched, tensored and cotensored. The exegesis is much the same whether we consider coalgebras over a field $k$, or graded $k$-coalgebras, or differential graded coalgebras, etc. In each case we need a notion of finiteness: finite $k$-dimension of the underlying $k$-vector space, finite dimension in each grade, etc. We denote by $Alg_f$ the category of (commutative associative) algebras that are finite.

The basic fact is that a coalgebra is the [[filtered colimit]] of its finite dimensional subcoalgebras. It follows from this that we can identify $Coalg$ with the category of finite-limit-preserving $Set$-valued functors on $Alg_f$.
This is because every such functor is a filtered colimit of representable functors, and for any finite algebra $A$ its $k$-vector-space dual $A^*$ is a finite coalgebra.

The product of coalgebras $C$ and $D$ is given by $C\otimes_k D$. The exponential $C\Rightarrow D$ is given by the functor taking $A\in Alg_f$ to $Hom_{Coalg}(A^*\otimes_k C,D)$. Note that $C\Rightarrow C$ has the structure of a cocommutative coassociative Hopf algebra.

For $X$ and $Y$ in $Alg$ we define $X\Rightarrow Y$ in $Coalg$ to be given by the functor taking $A\in Alg_f$ to $Hom_{Alg}(X,A\otimes_k Y)$.

For $C\in Coalg$ and $X\in Alg$ we denote by $C\Rightarrow X$ the algebra $Hom_k(C,X)$ with $k$-algebra structure induced by the coalgebra structure of $C$. We denote by $C\otimes X$ the quotient of the free $k$-algebra on $C\otimes_k X$ by the ideal generated by elements of the form

$c\otimes 1-\epsilon(c)$ and $c\otimes x_1 x_2 - \Sigma_i (c'_i\otimes x_1)(c''_i\otimes x_2)$

 where $\epsilon$ is the counit of $C$ and $\Sigma_i c'_i\otimes c''_i$ is the diagonal of $c$ in $C$.

The tensored, cotensored enrichment of $Alg$ over $Coalg$ can be extended to the case of commutative associative $k$-algebras in a topos. It is a consequence of work by N.J.Kuhn, _Generic representations of the Finite General Linear groups and the Steenrod Algebra_, that the mod 2 Steenrod algebra is the Hopf algebra $S(V)\Rightarrow S(V)$ where $S(V)$ is the free graded symmetric $\mathbb{Z}_2$-algebra on the **generic $\mathbb{Z}_2$-vectorspace**. Similar considerations apply to the mod p Steenrod algebra.

Recall that the mod 2 Steenrod Hopf algebra is the dual of the commutative Hopf algebra with generators $\xi_1, \ldots $ with diagonal taking $\xi_n$ to

 $\Sigma_{i+j=n} \xi_i^{2^j}\otimes \xi_j$

 where $\xi_0 = 1$. Its action on $S(V)$ is dual to the coaction taking a vector $v\in V$ to

 $\Sigma_i\xi_i\otimes v^{2^i}$

## Properties

### As filtered colimits of finite-dimensional pieces
  {#AsFilteredColimits}

+-- {: .num_theorem #CoalgebrasAreFilteredColimitsOfTheirFinDimSubalghebras}
###### Theorem
**(fundamental theorem of coalgebras)**

Every coalgebra is the [[filtered colimit]] of its finite-dimensional sub-coalgebras.

=--

This is maybe due to ([Sweedler 69](#Sweedler69)), for proof see also for instance ([Michaelis 03](#Michaelis03)). That this remains true for [[dg-coalgebras]] (see at _[dg-coalgebra -- As filtered colimit](differential+graded+coalgebra#AsFilteredColimits)_) is due to ([Getzler-Goerss 99](#GetzlerGoerss99)).

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

It follows that an [[associative algebra|algebra]], while not itself the filtered limit of its finite dimensional subalgebras in general, is, being the [[linear dual]] of a coalgebra, a "formal filtered limit", hence a [[pro-object]] in finite-dimensional algebras (e.g. [Abrams-Weibel 99, p. 7](#AbramsWeibel99)).

=--

## Related concepts

* [[CocommCoalg]]

* [[co-monoid]]

* [[cofree coalgebra]]

* [[dg-coalgebra]]

* [[Lie coalgebra]]

* [[model structure on coalgebras over a comonad]]

## References

* {#Sweedler69} [[Moss Sweedler]], _Hopf algebras_, 1969

* [[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]]]

* {#Michaelis03}  [[Walter Michaelis]], _Coassociative coalgebras_, Handbook of Algebra Volume 3, Elsevier (2003).

* Luzius Grunenfelder, [[Robert Paré]], _Families parametrized by coalgebras_,  J. Alg. __107__:2 (1987) 316-375 doi [pdf](https://core.ac.uk/download/pdf/82530098.pdf)

* {#AbramsWeibel99} [[Lowell Abrams]], [[Charles Weibel]], _Cotensor products of modules_ ([arXiv:math/9912211](http://arxiv.org/abs/math/9912211))

* {#GetzlerGoerss99} [[Ezra Getzler]], [[Paul Goerss]], _A model category structure for differential graded coalgebras_, 1999 ([ps](http://www.math.northwestern.edu/~pgoerss/papers/model.ps))

* [[Kathryn Hess]], [[Brooke Shipley]], _The homotopy theory of coalgebras over a comonad_ ([arXiv:1205.3979](http://arxiv.org/abs/1205.3979))

On [[locally presentable category|local presentability]] of [[CocommCoalg|categoris of coalgebras]] and their [[comodules]]:

* [[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)]

[[!redirects coalgebras]]

[[!redirects coassociative coalgebra]]
[[!redirects coassociative coalgebras]]

[[!redirects cocommutative coalgebra]]
[[!redirects cocommutative coalgebras]]

[[!redirects co-algebra]]
[[!redirects co-algebras]]

[[!redirects comultiplication]]
[[!redirects co-multiplication]]


coalgebra for an endofunctor

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

# Contents
* table of contents
{: toc}

## Idea

A _coalgebra over an endofunctor_ is like a [[coalgebra over a comonad]], but without a notion of [[associativity]].

The concept plays a role in [[computer science]] for models of state-based [[computation]] (see also [[monad (in computer science)]]).
The concept of the [[terminal coalgebra of an endofunctor]] is a way of encoding [[coinductive types]].

## Definition

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

For a [[category]] $C$ and [[endofunctor]] $F$, a **[[coalgebra]] of** $F$ is an [[object]] $X$ in $C$ together with a [[morphism]] $\alpha: X \to F(X)$.

Given two coalgebras $(x, \eta: x \to F x)$, $(y, \theta: y \to F y)$, a coalgebra [[homomorphism]] is a [[morphism]] $f: x \to y$ which respects the coalgebra structures:

$$\theta \circ f = F(f) \circ \eta$$

=--

(The object $X$ is sometimes called the __carrier__ of the coalgebra.)

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

The dual concept is an [[algebra for an endofunctor]]. Both [[algebras]] and coalgebras for endofunctors on $C$ are special cases of [[algebra for a C-C bimodule|algebras for C-C bimodules]].

=--

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

If $F$ is equipped with the structure of a [[monad]], then a coalgebra for it is equivalently an [[endomorphism]] in the corresponding [[Kleisli category]]. In this case the canonical [[monoidal category]] structure on endomorphisms induces a [[tensor product]] on those coalgebras.

=--

## Examples

### Coalgebras for endofunctors on $Set$

Each of the following examples is of the form $X\to F(X)$, (description of endofunctor $F\colon Set\to Set$) : (description of coalgebra). Where it appears, $A$ is a given fixed set.

* $X \to D(X)$, the set of [[probability distributions]] on $X$: Markov chain on $X$.
* $X \to \mathcal{P}(X)$, the [[power set]] of $X$: binary relation on $X$, and also the simplest type of [[Kripke frames]].
* $X \to X^A \times bool$, with $X^A$ the set of functions $A\to X$ and $bool = \{0,1\}$: [[deterministic automaton]].
* $X \to \mathcal{P}(X)^A\times bool$: [[nondeterministic automaton]].
* $X \to A \times X \times X$: labelled binary tree with labels from $A$.
* $X \to \mathcal{P}(A\times X)$: [[transition system|labelled transition system]] with labels from $A$.

See [[coalgebra]] for examples on categories of modules.

### The real line as a terminal coalgebra
 {#RealLine}

Let $Pos$ be the category of [[poset]]s. Consider the endofunctor

$$
  F_1 : Pos \to Pos
$$

that acts by [[ordinal product]] with $\omega$

$$
  F_1 : X \mapsto X \cdot \omega
  \,,
$$

where the right side is given the dictionary order, not the usual product order.

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

The terminal coalgebra of $F_1$ is order isomorphic to the non-negative [[real line]] $\mathbb{R}^+$, with its standard order.
=--

+-- {: .proof}
###### Proof
This is theorem 5.1 in

* D. Pavlovic, [[Vaughan Pratt]], _On coalgebra of real numbers_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204))
=--

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

The real interval $[0, 1]$ may be characterized, as a [[topological space]], as the terminal coalgebra for the functor on two-[[pointed object|pointed]] topological spaces which takes a space $X$ to the space $X \vee X$. Here, $X \vee Y$, for $(X, x_-, x_+)$ and $(Y, y_-, y_+)$, is the disjoint union of $X$ and $Y$ with $x_+$ and $y_-$ identified, and $x_-$ and $y_+$ as the two base points.
=--

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

This is discussed in

* [[Peter Freyd]], [cat list](http://www.mta.ca/~cat-dist/catlist/1999/realcoalg)

More information may be found at [[coalgebra of the real interval]].
=--

## Related concepts

* [[list of notable initial algebras and terminal coalgebras]]

* [[well-founded coalgebra]]

* [[algebra over a monad]], [[algebra over an endofunctor]], [[algebra over a profunctor]]

## References

* [[Michael Barr]], *Terminal coalgebras for endofunctors on sets*,  Theoretical Comp. Sci. **114** (1993) 299–315 [[pdf](https://www.math.mcgill.ca/barr/papers/trmclg.pdf), [[Barr-TerminalCoalgebras.pdf:file]]]

* {#Pattinson03} [[Dirk Pattinson]], *An Introduction to the Theory of Coalgebras* (2003) [[pdf](https://nasslli.sitehost.iu.edu/2003/datas/DirkPattinson.pdf), [[Pattinson-Coalgebras.pdf:file]]]

* [[Jiri Adamek]], *Introduction to coalgebras*, Theory and Applications of Categories **14** 8 (2005) 157-199 [[tac:14-08](http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/14/8/14-08.pdf)]

There are connections between the theory of coalgebras and [[modal logic]]  for which see

* [[Bart Jacobs]], _Introduction to Coalgebra. Towards Mathematics of States and Observations_ ([book pdf](http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro.pdf), [slides](http://cs.ioc.ee/ewscs/2011/jacobs/jacobs-slides.pdf))

and also

* Corina Cırstea, Alexander Kurz, [[Dirk Pattinson]], Lutz Schroder and Yde Venema, _Modal Logics are Coalgebraic_, from the Computer Journal 2011, [here](http://users.cecs.anu.edu.au/~dpattinson/Publications/cj2011.pdf).

and with [[quantum mechanics]], for which see this and

* [[Samson Abramsky]], _Coalgebras, Chu Spaces, and Representations of Physical Systems_ ([arXiv:0910.3959](http://arxiv.org/abs/0910.3959))

Here are two blog discussions of coalgebra theory:

* [[David Corfield]], _[_Coalgebraically Thinking_](http://golem.ph.utexas.edu/category/2008/11/coalgebraically_thinking.html)_

* [[David Corfield]], _[_The Status of Coalgebra_](http://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html)_

[[!redirects coalgebra of an endofunctor]]
[[!redirects coalgebras of an endofunctor]]
[[!redirects coalgebras of endofunctosr]]
[[!redirects coalgebra for an endofunctor]]
[[!redirects coalgebras for an endofunctor]]
[[!redirects coalgebras for endofunctors]]
[[!redirects coalgebra over an endofunctor]]
[[!redirects coalgebras over an endofunctor]]
[[!redirects coalgebras over endofunctors]]


coalgebra of the real interval

# Contents
* table of contents
{: toc}

## Introduction

This article is about structure on a closed interval of real numbers, generally taken to be $I = [0, 1]$, that is derivable from a [[coalgebra of an endofunctor|coalgebraic]] perspective. This topic was introduced by [[Peter Freyd]] in a [posting](#Freydcomment) about the [paper](#PavlovicPratt) by [[Dusko Pavlovic]] and [[Vaughan Pratt]].

## Coalgebraic description of $I$

For the moment we work [[classical mathematics|classically]], over the [[category of sets]]. A _bipointed set_ is a set equipped with two elements, that is a [[cospan]] of the form

$$\array{
1 & & & & 1 \\
 & ^\mathllap{x_0} \searrow & & \swarrow^\mathrlap{x_1} & \\
 & & X & &
}$$

where $x_0$ and $x_1$ might coincide. There is a [[monoidal category|monoidal product]] on $Cospan(1, 1)$ given by cospan composition (formed by taking [[pushouts]]); this monoidal product is denoted $\vee$. (Explicitly, $(X,x_0,x_1) \vee (Y,y_0,y_1)$ is $((X \uplus Y)/(x_1 \sim y_0), x_0, y_1)$.)  The monoidal unit is a $1$-element set with its unique bipointed structure. The category of such cospans or bipointed sets is denoted $Cos$.

Inside $Cos$ is the full subcategory of _two-pointed_ sets, where $x_0$ and $x_1$ are distinct. Let $Twop$ be the category of two-pointed sets. The monoidal product $\vee$ restricts to a functor

$$\vee \colon Twop \times Twop \to Twop$$

and one can define the square

$$sq = (Twop \stackrel{\Delta}{\to} Twop \times Twop \stackrel{\vee}{\to} Twop)$$

A **$sq$-coalgebra** is a two-pointed set $X$ together with a map $\xi: X \to X \vee X$. An example is given by $I = [0, 1]$, where $I \vee I$ is identified with the interval $[0, 2]$ and the coalgebra structure $I \to I \vee I$ is identified with multiplication by $2$, $[0, 1] \to [0, 2]$. This map will be denoted $\alpha$.

+-- {: .un_theorem}
###### Theorem (Freyd)
$(I, \alpha)$ is [[terminal coalgebra for an endofunctor|terminal]] in the category of $sq$-coalgebras.
=--

## Corecursively defined operations on $I$

We now define a number of operations on $I$. For $0 \leq x \leq 1$, define ${x \uparrow} \coloneqq \min(2 x, 1)$ and ${x \downarrow} \coloneqq \max(2 x - 1, 0)$. These give unary operations on $I$ which can also be defined as maps in $Cos$ using the coalgebra structure $\alpha$:

$$I \stackrel{{(-) \uparrow}}{\to} I = (I \stackrel{\alpha}{\to} I \vee I \stackrel{I \vee !}{\to} I \vee 1 \cong I)$$

$$\,$$

$$I \stackrel{{(-) \downarrow}}{\to} I  = (I \stackrel{\alpha}{\to} I \vee I \stackrel{! \vee I}{\to} 1 \vee I \cong I)$$

We similarly define unary operations ${(-)\uparrow}$, ${(-) \downarrow}$ for any $sq$-coalgebra $(X, \xi)$. For any coalgebra $X$ and $x \in X$, either ${x \downarrow} = x_0$ or ${x \uparrow} = x_1$. Moreover, if $i_0 \colon X \to X \vee X$ and $i_1 \colon X \to X \vee X$ are the evident pushout inclusions, we have $\xi(x) = i_0({x \uparrow})$ if ${x \downarrow} = x_0$, and $\xi(x) = i_1({x \downarrow})$ if ${x \uparrow} = x_1$. This means that coalgebra structures can be recovered from algebraic structures consisting of two constants $x_0, x_1$ and two unary operations $\uparrow$, $\downarrow$, although we must consider a coherent but non-algebraic axiom

$$\vdash {x \uparrow} = x_1 or {x \downarrow} = x_0$$

### Order-theoretic operations

Next, we define meet and join operations on $I$, making it a lattice, by exploiting corecursion. A slick corecursive definition of the order $\leq$ is that $x \leq y$

* if ${x \downarrow} = 0$ and ${x \uparrow} \leq {y \uparrow}$, or

* if ${y \uparrow} = 1$ and ${x \downarrow} \leq {y \downarrow}$.

If one prefers to work with operations, one could define the meet operation $\wedge \colon I \times I \to I$ by putting a suitable coalgebra structure on $I \times I$ and using terminality of the coalgebra $I$ to define $\wedge$ as a coalgebra map. A coalgebra structure

$$\xi \colon I \times I \to (I \times I) \vee (I \times I)$$

which works is

* $\xi(x, y) = i_0({x \uparrow}, {y \uparrow})$ if ${x \downarrow} = 0$ or ${y \downarrow} = 0$;

* $\xi(x, y) = i_1({x \downarrow}, {y \downarrow})$ if ${x \uparrow} = 1 = {y \uparrow}$.

### Midpoint operations

The general midpoint operation is not as easy to construct as one might think, but to start with we do have operations which take the midpoint between a given point and an endpoint. Namely, the left midpoint operation is the unary operation defined by

$$l = (I \stackrel{i_0}{\to} I \vee I \stackrel{\alpha^{-1}}{\to} I)$$

and the right midpoint operation is defined by

$$r = (I \stackrel{i_1}{\to} I \vee I \stackrel{\alpha^{-1}}{\to} I).$$

## Constructive aspects

There are two versions of this construction, classically equivalent, in [[constructive mathematics]], one of which produces the unit interval of [[MacNeille real numbers]], the other of which produces the unit interval of [[located Dedekind real numbers]].  Both start with $Cos$ (which is straightforward) but define $Twop$ differently, and the monoidal product $\vee$ is subtler.

For the MacNeille real numbers, define $Twop_M$ in the most obvious way,
$$ \neg\; x_0 = x_1 .$$
For the Dedekind real numbers, define $Two_D$ by requiring
$$ \forall x,\; \neg\; x = x_0 \;\vee\; \neg\; x = x_1 .$$
(The two statements are equivalent assuming the classical [[de Morgan law]] $\neg(p \wedge q) \Rightarrow \neg{p} \vee \neg{q}$, but the latter is stronger in [[intuitionistic logic]].)

In $Cos$, let $X \vee Y$ be a [[subset]] of $X \times Y$, with $(x,y) \in X \vee Y$ iff
$$ (\neg\; x = x_1 \;\Rightarrow\; y = y_0) \;\wedge\; (\neg\; y = y_0 \;\Rightarrow\; x = x_1) .$$
(A pushout can be constructed using $x = x_1 \;\vee\; y = y_0$ instead; this is stronger in intuitionistic logic but equivalent assuming [[excluded middle]].  If equality happens to be [[stable proposition|stable]] in $X$ and $Y$, then only the classical de Morgan law is needed to make this a pushout.)

As in the classical case, this makes $Twop_M$ and $Twop_D$ into "[[semigroup]]al" categories ([[monoidal categories]] but without the monoidal unit), since the unit object (which exists in $Cos$ as the [[singleton]]) does not exist in $Twop_M$ or in $Twop_D$, but we can still define $sq_M$ and $sq_D$ and consider their coalgebras.  As stated above, $[0,1]_M$ is the final coalgebra of $sq_M$, and $[0,1] = [0,1]_D$ is the final coalgebra of $sq_D$; both are equipped with $0$ and $1$.

Another approach to the Dedekind real interval looks simpler from the perspective of [[constructive analysis]].  Start with $Set_\#$, the category of $T_0$ [[point–point apartness spaces]] and [[strongly extensional functions]] between them.  (A point--point apartness space is a set equipped with an [[apartness relation]], that is a binary relation $\#$ satisfying properties [[de Morgan duality|dual]] to those of an equivalence relation; it is $T_0$ iff the apartness relation is [[tight relation|tight]]; and a function between such spaces is strongly extensional iff it reflects $#$.)  On $Cos_\#$, let $X \vee Y$ be the subset of $X \times Y$ satisfying
$$ (x \# x_1 \;\Rightarrow\; y = y_0) \;\wedge\; (y \# y_0 \;\Rightarrow\; x = x_1) .$$
Define $Twop_\#$ by
$$ x \# x_0 \;\vee\; x \# x_1 .$$
Now defining $sq_\#$ on $Twop_\#$, its final coalgebra is again the Dedekind unit interval (equipped with its endpoints).

## Computational aspects

... explain how the classical version corresponds to computation with (unsigned) binary digits ... the constructive (Dedekind) version corresponds to computation with signed binary digits ...

## References

* {#PavlovicPratt} [[Dusko Pavlovic]], [[Vaughan Pratt]], _On coalgebra of real numbers_, Electronic Notes in Theoretical Computer Science **19** (1999) 103-117, doi:[10.1016/S1571-0661(05)80272-5](https://doi.org/10.1016/S1571-0661%2805%2980272-5), [CiteSeerx](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204)

* {#Freydcomment} [[Peter Freyd]], _Real coalgebra_ Mailing to the categories list, Dec. 22, 1f999. ([link](http://www.mta.ca/~cat-dist/catlist/1999/realcoalg))

The constructive version is also due to Freyd:

* [[Peter Freyd]], _Reality Check_, post to the categories list, July 31, 2000 ([web](http://facultypages.ecc.edu/alsani/ct99-00%288-12%29/msg00200.html))

Freyd published 8 years later:

* [[Peter Freyd]], _Algebraic Real Analysis_, TAC **20** no.10 (2008) pp. 215--306. ([web](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

There is also a complete development in the [[Elephant]]:

* [[Peter Johnstone]], _Sketches of an Elephant_, following 4.7.14.

[[!redirects real coalgebra]]
[[!redirects coalgebra of the real interval]]
[[!redirects coalgebra of the real numbers]]
[[!redirects coalgebra of real numbers]]

[[!redirects Freyd interval]]


coalgebra over a comonad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _coalgebra_ or _comodule_ over a [[comonad]] $C$ on a [[category]] $A$ is an [[object]] $a\in A$ with a [[morphisms]] $a\to C a$ satisfying [[axioms]] [[formal dual|formally dual]] to those for an [[algebra over a monad]].  The category of coalgebras is called its (co-)[[Eilenberg-Moore category]] and satisfies a universal property dual to that of the [[Eilenberg-Moore object]] for a monad; it can thereby be internalized to any [[2-category]].  The [[forgetful functor]] from the category of coalgebras to the category $A$ is called a [[comonadic functor]].  Similarly, a comonad also has a [[co-Kleisli category]].

## Examples
 {#Examples}

* the modality ! (of course) in [[linear logic]] is modeled by a comonad and its coalgebras, which are also comonoids, allow us to model intuitionistic formulas in categorical logic

* [[partial differential equations]] are the coalgebras of a [[jet comonad]] (see [there](jet+comonad#References))

* well-behaved [[lenses (in computer science)]] are the coalgebras of the [[costate comonad]] (see [there](lens+in+computer+science#LensesAreCostateCoalgebras))

## Related concepts

* [[topos of coalgebras over a comonad]]

* [[coalgebra over an endofunctor]]

* [[cocommutative coalgebra]]

## References

Some introductory material on [[comonads]], [[coalgebras]] and [[co-Kleisli morphisms]] can be found in

* [[Valeria de Paiva]], _The Dialectica Categories_, Chapter 2. ([Ph Thesis](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf))

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 5. ([arXiv](http://arxiv.org/abs/1912.10642))

On [[model category]]-[[mathematical structure|structures]] on coalgebras over a comonad:

* [[Kathryn Hess]], [[Brooke Shipley]], *The homotopy theory of coalgebras over a comonad*, Proceedings of theLondon Mathematical Society **108** 2 (2014) [[arXiv:1205.3979](https://arxiv.org/abs/1205.3979), [doi:10.1112/plms/pdt038](https://doi.org/10.1112/plms/pdt038)]

[[!redirects coalgebras over a comonad]]
[[!redirects coalgebras over comonads]]

[[!redirects coalgebra of a comonad]]
[[!redirects coalgebras of a comonad]]

[[!redirects comodule over a comonad]]
[[!redirects comodules over a comonad]]
[[!redirects comodules over comonads]]

[[!redirects comodule for a comonad]]
[[!redirects comodules for a comonad]]
[[!redirects comodules for comonads]]


coalgebraic logic

* Adriana Balan, Alexander Kurz, Jirı Velebil, _Positive fragments of coalgebraic logics_,  [pdf](http://dep2.mathem.pub.ro/balan/papers/calco2013.pdf)

* [[davidcorfield:coalgebra]] and links therein
* [[modal logic]]


coalgebraic model for modal logic

##Idea

[[modal logic|Modal logics]] are said to be the logic of [[relational structures]].  The Kripke frame semantics of modal logics provides a clear picture of this:

Recall: A _frame_ for the basic modal language $\mathcal{L}_\omega(1)$ is a pair $\mathfrak{F} = (W,R)$ with $W$ a non-empty set and $R$ a binary relation on $W$.

At their 'crudest' the objects studied are just sets and binary relations on them.  In the basic 'many worlds' interpretation, a world $w$ satisfies a modal formula $\Box\phi$ if all worlds $w\prime$ accessible from $w$ (that is satisfying $wRw\prime$) satisfy $\phi$.

Now convert $R$ into a mapping $\rho : W\to \mathcal{P}(W)$, where $\rho(w)$ is the set of worlds accessible from $w$.  It is easy to rephrase the notion of 'satisfies' in terms of $\rho$, but $(W,\rho)$ is a [[coalgebra for an endofunctor|coalgebra]] for the power set  endofunctor on $Set$, so Kripke frames provide a simple example of a coalgebraic model for modal formulae. There are many other types of [[coalgebras for  endofunctors]] and many lead to modal logics.

##References

* [[Corina Cirstea]], [[Alexander Kurz]], Dirk Pattinson, Lutz Schröder and [[Yde Venema]], _Modal logics are coalgebraic_ ([pdf](http://eprints.soton.ac.uk/267144/1/ModalCoalgRev.pdf))


coarse structure

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#Contents#
* table of contents
{:toc}

## Definition

In [[coarse geometry]], a _coarse structure_ on a [[set]] $X$ is a [[subset]] $C\subset P(X\times X)$ (of the [[power set]] of the [[Cartesian product]] of $X$ with itself)  that contains the [[diagonal]] of $X$ and is closed under finite [[unions]], subsets, relational compositions, and relational inverses. (Here composition is given by the [[fiber product]] over $X$, whereas inverses are obtained by permuting the two factors of $X$.)

The elements of $C$ are called _entourages_ or _controlled subsets_.

## Related concepts

* [[bornological coarse space]]

[[!redirects coarse structures]]


coarse topology

>  For comparison of subsets of open subsets, see instead at _[[finer topology]]_.

# Coarse topology
* table of contents
{: toc}

## Idea

__Coarse topology__ (synonym: __coarse geometry__) is a branch of geometry/topology studying the asymptotic properties at large of metric spaces. While the topological
structures describe the space's local properties, the coarse structure describes the properties at large distances. One usually studies coarse structure by defining a "coarse category" of proper metric spaces where the morphisms are so called coarse maps.

## Coarse maps

....

## Terminology collision

Do not confuse the field of coarse topology with a comparative property of one topological structure on a set to be coarser than another which is then called finer. Finer or stronger topology has "more" open sets than coarser or weaker -- this is of course in the sense of the partial order with respect to inclusion of the topological structures considered as families of subsets.

## History

The field grew in part from the study of "wild" metric structures in study of Gromov hyperbolic groups, from attacks on [[Novikov conjecture]] and the
related [[Baum-Connes conjecture]]. Thus the operator algebras play a great role in the field.

## Literature

Related $n$Lab entries: [[asymptotic dimension]]

* _A short survey on coarse topology_, [pdf](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1126-9.pdf)

* J. Roe, _Index theory, coarse geometry, and topology of manifolds_, CBMS Regional Conference Series in Mathematics 90, Amer. Math. Soc. 1996.

* B. Hanke, D. Kotschick, J. Roe, T. Schick, _Coarse topology, enlargeability, and essentialness_, [arxiv:0707.1999](http://arxiv.org/abs/0707.1999)

* [[Mikhail Gromov]], _Hyperbolic groups_, Essays in group theory, Math. Sci. Res. Inst. Publ. __8__, Springer, New York, 1987, pp. 75–263. MR MR919829 (89e:20070) 61; _Asymptotic invariants of infinite groups_, in: Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. __182__, Cambridge Univ. Press 1993, pp. 1–295. MR1253544 (95m:20041)

* B. Grave, _Asymptotic dimension of coarse spaces_, New York J. Math. 12 (2006) 249–256

category: topology
[[!redirects coarse topology]]
[[!redirects coarse geometry]]


cobase change

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#### Limits and colimits
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=--
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#Contents#
* automatic table of contents goes here
{:toc}

## Definition

**Cobase change** is another term for [[pushout]] of a morphism:

if
$$
  \array{
    X &\to & U
    \\
    \downarrow && \downarrow
    \\
    Y &\to& Y \sqcup_X U
  }
$$
is a [[pushout]] square in some [[category]] then the [[morphism]] $U \to U \sqcup_X Y$ is the **cobase change** or **cobase extension** of $X \to Y$ along $X \to U$.

## Related concepts

* The [[duality|dual]] concept is [[base change]].

* [[base change]]

* [[homotopy co-base change]]

[[!redirects cobase changes]]


cobordism

+-- {: .rightHandSide}
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###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An (oriented) _cobordism_ $\Sigma$ from an (oriented) [[smooth manifold]] $X_{in}$ to an (oriented) smooth manifold $X_{out}$ is a smooth [[manifold with boundary]] such that its [[boundary]] is the [[disjoint union]]

$$
  \partial \Sigma \simeq X_{in} \coprod X_{out}
$$

with induced [[orientation]] agreeing with the given one on $X_{in}$ and being the opposite of that of $X_{out}$ ([Thom 54, Chapter IV, p. 64](#Thom54)). Hence by labelling disjoint components of the boundary of any [[manifold with boundary]] $\Sigma$ as either "incoming" or "outgoing", then $\Sigma$ becomes a cobordism from its incoming to its outgoing boundary components.

(While $X_{in} \coprod X_{out}$ is the [[boundary]] of $\Sigma$, conversely $\Sigma$ is the "co-boundary" of $X_{in}\coprod X_{out}$. This is part of the reason for the "co-" in "cobordism", but sometimes one just says _bordism_. The difference is more pronounced when distinguishing between [[bordism homology theory]] and [[cobordism cohomology theory]].)

With a few technical conditions on the boundary inclusions added, then $(n-1)$-dimensional manifolds with [[diffeomorphism]] classes of $n$-dimensional cobordisms between them form a [[category]] (a [[category of cobordisms]]) whose [[objects]] are the manifolds, whose [[morphisms]] are the cobordisms, and whose [[composition]] operation is the operation of gluing two cobordisms along a common boundary component.
Such a [[category of cobordisms]] $Bord_n$ of some [[dimension]] $n$  is naturally a  [[symmetric monoidal category]] $Bord_n^{\sqcup}$ with the [[tensor product]] being the [[disjoint union]] $\sqcup$ of manifolds.

The [[connected components]] in this category are called _cobordism classes_ of manifolds. Under [[cartesian product]] and [[disjoint union]], these form what is called the [[cobordism ring]] in the given dimension. The study of these classes, hence of manifolds "up to cobordisms", is a central topic in [[algebraic topology]].

A central insight connecting [[algebraic topology]] with [[mathematical physics]] is that a [[strong monoidal functor]] on a [[category of cobordisms]] with values in something like the category [[Vect]] (with its standard [[tensor product of modules]]) may be thought of as a formalized incarnation of what in [[physics]] is called a [[topological quantum field theory]]

$$
  Z \colon Bord_n^\sqcup \longrightarrow Vect^\otimes
  \,.
$$

Here one thinks of a cobordism $\Sigma$ as a piece of [[spacetime]] (or [[worldvolume]]) of dimension $n$, and of the $(n-1)$-dimensional manifolds that this goes between as a piece of [[space]] (or [[brane]]). A functor $Z$ as above is then thought of as sending each $(n-1)$-dimensional space $X$ to its [[space of quantum states]] $Z(X)$ and each spacetime $\Sigma$ between $X_{in}$ and $X_{out}$ to a [[linear map]] $Z(\Sigma)\colon Z(X_{in}) \longrightarrow Z(X_{out})$.

From the perspective of [[mathematics]], such a functor is a way to break up [[diffeomorphism]] [[invariants]] of [[closed manifolds]] of [[dimension]] $n$ into pieces and being able to reconstruct them from gluing of data associated to manifolds with $(n-1)$-dimensional boundary.

If one considers [[manifolds with corners]], then there is a fairly evident extension of the concept of cobordism that allows refinement of this gluing process to [[extended cobordisms]] of any dimension $k$ going between [[extended cobordisms]] of dimension $k-1$. Such extended cobordisms of maximal dimension $n$ form a [[symmetric monoidal (infinity,n)-category]] called, naturally, the [[(infinity,n)-category of cobordisms]]. The [[cobordism hypothesis]] asserts that this is a most fundamental object in [[higher category theory]] and [[higher algebra]], namely that it is the _[[free construction|free]]_ [[symmetric monoidal (infinity,n)-category]] [[(infinity,n)-category with duals|with duals]]. The corresponding extended concept of [[topological quantum field theory]] is accordingly called [[extended TQFT]] or similar.

## Properties

### Relation to Cohomotopy
 {#RelationToCohomotopy}

The [[Pontrjagin-Thom isomorphism]]  says that assigning [[Cohomotopy charge]] identifies suitable [[cobordism classes]]  of [[submanifolds]] or abstract [[manifolds]] with [[cocycles]] in [[Cohomotopy|unstable Cohomotopy]] (see [here](cohomotopy#RelationToCobordismGroup)) or [[stable Cohomotopy]].

The following graphics illustrates the [[Cohomotopy charge map]] of the pair creation/annihilation cobordism for 0-dimensional [[submanifolds]]:

> graphics grabbed from [SS 19](cohomotopy+charge#SatiSchreiber19)

(See also at _[[Cohomotopy charge]] -- [For charged points](Cohomotopy+charge#ForChargedPoints)_.)

## Related concepts

* [[handlebody]]

* [[h-cobordism]]

* [[movie]]

* [[cobordism theory]]

* [[cobordism group]], [[cobordism ring]]

* [[Thom spectrum]], [[Thom's theorem]]

* [[cobordism category]]

* [[(infinity,n)-category of cobordisms]], [[cobordism hypothesis]]

* [[cobordism cohomology theory]]

* [[extended cobordism]]

* [[B-bordism]]

* [[orbifold cobordism]]

* [[Lagrangian cobordism]]

* [[algebraic cobordism]]

* [[Sullivan chord diagram]]

* [[braid cobordism]]

## References

The notion originates with

* {#Thom54} [[René Thom]], p. 64 of: _[[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), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

* {#Pontrjagin55} [[Lev Pontrjagin]], Section III.6, p. 41 of: _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))

(Beware that Pontrjagin still wrote "homology" for "cobordism", following an earlier tradition that got abandoned.)

An early textbook account:

* {#Stong68} [[Robert Stong]], _Notes on Cobordism theory_, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [publisher page](http://press.princeton.edu/titles/6465.html))

Further review:

* [[Peter Landweber]], _A survey of bordism and cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society **100** 2 (1986) 207-223 [[doi:10.1017/S0305004100066032](https://doi.org/10.1017/S0305004100066032)]

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, Section 1.1 of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

* {#Kochmann96} [[Stanley Kochmann]], section 1.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[John Francis]] (notes by [[Owen Gwilliam]]), _[Topology of manifolds](http://math.northwestern.edu/~jnkf/classes/mflds/)_, _Lecture 2: Cobordism_ ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf))

* [[Manifold Atlas]], _[Bordism](http://www.map.mpim-bonn.mpg.de/Bordism)_

* Wikipedia, _[Cobordism](http://en.wikipedia.org/wiki/Cobordism)_

For more see at _[[cobordism cohomology theory]]_.

A relation to [[fixed point spaces]]:

* {#Prieto03} Carlos Prieto, _Fixed point theory and framed cobordism_, Topol. Methods Nonlinear Anal. Volume 21, Number 1 (2003), 155-169. ([euclid:tmna/1475266278](https://projecteuclid.org/euclid.tmna/1475266278))

[[!redirects cobordisms]]

[[!redirects bordism]]
[[!redirects bordisms]]

[[!redirects cobordant manifold]]
[[!redirects cobordant manifolds]]

[[!redirects bordant manifold]]
[[!redirects bordant manifolds]]

[[!redirects cobordism class]]
[[!redirects cobordism classes]]
[[!redirects bordism class]]
[[!redirects bordism classes]]

[[!redirects bordism theory]]


cobordism category

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#### Manifolds and Cobordisms
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[[!include manifolds and cobordisms - contents]]
=--
#### Functorial Quantum Field Theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of cobordism category is an abstract one intended to capture important features of (many variants of) the [[category]] of [[cobordism]]s and include in the same formalism cobordisms for closed manifolds with various kinds of structure.

The passage from a [[manifold]] $M$ to its [[boundary]] $\partial M$ has some formal properties which are preserved in the presence of [[orientation]], for manifolds with additional structure and so on. The category of [[compact space|compact]] smooth manifolds with boundary $D = Diff_c$ has finite [[coproducts]] and the boundary operator $\partial:D\to D$, $M\mapsto \partial M$ is an endofunctor commuting with coproducts. (Often these coproducts are referred to as [[direct sums]], and some say that $\partial$ is an [[additive functor]], but $D$ is not actually an [[additive category]]). The inclusions $i_M:\partial M\to M$ form a [[natural transformation]] of functors $i:\partial\to Id$. Finally, the isomorphism classes of objects in $D$ form a set, so $D$ is [[essentially small category|essentially small]] (svelte).

## Definition

### Axiomatization

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

A __cobordism category__ is a triple $(D,\partial,i)$ where

* $D$ is a [[svelte category]] (i.e. an [[essentially small category]])

  * with finite [[coproduct]]s (called [[direct sum]]s, often denoted by $+$),

  * including an [[initial object]] $0$ (also often denoted by $\emptyset$),

* $\partial:D\to D$ is an additive (direct-sum-preserving) [[functor]]

* and $i:\partial\to Id_D$ is a [[natural transformation]] such that $\partial\partial M = 0$ for all [[object]]s $M\in D$.

=--

Note that $i$ is *not* required to be a *[[subfunctor]]* of the identity, i.e. the components $i_M$ are not required to be [[monomorphism|monic]], which is however often the case in examples.

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

Two objects $M$ and $N$ in a cobordism category $(D,\partial,i)$ are said to be __cobordant__, written $M\sim_{cob} N$, if there are objects $U,V\in D$ such that $M+\partial U \cong N+\partial V$ where $\cong$ denotes the relation of being [[isomorphism|isomorphic]] in $D$.

=--

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

In particular, isomorphic objects are cobordant. Being cobordant is an [[equivalence relation]] and for any object $M$ in $D$, one has $\partial M\sim_{cob} 0$.

=--

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

Objects of the form $\partial M$ where $M$ is an object in $D$ are said to be *boundaries* and the objects $V$ such that $\partial V = 0$ are said to be *closed*.

=--

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

In particular, every boundary is closed. A direct sum of closed objects (resp. boundaries) is a closed object (resp. a boundary). If an object $M$ is a boundary and $M\cong N$ then $N$ is also a boundary.

=--

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

By the above, the relation of being cobordant is compatible with the direct sum, in the sense that the direct sum induces an associative commutative operation on the set of equivalence classes, which hence becomes a commutative [[monoid]] called the __cobordism semigroup__

$$
  \Omega(D,\partial,i)
  \,,
$$

of the cobordism category $(D,\partial,i)$.

=--

## Properties

### The homotopy type of the cobordism category
 {#GMTWTheorem}

#### Topological case

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

There is a [[weak homotopy equivalence]]

$$
  \Omega |Cob_d|
  \simeq
  \Omega^\infty(MTSO(d))
$$

between the [[loop space]] of the [[geometric realization]] of the $d$-cobordism category and the [[Thom spectrum]]-kind spectrum

$$
  \Omega^\infty MTSO(d)
   :=
  {\lim_\to}_{n \to \infty}  \Omega^{n+d} Th(U_{d,n}^\perp)
$$

where

$$
  U_{d,n}^\perp =
  \{
   ...
  \}
$$

=--

This is ([Galatius-Tillmann-Madsen-Weiss 06, main theorem](#GMWT)).

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

This statement may be thought of as a limiting case, of the [[cobordism hypothesis]]-theorem. See there for more.

=--

#### Geometric case

* ([Ayala](#Ayala))

The [[Thom group]] $\mathcal{N}_*$ of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for $D=Diff_c$.

## Examples
 {#Examples}

Examples of cobordism categories besides those of manifolds with [[(B,f)-structure|$(B,f)$-structure]]:

* "foams" used in [[link homology]] following [Khovanov 2004](Khovanov+homology#Khovanov04)

See also [MO:q/59677](https://mathoverflow.net/q/59677/381).

## Related concepts

* [[cobordism]], [[extended cobordism]]

* **category of cobordisms**

  * [[cobordism ring]]

  * [[Thom spectrum]]

  * [[(∞,n)-category of cobordisms]]

* [[conformal cobordism category]]

* [[cobordism hypothesis]]

* [[cobordism cohomology theory]]

## References

### General

A classical reference is

* [[Robert Stong]], _Notes on cobordism theory_, Princeton University Press 1968 (Russian transl., Mir 1973) ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [publisher page](http://press.princeton.edu/titles/6465.html))

The [GMTW theorem](#GMTWTheorem) about the [[homotopy type]] of the cobordisms category with topological structures on the cobordisms appears in

* {#GMWT} [[Søren Galatius]], [[Ib Madsen]], [[Ulrike Tillmann]], and [[Michael Weiss]], _The homotopy type of the cobordism category_, Acta Math. 202 (2009), no. 2, 195--239 ([arXiv:math/0605249](http://arxiv.org/abs/math/0605249))

A generalization to geometric structure on the cobordisms is discussed in

* {#Ayala} [[David Ayala]], _Geometric cobordism categories_ thesis (2009) ([arXiv:0811.2280](http://arxiv.org/abs/0811.2280))

### Embedded cobordism category

* [[Oscar Randal-Williams]], _Embedded Cobordism Categories and Spaces of Manifolds_, 	Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 ([arXiv:0912.2505](https://arxiv.org/abs/0912.2505))

On the [[homotopy groups]] of the [[embedded cobordism category]]:

* [[Marcel Bökstedt]], [[Anne Marie Svane]], _A geometric interpretation of the homotopy groups of the cobordism category_, Algebr. Geom. Topol. 14 (2014) 1649-1676 ([arXiv:1208.3370](https://arxiv.org/abs/1208.3370))

* [[Marcel Bökstedt]], [[Johan Dupont]], [[Anne Marie Svane]], _Cobordism obstructions to independent vector fields_, Q. J. Math. 66 (2015), no. 1, 13-61 ([arXiv:1208.3542](https://arxiv.org/abs/1208.3542))

[[!redirects cobordism categories]]

[[!redirects category of cobordisms]]
[[!redirects categories of cobordisms]]

[[!redirects Galatius-Madsen-Tillmann-Weiss theorem]]
[[!redirects Galatius-Tillmann-Madsen-Weiss theorem]]
[[!redirects GMWT theorem]]

[[!redirects embedded cobordism category]]
[[!redirects embedded cobordism categories]]


cobordism cohomology theory

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[algebraic topology]], a [[Whitehead-generalized cohomology theory]] [[Brown representability theorem|represented]] by a [[Thom spectrum]] is called a _cobordism cohomology theory_ ([Atiyah 61](#Atiyah61)), in [[duality]] with the corresponding [[generalized homology theory]] called _[[bordism homology theory]]_.

In both cases, a version of the [[Pontryagin-Thom construction]] identifies the (co)homology classes of these (co)homology theories with [[bordism]]-[[equivalence classes]] of [[manifolds]] (carrying some given [[extra structure]]), whence the name. For [[bordism homology theory]] this was understood since the very inception of the subject ([Thom 54](#Thom54)), while for cobordism cohomology theory this identification is made explicit in
[Atiyah 61, Sec. 3](#Atiyah61), [Quillen 71](#Quillen71) (relying on results from [Thom 54](#Thom54) nonetheless), see below at _[Geometric model via cobordism classes](#GeometricModel)_.

Accordingly, cobordism cohomology theories are fundamental concepts of [[bordism theory]] in [[differential topology]]. But in  addition they turn out to play a special role in the more abstract [[stable homotopy theory]] of [[complex oriented cohomology theories]] (with its variants such as [[quaternionic oriented cohomology|quaternionic-oriented theories]]) and in the resulting [[chromatic homotopy theory]], see for instance the [[universal complex orientation on MU]]. This way, cobordism cohomology embodies a remarkable confluence of the [[differential topology]] of [[smooth manifolds]] with deep issues in abstract [[homotopy theory]].

There are many different flavours of cobordism cohomology theories (see the list of [Examples](#Examples) below), depending on the [[tangential structure]] $f$ encoded in the representing [[Thom spectrum]] $M f$. Among the most commonly considered versions are these:

* The [[cohomology theory]] [[Brown representability theorem|represented]] by [[MO]] is the base case, in the sense that the [[orthogonal group]] $O(n)$ is [[homotopy equivalence|homotopy equivalent]] to the [[general linear group]] $GL_{\mathbb{R}}(n)$ (namely the [[maximal compact subgroup]]), so that $O$-[[tangential structure]] corresponds to no extra [[tangential structure]].

* The cohomology theory represented by [[MU]] is _[[complex cobordism cohomology]]_. Its [[periodic cohomology theory]] version is sometimes denoted [[MP]]. The cohomology theories [[MO]] and [[MU]] are unified by the [[equivariant cohomology theory]] called [[MR-theory]].

* On the other hand, [[framed manifold|framed]] cobordism cohomology theory [[MFr]] is equivalently _[[stable Cohomotopy theory]]_ (by the [[Pontryagin-Thom theorem]]). In its unstable version as plain [[Cohomotopy theory]] this -- and its relation to cobordism classes of [[normally framed submanifolds]] (the [Pontryagin construction](cohomotopy#RelationToCobordismGroup)) -- this was the historical origin of [[cobordism theory]] (in [Pontrjagin 38](#Pontrjagin38), [Pontrjagin 55](#Pontrjagin55)).

## Geometric model via cobordism classes
 {#GeometricModel}

We discuss a geometric model for the cobordism cohomology theory, due to [Quillen 71, Section 1](#Quillen71). We concentrate on the complex case, corresponding to the [[Thom spectrum]] [[MU]]:

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

For a [[smooth manifold]] $X$, the cobordism [[cohomology group]]
$\mathrm{M} \mathrm{U}^q(X) \;\coloneqq\; [\Sigma^\infty X_+, \Sigma^q MU]$
is equivalently the set of [cobordism classes](#CobordismClassesOfMaps) of [[proper maps|proper]] [complex-oriented](#ComplexOrientedMaps) maps $f \colon Z \to X$ of [[codimension]] $q$.

=--

([Quillen 71, Prop. 1.2](#Quillen71))

This uses the following definitions:

+-- {: .num_defn #ComplexOrientedMaps}
###### Definition
**(complex-oriented maps)**

Let $f \colon Z \to X$ be a [[smooth map]].

If the relative [[codimension]] of $f$ is even at all points of $Z$, then a _complex orientation_ is an [[equivalence class]] of factorizations of $f$
in the form

$$
  p \circ i
    \;\colon\;
  Z
    \longrightarrow
  E
    \longrightarrow
  X
  \,,
$$

where $p\colon E\to X$ is a [[complex vector bundle]] and $i \colon Z\to E$ is an [[embedding of topological spaces|embedding]] equipped with a [[complex structure]] on its [[normal bundle]].

Two such factorizations $(i,p)$ and $(i',p')$ are regarded as equivalent if there is another factorization $(i'',p'')$ together with embeddings of [[complex vector bundles]] $E\to E'$ and $E\to E''$ and a [[homotopy]]  $i''\colon X\times[0,1]\to E''\times [0,1]$ over $[0,1]$ equipped with a [[complex structure]] on its [[normal bundle]] that restricts to the corresponding complex structures on $X \times \{0\}$ and $X \times \{1\}$.

=--

+-- {: .num_defn #CobordismClassesOfMaps}
###### Definition
**(cobordism classes of maps)**

Here two [[proper map|proper]] [complex-oriented](#ComplexOrientedMaps) maps $f_i \colon Z_i \to X$ are called _cobordant_ if there is a proper complex-oriented map $b\colon W\to X\times\mathbf{R}$ such that $X\times\{0\}$ and $X\times\{1\}$ are [[transversal map|transversal]] to $b$ and pulling back $b$ to these submanifolds yields $f_0$ and $f_1$.

=--

([Quillen 71, p. 31](#Quillen71))

## Examples
 {#Examples}

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

## Related concepts

* the refinement of cobordism cohomology theory to [[differential cohomology]] is [[differential cobordism cohomology]].

* [[cobordism theory determining homology theory]]

* [[equivariant complex cobordism cohomology theory]]

* [[Adams-Novikov spectral sequence]]

\linebreak

[[!include chromatic tower examples - table]]

## References

### General

Original articles introducing cobordism as a [[Whitehead-generalized cohomology theory]]:

* {#Atiyah61} [[Michael Atiyah]], _Bordism and Cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 ([doi:10.1017/S0305004100035064](https://doi.org/10.1017/S0305004100035064), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahb.pdf))
  > (introducing the concept, with focus on[[MSO]])

* {#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))

  > (relating [[complex cobordism theory]] to [[topological K-theory]] and the [[e-invariant]])

* {#Quillen71} [[Daniel Quillen]], _Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations_, Advances in Mathematics 7 (1971), 29–56 (doi:10.1016/0001-8708(71)90041-7)

Early survey:

* [[Victor Buchstaber]], _Cobordisms in problems of algebraic topology_,  J Math Sci 7, 629–653 (1977) ([doi:10.1007/BF01084983](https://doi.org/10.1007/BF01084983))

* [[Peter Landweber]], _A survey of bordism and cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 2 September 1986, pp. 207-223 ([doi:10.1017/S0305004100066032](https://doi.org/10.1017/S0305004100066032))

Textbook accounts:

* {#Stong68} [[Robert Stong]], _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

* {#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))

The [[twisted cohomology|twisted]] and [[equivariant cohomology|equivariant]] versions:

* {#Cruickshank99} [[James Cruickshank]], _Twisted  Cobordism and its Relationship to Equivariant Homotopy Theory_, 1999 ([pdf](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf), [[Cruickshank99.pdf:file]])

  > ([[equivariant Pontrjagin theorem]] for [[free actions|free]] [[G-manifolds]])

* {#Cruickshank03} [[James Cruickshank]], _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (arXiv:10.1016/S0166-8641(02)00183-9)

[[!include Pontryagin-Thom construction -- references]]

### Relation to divisors

Relation of [[complex cobordism cohomology]] with [[divisors]], [[algebraic cycles]] and [[Chow groups]]:

* [[Burt Totaro]], _Torsion algebraic cycles and complex cobordism_, J. Amer. Math. Soc. 10 (1997), 467-493 ([doi:10.1090/S0894-0347-97-00232-4](https://doi.org/10.1090/S0894-0347-97-00232-4))

* [MO discussion](https://mathoverflow.net/a/272131/381)

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