cohomology with constant coefficients

[[!redirects constant coefficients]]

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

The [[cohomology]] of an [[object]] $X$ in an [[(∞,1)-topos]] $\mathbf{H}$ with coefficients in another object $A$ is the set of connected components of the [[hom-space]] from $X$ to $A$.

Notice that for every [[(∞,1)-sheaf (∞,1)-topos]] there is the terminal [[global section]] [[(∞,1)-geometric morphism]]

$$
 (LConst \dashv \Gamma) : \mathbf{H}
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
 \infty Grpd
  \,.
$$

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

In the case that $A = LConst \mathcal{A}$ for $\mathcal{A} \in $ [[∞Grpd]]  we say that

$$
  H(X,\mathcal{A}) := \pi_0 \mathbf{H}(X, LConst \mathcal{A})
  \,.
$$

the **cohomology of $X$ with constant coefficients**, constant on $\mathcal{A}$

=--

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

For $\mathbf{H}$ the [[(∞,1)-sheaf (∞,1)-topos]] over an [[(∞,1)-site]] $C$, we have that $LConst \mathcal{A}$ is the [[constant ∞-stack]] over $C$. Notice that this is the [[∞-stackification]] of the [[(∞,1)-presheaf]] that is literally constant (as an [[(∞,1)-functor]]) on $\mathcal{A}$. So unless over $C$ constant presheaves already satisfy [[descent]] (as for instance over an [[(∞,1)-cohesive site]]) the object $LConst \mathcal{A}$ is not itself given by a constant functor on $C^{op}$.

=--

## Properties

+-- {: .un_lemma}
###### Observation

If $\mathbf{H}$ is a [[locally ∞-connected (∞,1)-topos]] in that we have a further [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$ to $LConst$

$$
  (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd
$$

then by the [[adjunction]] hom-equivalence $\mathbf{H}(X, LConst \mathcal{A}) \simeq \infty Grpd(\Pi(X), \mathcal{A})$ we have that cohomology with constant coefficients in $\mathbf{H}$ is equivalently the cohomology of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi(X)$ in [[∞Grpd]] with coefficients in $\mathcal{A}$.

A [[cocycle]]

$$
  (\tilde \nabla : X \to LConst \mathcal{A}) \sim (\nabla : \Pi(X) \to \mathcal{A})
$$

in this cohomology may then be identified with what is called a [[local system]] on $X$ with coefficients in $\mathcal{A}$. So in this case we have

* _Cohomology with constant coefficients classifies local systems._

=--

## Examples

The relation between cohomology with local coefficients cohomology in [[∞Grpd]] $\simeq$ [[Top]] is discussed at [[nonabelian cohomology]] in the section nonabelian sheaf cohomology.


cohomotopy

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
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#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

### General

_Cohomotopy cohomology theory_ $\pi^\bullet$ is the ([[non-abelian cohomology|non-abelian]]) [[generalized cohomology theory]] whose [[cocycle spaces]] are [[spaces of maps]] into an [[n-sphere]], hence whose [[cohomology classes]] are [[homotopy classes]] of [[maps]] into an [[n-sphere]]:

$$
  \pi^n(X) \;\coloneqq\; Maps\big( X, S^n\big)/_{\sim_{homotopy}}
$$

So, dually to how [[homotopy groups]]

$$
  \pi_n(X)\coloneqq Maps\big( S^n, X\big)/_{\sim_{homotopy}}
$$

are [[groups]] of [[homotopy classes]] of maps _out_ of [[spheres]], Cohomotopy sets are sets of homotopy classes of maps _into_ spheres, whence the [[duality|dual]] name.

If instead one considers only the [[stable homotopy theory|stable]] aspect of Cohomotopy sets, by mapping into the [[stabilization]] of the spheres, hence into (some [[suspension spectrum|suspension]] of) the [[sphere spectrum]], then one speaks of _[[stable Cohomotopy]]_, written

$$
  \mathbb{S}^n(X)
    \;\coloneqq\;
  Maps
  \big(
    \Sigma^\infty X , \Sigma^{\infty} S^n
  \big)/_{\sim_{homotopy}}
  \,.
$$

In other words, the [[generalized (Eilenberg-Steenrod) cohomology]] theory which is [[Brown representability theorem|represented]] by the [[sphere spectrum]] is _[[stable Cohomotopy]]_.

+-- {: .num_remark #RemarkOnTerminology}
###### Remark
**(terminology)**

Therefore "Cohomotopy theory" is really shorthand for "Cohomotopy cohomology theory" and as such is [[duality|dual]] to _homotopy homology theory_, which in the stable case is known as _[[stable homotopy homology theory]]_.

In particular, cohomotopy theory is a [[concrete particular]] and not dual to the [[abstract general]] of [[homotopy theory]]; and is hence also not on par with the [[abstract general]] of [[cohomology theory]]. Rather, Cohomotopy theory is one _instance_ of a [[generalized cohomology|cohomology theory]], and as such is a sibling of [[ordinary cohomology theory]] ([[HR]]-theory)), [[K-theory]], etc.

To emphasize this, one might, in the [[stable cohomotopy theory|stable case]], say _$\mathbb{S}$-theory_ instead of "stable Cohomotopy theory"; where $\mathbb{S}$ denotes the [[sphere spectrum]]. In the unstable case there is no widely adopted notation, but one might consider saying "$\mathbf{\pi}$-theory" (with $\pi$ the established symbol for (co)[[homotopy groups]]) or _$S$-theory_ (with "$S$" for [[n-spheres]] $S^n$) for unstable Cohomotopy theory.

In any case, to highlight that Cohomotopy theory is a [[concrete particular]] and not an [[abstract general]], it makes good sense to capitalize the term and speak of _Cohomotopy cohomology theory_ or just _Cohomotopy theory_, for short.

The following table indicates the pattern:

| [[generalized cohomology theory]] |  [[generalized homology theory]] | [[classifying space]] | [[Brown representability theorem|representing]] [[spectrum]] |
|--|--|--|--|
| [[Cohomotopy theory]]
 ([[non-abelian cohomology|non-abelian]]) |  | [[n-sphere]]
 $S^n$ |  |
| [[stable Cohomotopy theory]]
 ([[generalized (Eilenberg-Steenrod) cohomology theory|abelian]]) | [[stable homotopy homology theory]] | [[sphere spectrum]]
 $\mathbb{S}$  | $\,$ | [[sphere spectrum]]
 $\mathbb{S}$ |
| [[ordinary cohomology]]:
 [[HR]]-[[generalized (Eilenberg-Steenrod) cohomology theory|cohomology theory]] |  [[ordinary homology]]: [[HR]]-[[generalized homology theory|homology theory]] | [[Eilenberg-MacLane space]] $K(R,n)$  |  [[Eilenberg-MacLane spectrum]]
 $H R$ |
| [[topological K-theory|K-cohomology theory]] | [[K-homology theory]] | [[stable unitary group]]
 [[stable unitary group|BU]]  | [[K-theory spectrum]]
 [[KU]], ... |

=--

As for any [[generalized cohomology theory]] there are immediate variants to plain Cohomotopy theory, as shown in the following table:

[[!include flavours of cohomotopy -- table]]

### As the absolute cohomology theory
 {#AsTheAbsoluteCohomologyTheory}

In some sense Cohomotopy is the most fundamental of all [[generalized cohomology theories]] ("[[The Music of the Spheres]]").

Concretely, [[stable Cohomotopy cohomology theory]] is the [[initial object]] among [[multiplicative cohomology theories]], in that the [[sphere spectrum]] is the [[initial object in an (infinity,1)-category|initial object]] in ([[E-infinity ring|E-infinity]]) [[ring spectra]]. This means that for any other [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology theory|cohomology theory]] $E$ there is an essentially unique multiplicative [[natural transformation]]

\[
  \label{BoardmanHomomorphism}
  \mathbb{S}^\bullet(X)
  \overset{\beta(X)}{\longrightarrow}
  E^\bullet(X)
\]

from Cohomotopy cohomology groups to $E$-cohomology groups -- the _[[Boardman homomorphism]]_.

Specifically for $E = K \mathbb{F}$ the [[algebraic K-theory]] of a [[field]] $\mathbb{F}$ (such as a [[prime field]] $\mathbb{F}_p$) there is such a comparison morphism; and another way how stable Cohomotopy is the most fundamental of all K-theories is that it is equivalently the [[algebraic K-theory]] over the "absolute base", namely over the "[[field with one element]]" $\mathbb{F}_1$ (see [there](stable+cohomotopy#AsAlgebraicKTheoryOverTheFieldWithOneElement) for more):

$$
  \mathbb{S}^\bullet(X)
  \;\simeq\;
  K\mathbb{F}_1 ^ \bullet(X)
  \,.
$$

For example, for $E = $ [[KU]] and in the case of $G$-[[equivariant cohomology theory]] ([[equivariant Cohomotopy theory]] and [[equivariant K-theory]]) the [[Boardman homomorphism]] (eq:BoardmanHomomorphism) gives the comparison map

$$
  \mathbb{S}^0_G(\ast)
  \simeq
  R_{\mathbb{F}_1}(G)
  \simeq
  A(G)
  \overset{\mathbb{C}[-]}{\longrightarrow}
  R_{\mathbb{C}}(G)
  \simeq
  KU^0_G(\ast)
$$

from the [[Burnside ring]] to the [[representation ring]] of the [[finite group]] $G$, by forming [[permutation representations]]; where we may think of the [[Burnside ring]] as being the [[representation ring]] over the "[[field with one element]]" (see e.g. [Chu-Lorscheid-Santhanam 10](field+with+one+element#ChuLorscheidSanthanam10)), as indicated above.

So far, this applies to [[stable Cohomotopy theory]], which historically has received almost all the attention. But, while [[stabilization]]  makes the immensely rich nature of [[homotopy theory]] a tad more tractable, it is only an approximation (just the first [[Goodwillie calculus|Goodwillie derivative]]!) of full unstable/[[non-abelian cohomology]].
Hence the one concept more fundamental than stable Cohomotopy theory is actual Cohomotopy theory.

For example, the classification of [[Yang-Mills instantons]] on $\mathbb{R}^4$ is typically regarded in the [[non-abelian cohomology]] theory represented by the [[classifying space]] $B SU(N)$ of the [[special unitary group]] (for $N \geq 2$, starting with [[SU(2)]])

$$
  (B SU(N))^0
  \Big(
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
  \Big)
  \;\coloneqq\;
  \Big\{
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
    \to
    B SU(N)
  \Big\}/_{\sim_{homotopy}}
  \;\simeq\;
  \underset{
    \mathclap{
    \color{blue}
    {\text{instanton} \atop \text{number}}
    }
  }{
    \mathbb{Z}
  }
  \,.
$$

But since the [[one-point compactification]] of 4d [[Euclidean space]] is the [[4-sphere]] $\big( \mathbb{R}^4\big)^{cpt} \simeq S^4$, this classification factors through one in unstable Cohomotopy theory, via the "unstable Boardman homomorphism" $S^4 \longrightarrow B SU(N)$ representing the generator of the 4th [[homotopy group]] of $B SU(N)$ (see [there](unitary+group#HomotopyGroups))

$$
  \pi^0
  \Big(
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}7
  \Big)
  \;\coloneqq\;
  \Big\{
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
    \to
    S^4
  \Big\}/_{\sim_{homotopy}}
  \;\simeq\;
  \underset{
    \mathclap{
    \color{blue}
    {\text{instanton} \atop \text{number}}
    }
  }{
    \mathbb{Z}
  }
  \,.
$$

(see [SS 19, p. 9-10](#SatiSchreiber19)) This is the tip of an iceberg. Which needs to be discussed elsewhere.

\linebreak

## Properties

### Hopf degree theorem

+-- {: .num_prop}
###### Proposition
**([[Hopf degree theorem]])**

Let $n \in \mathbb{N}$ be a [[natural number]] and $X \in Mfd$ be a [[connected topological space|connected]] [[orientation|orientable]] [[closed manifold]] of [[dimension]] $n$. Then the $n$th [[cohomotopy]] classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in [[bijection]] to the [[degree of a continuous function|degree]] $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

$$
  \pi^n(X)
    \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow}
   H^n(X,\mathbb{Z})
     \;\simeq\;
   \mathbb{Z}
$$

from $n$th [[cohomotopy]] to $n$th [[integral cohomology]] is a [[bijection]].

=--

(e.g. [Kosinski 93, IX (5.8)](#Kosinski93))

### Poincaré–Hopf theorem

* [[Poincaré–Hopf theorem]]

### Relation to Freudenthal suspension theorem

relation to the [[Freudenthal suspension theorem]] ([Spanier 49, section 9](#Spanier49))

### Smooth representatives

For $X$ a [[compact topological space|compact]] [[smooth manifold]], there is a [[smooth function]] $X \to S^n$ representing every cohomotopy class (with respect to the standard [[smooth structure]] on the [[sphere]] [[manifold]]).

### Reduced Cohomotopy
 {#ReducedCohomotopy}

For some purposes (such as in stating [[Pontryagin's theorem]] [below](#RelationToCobordismGroup)) it is more natural to use [[reduced cohomology|reduced]] Cohomotopy:

For $(X,x)$ a [[pointed homotopy type]]/cofibrant [[pointed topological space]], its **reduced $n$-Cohomotopy** is the set of connected components in the [[compact-open topology|mapping space]] of pointed functions to the [[n-sphere]] (for any choice of basepoint on the latter, then naturally thought of as $\infty \in (\mathbb{R}^n)^{cpt} \simeq S^n$):

\[
  \label{ReducedCohomotopySet}
  {\widetilde \pi}{}^n
  (X,x)
  \;\coloneqq\;
  \pi_0
  Maps^{\ast/}
  \big(
    (X,x),
    (S,\infty)
  \big)
  \,.
\]

However, the difference between reduced and unreduced Cohomotopy is small:

\begin{prop}
  For $n \geq 1$ a [[positive number|positive]] [[natural number]] and $(X,x)$ a [[pointed topological space]]/[[pointed homotopy type]], the canonical [[morphism]] from reduced (eq:ReducedCohomotopySet) to unreduced Cohomotopy of $X$ in degree $n$ is an [[isomorphism]]:

$$
  {\widetilde \pi}^{n \geq 1}(X,x)
    \overset{\;\;\simeq\;\;}{\longrightarrow}
  {\pi}^{n \geq 1}(X)
  \,.
$$
\end{prop}
(e.g. [Hatcher, 4A.2, 4A.3](algebraic+topology#Hatcher))
\begin{proof}
Observe that

$$
  Maps^{\ast/}\big((X,x),(S^n,\infty) \big)
    \overset{fib(ev_x)}{\longrightarrow}
  Maps\big(X,S^n \big)
    \overset{ev_x}{\longrightarrow}
  S^n
$$

(where $ev_x$ is the [[evaluation map]] at $x \in X$) is a [[homotopy fiber sequence]] as soon as $X$ is represented by a [[cell complex]] (e.g. [[CW-complex]]). This follows either from the [pullback-power axiom](enriched+model+category#PullbackPowerAxiom) in the [[classical model structure on topological spaces]], or, more abstractly from the characterization of [[derived hom-spaces]] in [[slice (infinity,1)-category|coslice (infinity,1)-categories]] [here](slice+infinity-category#HamSpacesInASlice).

Therefore we have the corresponding [[long exact sequence of homotopy groups]], which starts out as

\begin{xymatrix@C=20pt}
  \pi_1\big( S^n \big)
  \ar[r]
  \ar@{=}[d]
  &
  \pi_0 \mathrm{Maps}^{\ast/}\big((X,x),(S^n,\infty) \big)
  \ar[r]^{  }
  \ar@{=}[d]
  &
  \pi_0 \mathrm{Maps}\big(X,S^n \big)
  \ar[r]
  \ar@{=}[d]
  &
  \pi_0(S^n)
  \ar@{=}[d]
  \\
  {\overbrace{
  \begin{array}{ccc}
    \mathbb{Z} &\!\!\vert\!\!& n = 1
    \\
    \ast &\!\!\vert\!\!& n \geq 2
  \end{array}
  }}
  \ar[r]
  &
  {\widetilde \pi}{}^n(X,x)
  \ar[r]
  &
  {\pi}{}^n(X)
  \ar[r]
  &
  \ast
  \,.
\end{xymatrix}

By [[exact sequence|exactness]], this yields the claim for $n \geq 2$; while for $n = 1$ it gives the desired isomorphism only up to a possibly non-trivial [[quotient set|quotient]] by an [[action]] of $\mathbb{Z} \simeq \pi_1(S^1)$:

$$
  {\widetilde \pi}{}^1(X,x)/\pi_1(S^1)
  \simeq
  \pi^1(X)
  \,.
$$

Hence it is now sufficient to see that the $\pi_1(S^1)$-[[action]] on reduced 1-Cohomotopy is always trivial:

Now, the action of $n \in \pi_1(S^1)$ takes $\big[ (X,x) \overset{ \;c\; }{\longrightarrow} (S^1,\ast) \big]$ to any $\big[ (X,x) \overset{ \;c'\; }{\longrightarrow} (S^1,\ast) \big]$ such that $[X \overset{c}{\longrightarrow} S^1] \overset{\eta}{\Rightarrow} [X \overset{c'}{\longrightarrow} S^1]$ in $\pi^1(X)$, with $[\eta(x,-)] = n \in \pi_1(S^1)$.
But using the [[group]]-structure on $S^1 = \mathbb{R}/\mathbb{Z}$  we have that one such $\eta$ is given by $\eta(x,t) \coloneqq c(x) + n t \,mod\, \mathbb{Z}$, which yields $c'(x) = c(x) + n \,mod\, \mathbb{Z}$ and hence $c' = c$.
\end{proof}

### Pontryagin isomorphism between Cohomotopy and framed Cobordism
  {#RelationToCobordismGroup}

_[[Pontryagin's theorem]]_ ([Pontryagin 38](#Pontryagin38), [Pontryagin 55](#Pontryagin55)) is an identification of [[cobordism classes]] of [[normally framed submanifolds]] of [[codimension]] $n$ in a [[closed manifold]] $X^d$ with the $n$-[[Cohomotopy]] of that manifold -- the [[homotopy classes]] of [[maps]] into the [[n-sphere]], which is exhibited by the _[[Cohomotopy charge map]]_, often known as the [[Pontryagin-Thom collapse construction]]_:

$$
  Cob^n_Fr(X)
  \overset{\simeq}{\longrightarrow}
  \pi^n(X)
  \,.
$$

The analogous statement identifying [[cobordism classes]] of [[normally oriented submanifolds]] with [[homotopy classes]] of maps into the [[universal vector bundle|universal]] [[special orthogonal group|special orthogonal]] [[Thom space]] $M SO(n)$ is _[[Thom's theorem]]_ ([Thom 54](#Thom54)). Both statements, as well as their joint generalization to other [[tangential structures]] and notably their [[stabilization]] to [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[Cobordism cohomology theory]], have come to be widely known as the _[[Pontryagin-Thom collapse construction]]_, or similar, and form the basis of modern [[cobordism theory]] and its application in [[stable homotopy theory]].

\linebreak

For $X$ a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] $D$, the assignment of [[Cohomotopy charge]] (the _[[Pontryagin-Thom construction]]_, in this unstable and framed version due to [Pontrjagin 55](#Pontrjagin55), see review in [Kosinski 93, IX.5](#Kosinski93), [Milnor 97](#Milnor97)) identifies the [[set]]

$$
  SubMfd_{/bord}^{d}(X)
$$

of [[cobordism classes]] of  [[closed manifold|closed]] and [[normally framed submanifolds]] $\Sigma \overset{\iota}{\hookrightarrow} X$ of [[dimension]] $d$ inside $X$ with the [[cohomotopy]] $\pi^{D-d}(X)$ of $X$ in degree $D- d$

$$
  SubMfd_{/bord}^{d}(X)
    \underoverset{\simeq}{\;\;\;PT\;\;\;}{\longrightarrow}
  \pi^{D-d}(X)
  \,.
$$

(e.g. [Kosinski 93, IX Theorem (5.5)](#Kosinski93))

In particular, by this [[bijection]] the canonical [[group]] [[structure]] on [[cobordism groups]] in sufficiently high [[codimension]] (essentially given by [[disjoint union]] of [[submanifolds]]) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

{#PontrjaginThomConstructionGraphics}$\,$

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

Here the [[normal framing]] of the submanifolds plays the role of the [[charge]] in [[Cohomotopy]] which they carry:

{#nCohomotopyChargeOfSubmanifoldsIsNormalFraming}$\,$

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

For example:

{#nCohomotopyOfEuclideanNSpaceIllustration}$\,$

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

This construction generalizes to [[equivariant cohomotopy]], see [there](equivariant+cohomotopy#PontryaginThomConstruction).

With the [[equivariant Hopf degree theorem]] the above example has the following $\mathbb{Z}_2$-[[equivariant homotopy theory|equivariant]] version (see [there](equivariant+Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationSpheres)):

{#EquivariantnCohomotopyOfEuclideanNOrientifoldIllustration}$\,$

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

Further by the [[equivariant Hopf degree theorem]] (see [there](equivariant+Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationTori)), this example generalizes to [[equivariant cohomotopy]] of [[toroidal orbifold|toroidal]] [[orientifolds]]:

{#EquivariantnCohomotopyOfToroidalNOrientifoldIllustration}$\,$

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

\linebreak

### Composition/product in Cohomotopy
  {#ProductInCohomotopy}

Under the [[Pontryagin-Thom isomorphism]] ([above](#RelationToCobordismGroup)) the product in Cohomotopy, i.e. the [[composition]] operation

$$
  \array{
    \pi^{n_1}
    \big(
      S^{n_2 + n_1}
    \big)
    \times
    \pi^{n_2 + n_1}(X)
    &\overset{}{\longrightarrow}&
    \pi^{n_1}(X)
    \\
    (c_1, c_2) &\mapsto& c_1 \circ c_2
  }
  \,,
$$

corresponds to forming [[product manifolds]] of submanifolds, in that (see also [Kosinski 93, Section IX 6.1](#Kosinski93)):

$$
    \mathrm{PT}^{-1}(c_1 \circ c_2)
    \;\simeq\;
    \Big[
      \mathrm{PT}^{-1}(c_2) \times \mathrm{PT}^{-1}(c_1)
      \;\subset\;
      \mathrm{PT}^{-1}(c_2) \times \mathbb{R}^{n_2 + n_1}
      \;\subset\;
      X
    \Big]
  \,.
$$

This is exhibited by the following [[pasting diagram]] of [[pullbacks]]/[[fiber products]] (repeatedly using the [[pasting law]]):

\begin{imagefromfile}
        "file_name": "ProductInCohomotopyUnderPT.jpg",
        "web": "nlab",
        "width": 600,
        "unit": "px",
        "margin": {
            "top": -20,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "product in Cohomotopy under the PT isomorphism",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Here the vertical inclusions are the defining ones of the [[submanifolds]] or of their [[normal bundles]], identified with some choice of [[tubular neighbourhoods]].

\linebreak

### Cohomotopy charge map and Relation to configuration spaces
 {#RelationToConfigurationSpaces}

The _[[Cohomotopy charge map]]_ is the [[function]] that assigns to a [[configuration space of points|configuration of points]] their total [[charge]] as measured in [[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

This is alternatively known as the "electric field map" ([Salvatore 01](Cohomotopy+charge#Salvatore01) following [Segal 73, Section 1](Cohomotopy+charge#Segal73), see also [Knudsen 18, p. 49](Cohomotopy+charge#Knudsen18)) or the "scanning map" ([Kallel 98](Cohomotopy+charge#Kallel98)).

For $D \in \mathbb{N}$ the _Cohomotopy charge map_ is the  [[continuous function]]

\[
  \label{CohomotopyChargeMapOnEuclideanSpace}
  Conf\big( \mathbb{R}^D \big)
  \overset{cc}{\longrightarrow}
  \mathbf{\pi}^D
  \Big(
    \big( \mathbb{R}^D \big)^{cpt}
  \Big)
  =
  Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big)
  =
  \Omega^{D} S^D
\]

from the [[configuration space of points]] in the [[Euclidean space]] $\mathbb{R}^D$ to the $D$-[[Cohomotopy]] [[cocycle space]] [[vanishing at infinity]] on the [[Euclidean space]](which is equivalently the [[space of maps|space of pointed maps]] from the [[one-point compactification]] $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]]), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit [[charge]] to their total [[charge]] as measured in  [[Cohomotopy]]-[[generalized cohomology|cohomology theory]] ([Segal 73, Section 3](Cohomotopy+charge#Segal73)).

This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeleed configurations and to [[equivariant Cohomotopy]]. The following graphics illustrates the Cohomotopy charge map on [[G-space]] [[tori]] for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-[[equivariant Cohomotopy]]:

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

In some situations the [[Cohomotopy charge map]] is a [[weak homotopy equivalence]] and hence exhibits, for all purposes of [[homotopy theory]], the [[Cohomotopy]] [[cocycle space]] of Cohomotopy charges as an equivalent reflection of the [[configuration space of points]].

+-- {: .num_prop #GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace}
###### Proposition
**([[group completion]] on [[configuration space of points]] is [[iterated based loop space]])**

The [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    cc
  }{\longrightarrow}
  \Omega^D S^D
$$

from the full unordered and unlabeled configuration space ([here](configuration+space+of+points#eq:UnorderedUnlabeledConfigurationSpace)) of [[Euclidean space]] $\mathbb{R}^D$  to the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]], exhibits the [[group completion]] ([here](configuration+space+of+points#eq:GroupCompletionOfConfigurationSpaceMonoid)) of the configuration space monoid

$$
  \Omega B_{{}_{\sqcup}\!}
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    \simeq
  }{\longrightarrow}
  \Omega^D S^D
$$

=--

([Segal 73, Theorem 1](Cohomotopy+charge#Segal73))

+-- {: .num_prop #CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace}
###### Proposition
**([[Cohomotopy charge map]] is [[weak homotopy equivalence]] on sphere-labeled [[configuration space of points]])**

For $D, k \in \mathbb{N}$ with $k \geq 1$, the [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D,
    S^k
  \big)
  \underoverset{\simeq}{cc}{\longrightarrow}
  \Omega^D S^{D + k}
$$

is a [[weak homotopy equivalence]] from the configuration space ([here](configuration+space+of+points#eq:UnorderedLabeledCOnfigurationSpace)) of unordered points with labels in $S^k$ and vanishing at the base point of the label space to the $D$-fold [[iterated loop space]] of the [[n-sphere|D+k-sphere]].

=--

([Segal 73, Theorem 3](Cohomotopy+charge#Segal73))

The May-Segal theorem \ref{ScanningMapEquivalenceOverCartesianSpace} generalizes from [[Euclidean space]] to [[closed manifold|closed]] [[smooth manifolds]] if at the same time one passes from plain [[Cohomotopy]] to [[twisted Cohomotopy]], twisted, via the [[J-homomorphism]], by the [[tangent bundle]]:

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

Let

1. $X^n$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ J-twisted Cohomotopy space}}
  }{
    Maps_{{}_{/B O(n)}}
    \Big(
      X^n
      \;,\;
      S^{
        \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}}
      }
      \!\sslash\!
      O(n)
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. the [[J-homomorphism|J]]-[[twisted Cohomotopy|twisted (n+k)-Cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

([Bödigheimer 87, Prop. 2](Cohomotopy+charge#Boedigheimer87), following [McDuff 75](Cohomotopy+charge#McDuff75))

+-- {: .num_prop #ScanningMapEquivalenceOverClosedFramedManifold}
###### Remark

In the special case that the [[closed manifold]] $X^n$ in Prop. \ref{ScanningMapEquivalenceOverClosedManifold} is [[parallelizable manifold|parallelizable]], hence that its [[tangent bundle]] is [[trivial bundle|trivializable]], the statement of Prop. \ref{ScanningMapEquivalenceOverClosedManifold} reduces to this:

Let

1. $X^n$ be a [[parallelizable manifold|parallelizable]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ Cohomotopy space}}
  }{
    Maps
    \Big(
      X^n
      \;,\;
      S^{
       n + k
      }
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. $(n+k)$-[[Cohomotopy]] space of $X^n$, hence the [[space of maps]] from $X$ to the [[n-sphere|(n+k)-sphere]]

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

\linebreak

### Complex-rational Cohomotopy and moduli space of Yang-Mills monopoles

The assignment of [[scattering amplitudes]] of [[monopoles]] in [[SU(2)]]-[[Yang-Mills theory]] is a  [[diffeomorphism]]

$$
  \mathcal{M}_k
  \overset{ S }{\longrightarrow}
  R_k
$$

identifying the [[moduli space of monopoles]] of number $k$ with the space of complex-[[rational functions]] form the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k$ (hence the [[cocycle space]] of complex-rational 2-[[Cohomotopy]]).

([Atiyah-Hitchin 88, Theorem 2.10](moduli+space+of+monopoles#AtiyahHitchin88))

{#Illustration} $\,$

This is a [[non-abelian group|non-abelian]] analog of the [[Dirac charge quantization]] of the [[electromagnetic field]], with [[ordinary cohomology]] replaced by [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]].

## Examples

### Cohomotopy in the lowest dimensions

Since the [[0-sphere]] is the disjoint union of two points, $0$-cohomotopy corresponds to the powerset of the connected components of a space.

Further, the $(-1)$-sphere is understood as the [[empty space]]. Since the only map to the empty space is the identity map from the empty space, the $(-1)$-cohomotopy of a space is measuring whether that space is empty. In the context of [[homotopy type theory]], this is the same as [[negation]].

### Of projective spaces

Cohomotopy sets of [[projective spaces]] are computed in [West 71](#West71)

### Of 4-Manifolds
 {#On4Manifolds}

Let $X$ be a [[4-manifold]] which is [[connected topological space|connected]] and [[orientation|oriented]].

The [[Pontryagin-Thom construction]] as [above](#RelationToCobordismGroup) gives for $n \in \mathbb{Z}$ the [[commuting diagram]] of sets

$$
  \array{
     \pi^n(X)
       &\overset{\simeq}{\longrightarrow}&
     \mathbb{F}_{4-n}(X)
     \\
     {}^{ \mathllap{h^n} }
     \downarrow
       &&
     \downarrow^{ h_{4-n} }
     \\
     H^n(X,\mathbb{Z})
       &\underset{\simeq}{\longrightarrow}&
     H_{4-n}(X,\mathbb{Z})
     \,,
  }
$$

where $\pi^\bullet$ denotes cohomotopy sets, $H^\bullet$ denotes [[ordinary cohomology]], $H_\bullet$ denotes [[ordinary homology]] and $\mathbb{F}_\bullet$ is [[normal bundle|normally]] [[framing|framed]] [[cobordism classes]] of [[normal bundle|normally]] [[framing|framed]] [[submanifolds]]. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of [[Eilenberg-MacLane spaces]]):

$$
  h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z}
  \,.
$$

Now

* $h^0$, $h^1$, $h^4$ are [[isomorphisms]]

* $h^3$ is an isomorphism if $X$ is "odd" in that it contains at least one closed oriented [[surface]] of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-[[quotient group]] of $\pi^3(X)$ (which is a group via the [[group]]-[[structure]] of the [[3-sphere]] ([[special unitary group|SU(2)]]))

([Kirby-Melvin-Teichner 12](#KirbyMelvinTeichner12))

\linebreak

## Related concepts

* [[unstable topological K-theory]]

[[!include flavours of cohomotopy -- table]]

* [[differential cohomotopy]]

\linebreak

[[!include Segal completion -- table]]

\linebreak

* [[Cohomotopy charge map]]

## References

### General

Original articles:

* {#Borsuk36} [[Karol Borsuk]], *Sur les groupes des classes de transformations continues*, CR Acad. Sci. Paris **202** 1400-1403 (1936) 2-3

* {#Pontryagin38a} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication I_, Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

* {#Spanier49} [[Edwin Spanier]], _Borsuk's Cohomotopy Groups_, Annals of Mathematics Second Series, Vol. 50, No. 1 (Jan., 1949), pp. 203-245 ([jstor:1969362](http://www.jstor.org/stable/1969362))

* [[Franklin Peterson]], _Some Results on Cohomotopy Groups_, American Journal of Mathematics Vol. 78, No. 2 (Apr., 1956), pp. 243-258 ([jstor:2372514](https://www.jstor.org/stable/2372514))

* [[Jan Jaworowski]], _Generalized cohomotopy groups as limit groups_, Fundamenta Mathematicae 50 (1962), 393-402 ([doi:10.4064/fm-50-4-333-340](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/4), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50133.pdf))

See also

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

* [[eom]], _[Cohomotopy group](https://www.encyclopediaofmath.org/index.php/Cohomotopy_group)_

Further discussion:

* [[Laurence Taylor]], _The principal fibration sequence and the second cohomotopy set_, Proceedings of the Freedman Fest, 235251, Geom. Topol. Monogr., 18, Coventry, 2012 ([arXiv:0910.1781](https://arxiv.org/abs/0910.1781), [doi:10.2140/gtm.2012.18.235](http://dx.doi.org/10.2140/gtm.2012.18.235))

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

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

### Examples

Cohomotopy sets of [[projective spaces]]:

* {#West71} Robert West, _Some Cohomotopy of Projective Space_, Indiana University Mathematics Journal Vol. 20, No. 9 (March, 1971), pp. 807-827 ([jstor:24890146](https://www.jstor.org/stable/24890146))

Cohomotopy sets of [[4-manifolds]]:

* Nobuo Shimada, _Homotopy classification of mappings of a 4-dimensional complex into a 2-dimensional sphere_, Nagoya Math. J., Volume 5 (1953), 127-144 ([euclid:nmj/1118799399](https://projecteuclid.org/euclid.nmj/1118799399))

* [[Daniel Freed]], [[Karen Uhlenbeck]], Appendix B of: _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

* {#KirbyMelvinTeichner12} [[Robion Kirby]], [[Paul Melvin]], [[Peter Teichner]], _Cohomotopy sets of 4-manifolds_, Geometry & Topology Monographs 18 (2012) 161–190 ([arXiv:1203.1608](https://arxiv.org/abs/1203.1608), [doi:10.2140/gtm.2012.18.161](http://dx.doi.org/10.2140/gtm.2012.18.161))

Cohomotopy sets of [[Thom spaces]]:

* [[Thomas Rot]], _Homotopy classes of proper maps out of vector bundles_,  Arch. Math. 114, 107–117 (2020 ([arXiv:1808.08073](https://arxiv.org/abs/1808.08073), [doi:10.1007/s00013-019-01372-z](https://doi.org/10.1007/s00013-019-01372-z))

In relation to [[quaternionic line bundles]] over [[5-manifolds]]:

* [[Panagiotis Konstantis]], _Vector bundles and cohomotopies of spin 5-manifolds_, Homology, Homotopy and Applications Volume 23 (2021) Number 1 ([arXiv:1812.06547](https://arxiv.org/abs/1812.06547), [doi:10.4310/HHA.2021.v23.n1.a9](https://dx.doi.org/10.4310/HHA.2021.v23.n1.a9))

Cohomotopy sets of [[spin structure|spin]]-manifolds in co-degree 1:

* [[Panagiotis Konstantis]], _A counting invariant for maps into spheres and for zero loci of sections of vector bundles_,  Abh. Math. Semin. Univ. Hambg. (2020) ([arXiv:1911.03214](https://arxiv.org/abs/1911.03214), [doi:10.1007/s12188-020-00228-6](https://doi.org/10.1007/s12188-020-00228-6))

### Cohomotopy cocycle spaces

Discussion of Cohomotopy [[cocycle spaces]] (i.e. [[spaces of maps]] into an [[n-sphere]]):

* {#Hansen74} [[Vagn Lundsgaard Hansen]], _The homotopy problem for the components in the space of maps on the $n$-sphere_, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 ([DOI:10.1093/qmath/25.1.313](https://doi.org/10.1093/qmath/25.1.313))

* {#Hansen81} [[Vagn Lundsgaard Hansen]], _On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere_, Transactions of the American Mathematical Society, Vol. 265, No. 1 (May, 1981), pp. 273-281 ([jstor:1998494](https://www.jstor.org/stable/1998494))

* [[Victor Vassiliev]], _Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials_ ([arXiv:1809.05632](https://arxiv.org/abs/1809.05632))

* [[Douglas Ravenel]], *What we still don't know about loop spaces of spheres*, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), *Homotopy Theory via Algebraic Geometry and Group Representations*, Contemporary Mathematics **220**, AMS 1998  ([pdf](https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf), [[Ravenel_LoopSpacesOfSpheres.pdf:file]], [doi:10.1090/conm/220](http://dx.doi.org/10.1090/conm/220))

Discussion of [[cocycle spaces]] for [[rational Cohomotopy]] (see also at _[[rational model of mapping spaces]]_):

* {#MollerRaussen85} [[Jesper Møller]], [[Martin Raussen]], _Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces_, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 ([jstor:2000242](https://www.jstor.org/stable/2000242))

### Relation to knots and links

 Relation to [[knots]] and [[links]]:

* Maths.SE, _[Framed Cobordism Classes of links in $\mathbb{R}^3$](https://math.stackexchange.com/q/426482/58526)_

### In computational topology

Discussion of [[Cohomotopy]]-sets in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Francis Sergeraert]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Computing all maps into a sphere_, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 ([arxiv:1105.6257](https://arxiv.org/abs/1105.6257))

* [[Lukáš Vokřínek]], _Decidability of the extension problem for maps into odd-dimensional spheres_,  Discrete Comput Geom (2017) 57: 1 ([arXiv:1401.3758](https://arxiv.org/abs/1401.3758))

[[!include cohomotopy in topological data analysis -- references]]

### Equivariant Cohomotopy

Discussion of the [[stable cohomotopy|stable]] cohomotopy ([[framed manifold|framed]] [[cobordism cohomology theory]]) in the [[equivariant cohomology]]-version of cohomotopy ([[equivariant cohomotopy]]):

* {#Wasserman69} [[Arthur Wasserman]], section 3 of _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

* {#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)

and in the [[twisted cohomology]]-version ([[twisted cohomotopy]])

* {#Cruickshank03} [[James Cruickshank]], Section 7 of _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)

Discussion of [[M-brane]] physics in terms of [[equivariant rational homotopy theory|rational equivariant]] cohomotopy:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

and in terms of [[twisted cohomotopy]]:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

### In spectral geometry

Discussion of [[smooth functions]] into the [[4-sphere]] in the context of [[Connes-Lott models]] in spectral [[non-commutative geometry]]:

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471))

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))

* {#Connes17} [[Alain Connes]], section 4 of _Geometry and the Quantum_, in _Foundations of Mathematics and Physics One Century After Hilbert_, Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125))

[[!redirects Cohomotopy]]

[[!redirects cohomotopy theory]]
[[!redirects Cohomotopy theory]]

[[!redirects cohomotopy set]]
[[!redirects cohomotopy sets]]
[[!redirects Cohomotopy set]]
[[!redirects Cohomotopy sets]]

[[!redirects unstable cohomotopy]]
[[!redirects unstable Cohomotopy]]

[[!redirects cohomotopy group]]
[[!redirects cohomotopy groups]]
[[!redirects Cohomotopy group]]
[[!redirects Cohomotopy groups]]

[[!redirects cohomotopy cohomology ]]
[[!redirects cohomotopy cohomology theory]]
[[!redirects Cohomotopy cohomology ]]
[[!redirects Cohomotopy cohomology theory]]


cohomotopy charge map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

The _Cohomotopy charge map_ is the [[function]] that assigns to a [[configuration space (mathematics)|configuration]] of [[normally framed submanifolds]] of [[codimension]] $n$ their total [[charge]] as measured in $n$-[[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

Concretely, this is the function which assigns to a [[normally framed submanifold]] its **asymptotic normal distance function**, namely the [[distance]] from the [[submanifold]] measured

1. in [[direction vector|direction]] [[orthogonality|perpendicular]] to the submanifold, as encoded by the [[normal framing]];

1. asymptotically, regarding all points outside a [[tubular neighbourhood]] as being [[one-point compactification|at infinity]].

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

For general $n$ this is known as the "[[Pontrjagin-Thom collapse construction]]".

### For charged points
 {#ForChargedPoints}

For maximal [[codimension]] $n$ inside an [[orientation|oriented]] [[smooth manifold|manifold]], hence for 0-dimensional submanifolds, hence for [[configuration space of points|configurations of points]] and with all points regarded as equipped with positive normal framing, the Cohomotopy charge map is alternatively known as the "electric field map" ([Salvatore 01](#Salvatore01) following [Segal 73, Section 1](#Segal73), see also [Knudsen 18, p. 49](#Knudsen18)) or the "scanning map" ([Kallel 98](#Kallel98), [Manthorpe-Tillmann 13](#ManthorpeTillmann13)):

In maximal [[codimension]] $D \in \mathbb{N}$, the _Cohomotopy charge map_ is thus the  [[continuous function]]

\[
  \label{CohomotopyChargeMapOnEuclideanSpace}
  Conf\big( \mathbb{R}^D \big)
  \overset{cc}{\longrightarrow}
  \mathbf{\pi}^D
  \Big(
    \big( \mathbb{R}^D \big)^{cpt}
  \Big)
  =
  Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big)
  =
  \Omega^{D} S^D
\]

from the [[configuration space of points]] in the [[Euclidean space]] $\mathbb{R}^D$ to the $D$-[[Cohomotopy]] [[cocycle space]] [[vanishing at infinity]] on the [[Euclidean space]](which is equivalently the [[space of maps|space of pointed maps]] from the [[one-point compactification]] $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]]), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit [[charge]] to their total [[charge]] as measured in  [[Cohomotopy]]-[[generalized cohomology|cohomology theory]] ([Segal 73, Section 3](#Segal73)).

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

(See also at _[[cobordism]] -- [Relation to Cohomotopy](cobordism#RelationToCohomotopy)_.)

This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to [[equivariant Cohomotopy]]. The following graphics illustrates the Cohomotopy charge map on [[G-space]] [[tori]] for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-[[equivariant Cohomotopy]]:

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

## Properties

### Characterization of cobordism classes by their Cohomotopy charge

The unstable [[Pontrjagin-Thom theorem]] states that Cohomotopy charge faithfully reflects [[configuration space (mathematics)|configurations]] of [[normally framed submanifolds]] up to normally framed embedded [[cobordism]], hence that the [[Pontrjagin-Thom collapse construction]] induces a [[bijection]] between [[cobordism classes]] of [[normally framed submanifolds]] and the [[Cohomotopy set]] in degree the respective [[codimension]]:

$$
  \left\{
    {
      {
        \text{normally framed submanifolds}
      }
      \atop
      {
        \text{in}\;X\;\text{of codimension}\; n
      }
    }
  \right\}
  \Big/_{\sim_{cobordism}}
  \underoverset{\simeq}{cc}{\longrightarrow}
  \underset{
     \mathclap{
       \color{blue}
       {
         \text{Cohomotopy}
         \atop
         \text{set}
       }
     }
  }{
    \pi^n\big( X \big)
  }
$$

For more details see [here](cohomotopy#RelationToCobordismGroup).

In good situations this [[bijection]] of [[sets]] of [[homotopy classes]] enhances to a [[weak equivalence]] of [[configuration space (mathematics)|configuration spaces]]/[[cocycle spaces]]. See _[Characterization of point configurations by their Cohomotopy charge](#CharacterizationOfPointConfigurations)_ below.

\linebreak

### Characterization of point configurations by their Cohomotopy charge
 {#CharacterizationOfPointConfigurations}

In some situations the [[Cohomotopy charge map]] is a [[weak homotopy equivalence]] and hence exhibits, for all purposes of [[homotopy theory]], the [[Cohomotopy]] [[cocycle space]] of Cohomotopy charges as an equivalent reflection of the [[configuration space of points]].

#### On Euclidean spaces via plain Cohomotopy

+-- {: .num_prop #GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace}
###### Proposition
**([[group completion]] on [[configuration space of points]] is [[iterated based loop space]])**

The [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    cc
  }{\longrightarrow}
  \Omega^D S^D
$$

from the full unordered and unlabeled configuration space ([here](configuration+space+of+points#eq:UnorderedUnlabeledConfigurationSpace)) of [[Euclidean space]] $\mathbb{R}^D$  to the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]], exhibits the [[group completion]] ([here](configuration+space+of+points#eq:GroupCompletionOfConfigurationSpaceMonoid)) of the configuration space monoid

$$
  \Omega B_{{}_{\sqcup}\!}
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    \simeq
  }{\longrightarrow}
  \Omega^D S^D
$$

=--

([Segal 73, Theorem 1](#Segal73))

+-- {: .num_prop #CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace}
###### Proposition
**([[Cohomotopy charge map]] is [[weak homotopy equivalence]] on sphere-labeled [[configuration space of points]])**

For $D, k \in \mathbb{N}$ with $k \geq 1$, the [[Cohomotopy charge map]] (eq:CohomotopyChargeMapOnEuclideanSpace)

$$
  Conf
  \big(
    \mathbb{R}^D,
    S^k
  \big)
  \underoverset{\simeq}{cc}{\longrightarrow}
  \Omega^D S^{D + k}
$$

is a [[weak homotopy equivalence]] from the configuration space ([here](configuration+space+of+points#eq:UnorderedLabeledCOnfigurationSpace)) of unordered points with labels in $S^k$ and vanishing at the base point of the label space to the $D$-fold [[iterated loop space]] of the [[n-sphere|D+k-sphere]].

=--

([Segal 73, Theorem 3](#Segal73))

#### On closed manifolds via twisted Cohomotopy

The May-Segal theorem \ref{ScanningMapEquivalenceOverCartesianSpace} generalizes from [[Euclidean space]] to [[closed manifold|closed]] [[smooth manifolds]] if at the same time one passes from plain [[Cohomotopy]] to [[twisted Cohomotopy]], twisted, via the [[J-homomorphism]], by the [[tangent bundle]]:

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

Let

1. $X^n$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ J-twisted Cohomotopy space}}
  }{
    Maps_{{}_{/B O(n)}}
    \Big(
      X^n
      \;,\;
      S^{
        \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}}
      }
      \!\sslash\!
      O(n)
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. the [[J-homomorphism|J]]-[[twisted Cohomotopy|twisted (n+k)-Cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

([Bödigheimer 87, Prop. 2](#Boedigheimer87), following [McDuff 75](#McDuff75))

+-- {: .num_prop #ScanningMapEquivalenceOverClosedFramedManifold}
###### Remark

In the special case that the [[closed manifold]] $X^n$ in Prop. \ref{ScanningMapEquivalenceOverClosedManifold} is [[parallelizable manifold|parallelizable]], hence that its [[tangent bundle]] is [[trivial bundle|trivializable]], the statement of Prop. \ref{ScanningMapEquivalenceOverClosedManifold} reduces to this:

Let

1. $X^n$ be a [[parallelizable manifold|parallelizable]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ Cohomotopy space}}
  }{
    Maps
    \Big(
      X^n
      \;,\;
      S^{
       n + k
      }
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. $(n+k)$-[[Cohomotopy]] space of $X^n$, hence the [[space of maps]] from $X$ to the [[n-sphere|(n+k)-sphere]]

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

## References

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

### For point configurations

The theorem that, with due care, for [[configuration space of points|point configurations]] the Cohomotopy charge map is in fact a [[weak homotopy equivalence]] between the [[configuration space of points]] and the [[Cohomotopy]] [[cocycle space]] originates with

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

* {#Segal73} [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. __21__ (1973), 213–221. MR 0331377 ([pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf))

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

with comprehensive review in

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]])

See also:

* {#Kallel98} [[Sadok Kallel]], _Spaces of particles on manifolds and Generalized Poincaré Dualities_, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 ([doi:10.1093/qjmath/52.1.45](https://doi.org/10.1093/qjmath/52.1.45))

* {#Salvatore01} [[Paolo Salvatore]], _Configuration spaces with summable labels_, In: Aguadé J., Broto C., [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 ([arXiv:math/9907073](https://arxiv.org/abs/math/9907073))

* [[Oscar Randal-Williams]], section 10 of: _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))

* {#ManthorpeTillmann13} Richard Manthorpe, [[Ulrike Tillmann]], _Tubular configurations: equivariant scanning and splitting_, Journal of the London Mathematical Society, Volume 90, Issue 3 ([arxiv:1307.5669](https://arxiv.org/abs/1307.5669), [doi:10.1112/jlms/jdu050](https://doi.org/10.1112/jlms/jdu050))

* {#Knudsen18} [[Ben Knudsen]], _Configuration spaces in algebraic topology_ ([arXiv:1803.11165](https://arxiv.org/abs/1803.11165))

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

[[!redirects cohomotopy charge map]]
[[!redirects cohomotopy charge maps]]

[[!redirects Cohomotopy charge map]]
[[!redirects Cohomotopy charge maps]]

[[!redirects scanning map]]
[[!redirects scanning maps]]

[[!redirects scanning map equivalence]]
[[!redirects scanning map equivalences]]

[[!redirects cohomotopy charge]]
[[!redirects cohomotopy charges]]

[[!redirects Cohomotopy charge]]
[[!redirects Cohomotopy charges]]


cohomotopy in topological data analysis -- references

### Cohomotopy in topological data analysis
  {#CohomotopyInTopologicalDataAnalysisReferences}

Introducing [[persistent cohomotopy]] as a tool in [[topological data analysis]], improving on the use of [[well groups]] from [[persistent homology]]:

* {#FranekKrcal16} [[Peter Franek]], [[Marek Krčál]], _On Computability and Triviality of Well Groups_, Discrete Comput Geom **56** (2016) 126 ([arXiv:1501.03641](https://arxiv.org/abs/1501.03641), [doi:10.1007/s00454-016-9794-2](https://doi.org/10.1007/s00454-016-9794-2))

* {#FranekKrcal17} [[Peter Franek]], [[Marek Krčál]], _Persistence of Zero Sets_, Homology, Homotopy and Applications, **19** 2 (2017)  ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))

* {#FranekKrcalWagner18} [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology **1** (2018) 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))

Review:

* {#FranekKrčál15} [[Peter Franek]], [[Marek Krčál]], _Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory_, talk at ACAT meeting 2015 $[$[[Franek-Krcal-CohomotopyRobustProperties.pdf:file]]$]$

* {#SS22} [[Urs Schreiber]] on joint work with [[Hisham Sati]]: *[[schreiber:New Foundations for TDA -- Cohomotopy]]*, (May 2022)


coideal

Given a [[commutative unital ring]] $k$ and a [[coassociative coalgebra|coassociative]] $k$-[[coalgebra]] $C = (C,\Delta,\epsilon)$ with comultiplication $\Delta$ and counit $\epsilon\colon C\to k$, a $k$-[[submodule]] $I\subseteq C$ is

* a __right coideal__ if $\Delta(I)\subseteq \overline{I\otimes C}$,

* a __left coideal__ if $\Delta(I)\subseteq \overline{C\otimes I}$,

* a __coideal__ if $\Delta(I)\subseteq \overline{I\otimes C}+\overline{C\otimes I}$ and $\epsilon(I)=0$.

Here, $\overline{I \otimes C}$ denotes the image of $I \otimes C \to C \otimes C$ (this distinction is not necessary when $C$ is a flat $k$-module, for example when $k$ is a field).

In other words, a left coideal is simply a left subcomodule in $C$ with [[coaction]] being the comultiplication and a coideal is a sub[[bicomodule]] in $C$, where the comultiplication plays simultaneously the roles of left and right $C$-coactions on itself.

If $I\subseteq C$ is a coideal, then the [[quotient module|quotient]] $k$-module $C/I$ is equipped with the canonical structure of a coassociative $k$-coalgebra, the __quotient co(al)gebra__; the comultiplication is induced via choosing an arbitrary representative in each class, namely

$$
\Delta_{C/I}(c+I):= \Delta_C(c) + I\otimes C + C\otimes I
\in C/I \otimes C/I,
$$
or, in [[Sweedler notation]],
$\Delta_{C/I}(c) = \sum (c_{(1)}+I)\otimes (c_{(2)}+I)$ where $\Delta_C(c)= \sum c_{(1)}\otimes c_{(2)}$.

[[!redirects coideal]]
[[!redirects coideals]]
[[!redirects left coideal]]
[[!redirects left coideals]]
[[!redirects right coideal]]
[[!redirects right coideals]]
[[!redirects quotient cogebra]]
[[!redirects quotient coalgebra]]
[[!redirects quotient cogebras]]
[[!redirects quotient coalgebras]]


coimage

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

* **coimage**

* [[image]]

***

# Coimages
* table of contents
{: toc}

## Idea

The _coimage_ of a morphism is the notion [[duality|dual]] to its [[image]].

Under certain conditions (the [[codomain]] of) the coimage coincides with (the [[domain]] of) the image, and even if not, often the _coimage_ is what one calls (or wants to think of as) the image. The two notions tend to go hand in hand, with the coimage being an [[epi]] and the image being a [[mono]] in some [[(epi, mono) factorization]]. For more of the general theory see [[image]].

## Definition

The **coimage** of a [[morphism]] $f : c \to d$ in a [[category]] $C$ is the [[image]] of the corresponding morphism in the [[opposite category]] $C^{op}$.

### In terms of colimits

If $C$ has finite [[limit]]s and [[colimit]]s, then the coimage of a morphism $f : c \to d$ is the [[coequalizer]] of its [[kernel pair]]:

$$
  coim f \simeq colim ( c \times_d c \stackrel{\to}{\to} c)
  \,,
$$

This is isomorphic to the [[pushout]] $c \sqcup_{c\times_d c} c$

$$
  coim f \simeq c \sqcup_{c\times_d c} c
  \,.
$$

So in

$$
  \array{
    c \times_d c &&\to&& c
    \\
     &&& \swarrow
    \\
    \downarrow^f && coim f && \downarrow^f
    \\
    & \nearrow && \searrow
    \\
    C && \stackrel{f}{\to} && d
  }
$$

the outer square is a [[pullback]] square while the inner is a [[pushout]].

Notice that being a coequalizer, the morphism

$$
  c \to coim f
$$

is an [[epimorphism]] and in fact a [[regular epimorphism]].

### In an $(\infty,1)$-category
 {#InInfCat}

In an [[(∞,1)-category]] $C$ with [[(∞,1)-limit]]s and -colimits, the colimit-definition of coimages generalizes as follows:

for $f : c \to d$ a morphism in $C$, let

$$
  C(f) =
  \left(
    \cdots \stackrel{\to}{\stackrel{\to}{\to}} c \times_d c \stackrel{\to}{\to} c
  \right)
$$

be the [[Cech nerve]] of $f$. This is the [[groupoid object in an (∞,1)-category]] that resolves the [[kernel pair]] [[equivalence relation]]: where $c \times_d \stackrel{\to}{\to} c$ is the relation that makes two [[generalized element]]s of $c$ _equal_ if their image in $d$ is equal,  the full Cech nerve is the internal [[∞-groupoid]] where there is just an equivalence between such two elements.

The Cech nerve is a [[simplicial object|simplicial]] [[diagram]]

$$
  C(f) : \Delta^{op} \to C
$$

The **coimage** of $f$ is the [[(∞,1)-colimit]] over this diagram

$$
  coim(f) := \lim_\to
    \left(
    \cdots \stackrel{\to}{\stackrel{\to}{\to}} c \times_d c \stackrel{\to}{\to} c
  \right)
$$

See also at _[infinity-image -- As the ∞-colimit of the kernel ∞-groupoid](infinity-image#ViaColimitOfCechNerve)_.

## Properties

* Morphisms for which image and coimage coincide (in a certain sense) are [[strict morphism]]s.

## Examples

### For $(\infty,1)$-coimages

Let $G$ be a [[group]]. In the [[(∞,1)-category]] $C =$ [[∞-Grpd]] we have $G$ as a 0-[[truncated]] [[∞-group]] object as well as its [[delooping]] $\mathbf{B}G$, which is the one-object [[groupoid]] with $G$ as its morphisms.

Then: the coimage of the point inclusion $f : * \to \mathbf{B}G$ is $\mathbf{B}G$ itself.

Because the homotopy-[[Cech nerve]] of the point inclusion is the usual simplicial incarnation of $G$

$$
  C(f) = \left(
    \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
$$

now regarded as a simplicial object in [[∞Grpd]]. Its homotopy colimit is again $\mathbf{B}G$. This follows for instance abstractly from the fact that [[∞Grpd]] is an [[(∞,1)-topos]] and therefore satisfies [[Giraud's axioms]], which say that every [[groupoid object in an (∞,1)-category]] is _effective_ in an [[(∞,1)-topos]].

[[!redirects coimage]]
[[!redirects coimages]]


coinduction

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

#Contents#
* table of contents
{:toc}

## Idea

Coinduction is [[duality|dual]] to [[induction]].  It generalises to [[corecursion]].

It is a method of [[proof]] which relies on the fact that any two states of the [[terminal coalgebra]] [[algebra for an endofunctor|for an endofunctor]] $H$ must be equal if they are indistinguishable under repeated operations of $H$. That is, there are no proper coalgebra [[quotient objects]]. Generally, we show the existence of a [[bisimulation]] between states of terminal coalgebra, that is a relation between states, such that when the coalgebra function is
applied, the respective outputs are still related. Since any bisimulation must be contained within the identity relation, we can then conclude that the states are equal.

## Examples {#Examples}

+-- {: .num_example}
###### Example
Let $\bar{\mathbb{N}}$ be the set of [[conatural numbers]] and take $add\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}} \times \bar{\mathbb{N}}$. We can then establish a bisimulation between the terms $(n + m)$ and $(m + n)$, from which we can conclude that this addition is commutative. (See p. 52 of Rutten [Universal coalgebra: a theory of systems](http://homepages.cwi.nl/~janr/papers/files-of-papers/universal_coalgebra.pdf).)
=--

+-- {: .num_example}
###### Example
Many theorems of calculus involving formal [[power series]] $f(x) = \sum_{n \geq 0} \frac{a_n x^n}{n!}$ are naturally viewed as proofs by coinduction. The collection of power series $k[ [x] ]$ over a [[field]] $k$ (say of [[characteristic]] $0$) may be viewed as a *stream coalgebra* $k^\mathbb{N}$, i.e., the [[terminal coalgebra of the endofunctor]] $k \times -:\; Set \to Set$, with the coalgebra structure

$$\array{
\langle eval_0, D = \frac{d}{d x} \rangle: k[ [x] ] & \to & k \times k[ [x] ] \\
\sum_{n \geq 0} \frac{a_n x^n}{n!} & \mapsto & (a_0, \sum_{n \geq 0} \frac{a_{n+1} x^n}{n!})
}$$

Thus if we set up a bisimulation $\sim$ between two power series $f, g$, establishing $f(0) = g(0)$ and $(D f) \sim (D g)$, we may conclude by coinduction that $f = g$ (else, $k[ [x] ]/\sim$ would be a proper quotient).

As an illustration: consider a field $k$ of characteristic $0$ and, for $r \in k$, define $(1 + x)^r$ to be the power series expansion of $\exp(r \cdot \log(1 + x))$, where $\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$ and $\log(1 + x) = \sum_{n \geq 1} (-1)^{n+1}\; \frac{x^n}{n}$ is the inverse of $\exp(x)-1$. To prove the generalized [[binomial theorem]]

$$(1 + x)^r = f(r) \coloneqq \sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}$$

where $r^\underline{k}$ is the [[falling factorial|falling power]] $r(r-1)\ldots (r-k+1)$, we introduce a bisimilarity

$$\sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}\; \sim\; \exp(r \cdot \log(1 + x))$$

and observe first that the [[constant coefficients]] of $f(x)$ and $(1+x)^r$ are both $1$, and that

$$D\left(\sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}\right) = r \sum_{k \geq 0} \frac{(r-1)^\underline{k} x^k}{k!}$$

(because $r^{\underline{k+1}} = r \cdot (r-1)^{\underline{k}}$) and similarly

$$D\left(\exp(r \cdot \log(1 + x))\right) = r \cdot \exp((r-1) \cdot \log(1 + x))$$

by the [[chain rule]], whence $D(f/r) \sim D((1+x)^r/r)$. Hence $f/r$ is bisimilar to $(1+x)^r/r$, and we conclude in the terminal coalgebra that $f(x) = (1+x)^r$.
=--

## See also

* [[directed (n,r)-graph]]

* [[bisimulation]]

## References
 {#References}

* [[Bart Jacobs]], [[Jan Rutten]], _A tutorial on (Co)Algebras and (Co)Induction_ ([pdf](http://www.cs.ru.nl/~bart/PAPERS/JR.pdf))

* Davide Sangiorgi, _Introduction to Bisimulation and Coinduction_, Cambridge Universtity Press (2012)  ([web](http://www.cs.unibo.it/~sangio/IntroBook.html))

* Davide Sangiorgi, Jan Rutten (eds.), _Advanced Topics in Bisimulation and Coinduction_, Cambridge Universtity Press (2012) ([web](http://www.cs.unibo.it/~sangio/AdvancedBook.html))

Discussion of [[differential calculus]] in terms of coinduction is in

* Martin Escardo, Duško Pavlović, _Calculus in coinductive form_ (1998) ([web](https://ieeexplore.ieee.org/abstract/document/705675) or [pdf](http://dusko.org/wp-content/uploads/2020/03/current.pdf))

About coinduction in [[cubical type theory]]:

* [[Magnus Baunsgaard Kristensen]], [[Rasmus Ejlers Møgelberg]], [[Andrea Vezzosi]], *Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks* ([arXiv:2102.01969](https://arxiv.org/abs/2102.01969))

[[!redirects coinductions]]

[[!redirects coinductive]]

[[!redirects coinductive definition]]

[[!redirects coinductively]]


coinductive definition

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

#Contents#
* table of contents
{:toc}

## Idea

A _coinductive definition_ is a [[definition]] by [[coinduction]].

## Definition

See at _[[coinductive type]]_.

## Related concepts

* [[inductive definition]]

[[!redirects coinductiv definitions]]


coinductive type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _coinductive types_ is [[duality|dual]] to that of _[[inductive types]]_.

## Examples

Examples of coinductive types include:

* streams (infinite sequences);
* [[trees]] with infinite branches;
* [[conatural numbers]].

## Properties

### Categorical semantics

Where the [[categorical semantics]] of an [[inductive type]] is an [[initial algebra for an endofunctor]], the semantics of a coinductive type is a [[terminal coalgebra of an endofunctor]].

### Coinductive type formation in homotopy type theory

There is an obstacle to the complete [[dualization]] of the usual rules for [[inductive types]] in [[homotopy type theory]], including dualizing the [[induction principle]] to a “coinduction principle”. For this some form of “codependent types” would be needed.

> The universal property defining (internal) coinductive types in HoTT is dual to the one defining (internal) inductive types.  One might hence assume that their existence is equivalent to a set of type-theoretic rules dual (in a suitable sense) to those given for external W-types...  However, the rules for external W-types cannot be dualized in a naïve way, due to some asymmetry of HoTT related to dependent types as maps into a “type of types” (a universe) ([ACS15](#ACS15))

However, it is possible instead to dualize the alternative characterization of inductive types as [[initial algebra for an endofunctor|initial algebras]] for a notion of coinductive types as [[terminal coalgebra of an endofunctor|terminal coalgebras]], and that can be formulated (and often constructed) in ordinary HoTT. In ([ACS15](#ACS15)) the authors proceed to construct coinductive types from indexed inductive types.

## Related concepts

* [[coinduction]], [[corecursion]]

* [[coinductive definition]]

* [[inductive type]]

* [[codependent type theory]]

[[!redirects coinductive types]]
[[!redirects codata]]

## References

* {#ACS15} [[Benedikt Ahrens]], [[Paolo Capriotti]], [[Régis Spadotti]], *Non-wellfounded trees in Homotopy Type Theory*, in 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 17-30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015) ([doi:10.4230/LIPIcs.TLCA.2015.17](https://doi.org/10.4230/LIPIcs.TLCA.2015.17), [arXiv:1504.02949](https://arxiv.org/abs/1504.02949))

* _coinductives_, [discussion](https://groups.google.com/forum/#!msg/homotopytypetheory/tYRTcI2Opyo/PIrI6t5me-oJ) on Homotopy Type Theory group.


coinvariant

# Contents
* table of contents
{: toc}

## Idea

The notion of _coinvariant_ is [[duality|dual]] to that of *[[invariant]]*.

## Definitions

### Of group representations

Let $G$ be a [[discrete group]], $k$ a [[commutative unital ring]], $k[G]$ the [[group ring]] of $G$ and $V$ a left $k[G]$-[[module]], hence a linear $G$-[[representation]] over $k$.

Then there is well-defined [[quotient]] $k$-module $V_G = V/\langle{v - g v}\rangle$ called the module of __$G$-coinvariants__. Here $\langle{v - g v}\rangle$ denotes the smallest $k$-[[submodule]] of $V$ containing all expressions of the form $v - g v$ where $g\in G$ and $v\in V$.

Notice that here $v$ and $g v$ are two points on same [[orbit]] of $G$ and so the coinvariants are essentially the [[orbits]] of $G$ in $V$.

### Of coalgebras

Let $C$ be a $k$-[[coalgebra]], $\chi$ a [[group-like element]], that is, an element such that $\Delta_C(\chi)=\chi\otimes\chi$, and $\rho:V\to V\otimes C$ a right $C$-[[coaction]]. Any element $v\in V$ such that $\rho(v)=v\otimes\chi$ is called a __$(\rho,\chi)$-coinvariant element__ in the $C$-comodule $(V,\rho)$. Suppose $H$ is a [[bialgebra]], $A$ an algebra and $\rho:A\to A\otimes H$ a coaction making $A$ into a right $H$-comodule algebra. The unit element $1_H$ is a group-like element, and we call $(\rho,1)$-coinvariants simply __$\rho$-coinvariants__. The subset of $\rho$-coinvariants in $A$ is a subalgebra, called the __subalgebra of coinvariants__.

## Related concepts

* [[invariant]]

* [[norm map]]

* [[homotopy coinvariant functor]]

[[!include homotopy type representation theory -- table]]

[[!redirects coinvariant]]
[[!redirects coinvariants]]


coisotropic calculus

## References

* [[Alan Weinstein]], _Coisotropic calculus and Poisson groupoids_  J. Math. Soc. Japan Volume 40, Number 4 (1988), 705-727. ([EUCLID](http://projecteuclid.org/euclid.jmsj/1230129807))


coisotropic submanifold

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#### Differential geometry
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#Contents#
* table of contents
{:toc}

## Definition

For $(X, \pi)$ a [[Poisson manifold]], a [[submanifold]] $S \hookrightarrow X$ is called **coisotropic** if the restriction of the contraction map with the [[Poisson tensor]]

$$
  \pi \;\colon \; T^* X \to T X
$$

to the [[conormal bundle]] $N^* S \hookrightarrow T^* X|_S$ factors through the [[tangent bundle]] $T S$

$$
  \pi|_{N^* S} \;\colon\; {N^* S} \to T S \hookrightarrow T X
  \,.
$$

Equivalently, $S\hookrightarrow X$ is coisotropic if the subalgebra of $C^\infty(X)$ of functions vanishing on $S$ is closed under the Poisson bracket.

## Properties

### Relation to Poisson Lie algebroids
 {#RelationToPoissonLieAlgebroids}

A [[Poisson manifold]] induces a [[Poisson Lie algebroid]], which is a [[symplectic Lie n-algebroid]] for $n = 1$. Its coisotropic submanifolds correspond to the [[Lagrangian dg-submanifolds]] (see there) of this Poisson Lie algebroid.

## Related concepts

* [[isotropic submanifold]]

* [[Dirac structure]]

* [[coisotropic calculus]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

Surveys include

* Aïssa Wade, _On the geometry of coisotropic submanifolds of Poisson
manifolds_ ([pdf](http://www.math.psu.edu/wade/CAARMS13-proceedings-Jan08.pdf))

The relation to the [[Poisson sigma-model]] is discussed in

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model_,  	Lett. Math. Phys. 69 (2004) 157-175 ([arXiv:math/0309180](http://arxiv.org/abs/math/0309180))
 {#CattaneoFelder03}

Characterization in terms of leaves of [[Lagrangian submanifold|Lagrangian]] [[foliation of a Lie algebroid|foliation of]] the [[Poisson Lie algebroid]] is mentioned in

* [[Pavol Ševera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_ ([arXiv:0105080](http://arxiv.org/abs/math/0105080))
 {#Severa}

and discussed in more detail in section 7.2 of

* [[Eli Hawkins]], _A groupoid approach to quantization_,  J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. ([arXiv:math.SG/0612363](http://arxiv.org/abs/math.SG/0612363))
  {#EH}

Comments on [[higher algebra]] aspects are in the slides

* [[Florian Schätz]], _Homotopical algebra of coisotropic submanifolds_ ([pdf](http://www.math.ist.utl.pt/~fschaetz/poster-Lausanne.pdf))

[[!redirects coisotropic submanifolds]]


coisotropic subspace

## Related concepts

[[!include (co)isotropic subspaces - table]]

[[!redirects coisotropic subspaces]]
[[!redirects co-isotropic subspace]]
[[!redirects co-isotropic subspaces]]


Coisson algebra

#Contents#
* table of contents
{:toc}

## Idea

A _Coisson algebra_ or _vertex Poisson algebra_ is essentially a [[Poisson algebra]] [[internalization|internal to]] [[D-module]]s.

## References

Section 2.6 of

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

[[!redirects Coisson algebras]]

[[!redirects vertex Poisson algebra]]
[[!redirects vertex Poisson algebras]]


cokernel

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

## Idea

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

In a [[category]] with a [[terminal object]] $1$, the **cokernel** of a [[morphism]] $f : A \to B$ is the [[pushout]]

$$
  coker(f)
    \;\coloneqq\;
  1 \underset{A}{\sqcup} B
  \phantom{AAAAAA}
  \array{
    A &\stackrel{\phantom{A}f\phantom{A}}{\longrightarrow}& B
    \\
    \big\downarrow &{}^{(po)}& \big\downarrow
    \\
    1
      &\longrightarrow&
    coker(f)
  }
$$

=--

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

In the case when the terminal object is in fact [[zero object]], one can, more explicitly, characterize the object $coker(f)$ as [[generalized the|the]] object (unique up to unique [[isomorphism]]) that satisfies the following [[universal property]]:

for every object $C$ and every morphism $h \colon B \longrightarrow C$ such that $h \circ f = 0$ is the [[zero morphism]], there is a unique morphism $\phi \colon coker(f) \to C$ such that $h = \phi \circ i$.

=--

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

The notion of cokernel is [[duality|dual]] to that of _[[kernel]]_. A **cokernel** in a [[category]] $\mathcal{C}$ is a [[kernel]] in the [[opposite category]] $\mathcal{C}^{op}$.

=--

## Properties

### Exactness properties

* Taking cokernels is a [[right exact functor]] on arrow-categories.

## Examples

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

In the category [[Ab]] of [[abelian groups]] the cokernel of a morphism $f : A \to B$ is the [[quotient]] of $B$ by the [[image]] (of the underlying morphism of [[sets]]) of $f$.

=--

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

More generally, for $R$ any [[ring]], this is true in the category $R$[[Mod]] of [[modules]]: the cokernel of a morphism is the quotient by its set-theoretic image.

=--

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

In the category [[Grp]] of general (not necessarily abelian) groups, the cokernel is instead the [[quotient group]] by the [[normal closure]] of the image.

=--

The following example is by the very definition of _[[abelian category]]_.

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

In an [[abelian category]] the [[coimage]] of any morphism $f$ is the cokernel of its [[kernel]]

$$
  coim(f) = coker(ker(f))
  \,.
$$

=--

## Related concepts

* [[kernel]]

* [[cocone]]

* [[cofiber]], [[homotopy cofiber]]

[[!redirects cokernels]]


cokernel pair

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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The _cokernel pair_ of a [[morphism]] in a [[category]] is the [[pushout]] of the morphism along itself.

The dual notion is that of [[kernel pair]].

[[!redirects cokernel pairs]]


colax-distributive rig category

# Colax-distributive rig categories

* table of contents
{: toc}

## Idea

A *colax-distributive rig category* is like a [[rig category]], but where the distributive laws hold only laxly.

## Definition

A **right colax-distributive rig category** is a category $C$ equipped with two monoidal structures $(C,\otimes,I)$ and $(C,\oplus,J)$ and natural transformations
$$ (X\oplus Y) \otimes Z \to (X\otimes Z) \oplus (Y\otimes Z) $$
and
$$ J \otimes Z \to J $$
making the functor $(-\otimes Z)$ [[colax monoidal functor|colax monoidal]] with respect to $\oplus$, and perhaps some other coherence equations.  A **left colax-distributive** rig category has instead transformations
$$ Z \otimes  (X\oplus Y) \to (Z\otimes X) \oplus (Z\otimes Y) $$
and
$$ Z \otimes J \to J$$
making $(Z\otimes -)$ colax monoidal, and a simply **colax-distributive** rig category has both.

## Examples

* Of course, a [[rig category]] is a colax-distributive rig category where the distributivity transformations are isomorphisms.

* If $(C,\otimes,I)$ is *any* monoidal category with finite products, then it becomes a colax-distributive rig category with $(C,\oplus,J)$ the cartesian monoidal structure and the distributivity transformations being induced by the universal property of products.

* If $(D,\diamond,\star)$ is a [[duoidal category]] in which $\star$ is compatibly braided (or perhaps symmetric), then the category $CComon_\star(D)$ of cocommutative comonoids in $D$ inherits a monoidal structure from $\diamond$ and has finite products; hence it is a colax-distributive rig category.

## Structures in colax-distributive rig categories

### Rings and near-rings

In a right colax-distributive rig category $(C,\otimes,I,\oplus,J)$, a **[[near-rig]]** is an object $R$ equipped with an $\oplus$-monoid structure $add:R\oplus R \to R$ called "addition", and an $\otimes$-monoid structure $mult : R\otimes R \to R$ called "multiplication", such that the right distributive law:
$$ \array{
    (R\oplus R) \otimes R & \xrightarrow{add\otimes id} &
    R\otimes R & \xrightarrow{mult} &
    R\\
    \downarrow & & & &
    \uparrow^{add}\\
    (R\otimes R) \oplus (R\otimes R) && \xrightarrow{mult \oplus mult} &&
    R\oplus R
  }
  $$
and the right absorption law:
  $$ \array{
    J\otimes R & \to &
    J & \xrightarrow{0} &
    R\\
    & _{0\otimes id}\searrow && \nearrow_\mult \\
    && R\otimes R
  }
  $$
hold.

If also the left distributive and absorption laws hold (which requires $C$ to also be left colax-distributive), then $R$ is a **rig**.  And if $\oplus$ is cartesian and the additive monoidal structure is a group structure, then $R$ is a **(near-)ring**.

In the right colax-distributive category $CComon_\star(D)$ of cocommutative comonoids in a [[duoidal category]] $D$, these agree with the definitions of (cocommutative) (near)-ri(n)gs in $D$ (see [[duoidal category]] for these).

On the other hand, if $C$ is a [[preadditive category|preadditive]] [[distributive monoidal category]] and $\oplus$ is the cartesian monoidal structure (which is also the *cocartesian* structure, so that $C$ is in fact a [[rig category]]), then every object is an $\oplus$-monoid in a unique way, and the distributive and absorption laws are also automatic.  Thus, in this case near-rigs and rigs coincide simply with $\otimes$-monoids.  If $C$ is [[additive category|additive]], then near-rings and rings also coincide with $\otimes$-monoids.

## Related pages

* [[distributivity for monoidal structures]]

[[!redirects colax-distributive rig category]]
[[!redirects colax-distributive rig categories]]
[[!redirects right colax-distributive rig category]]
[[!redirects right colax-distributive rig categories]]
[[!redirects left colax-distributive rig category]]
[[!redirects left colax-distributive rig categories]]

[[!redirects colax-distributive category]]
[[!redirects colax-distributive categories]]
[[!redirects right colax-distributive category]]
[[!redirects right colax-distributive categories]]
[[!redirects left colax-distributive category]]
[[!redirects left colax-distributive categories]]


colax/lax comma object

# Colax/lax comma object

* table of contents
{: toc}

## Idea

For a [[2-monad]] $T$, the 2-category $T Alg$ of $T$-algebras and pseudo $T$-morphisms has weak [[2-limits]] (bilimits), and more precisely [[pie-limits]].  The 2-categories $T Alg_l$ and $T Alg_c$ of [[lax morphism|lax and colax]] $T$-morphisms do not have all 2-limits (even weak ones), but they do have some, particularly when some of the morphisms involved in the diagram are strict or pseudo; see [[rigged limit]] for a characterization of these.  There is no 2-category containing both lax and colax morphisms, but nevertheless some limits of diagrams involving lax and colax morphisms can be given a $T$-algebra structure.  Of these one of the most commonly encountered is a [[comma object]] $(f/g)$ where $f$ is colax and $g$ is lax.  It is unclear exactly how to state a universal property for this comma object, but it is probably related to the [[double category of algebras]].

## Definition

Let $T$ be a (strict, for simplicity) [[2-monad]] on a ([[strict 2-category|strict]], for simplicity) [[2-category]] $K$, and let $f:A\to C$ be a colax $T$-morphism and $g:B\to C$ a [[lax morphism|lax]] $T$-morphism.  Suppose that the ([[strict 2-limit|strict]], for simplicity) [[comma object]] $(f/g)$ exists in $K$; thus it is equipped with projections $p:(f/g)\to A$ and $q:(f/g)\to B$ (which are, so far, only morphisms in $K$) and a 2-cell

$$\array{ (f/g) & \xrightarrow{p} & A\\
^q\downarrow & \swArrow_\alpha & \downarrow^f \\
B & \xrightarrow{g} & C
}$$

that is universal among such 2-cells.  Now consider the following [[pasting diagram|pasting composite]]:

$$\array{
T(f/g) & \to & T A \\
\downarrow & \swArrow_{T\alpha} & \downarrow^{T f} & \searrow \\
T B & \to & T C & \swArrow_{\bar{f}} & A \\
& \searrow & \swArrow_{\bar{g}} & \searrow &\downarrow^f \\
&& B & \xrightarrow{g} & C
}$$

Here $\bar{f}$ is the colax $T$-morphism constraint of $f$, while $\bar{g}$ is the lax $T$-morphism constraint of $g$.  Notice that these go in exactly the right directions for the above pasting to be well-defined.  Now by the universal property of $(f/g)$, there is a unique morphism $T(f/g) \to (f/g)$ such that the above pasting composite is equal to the following one:

$$ \array{
T(f/g) & \to & T A \\
\downarrow & \searrow &  & \searrow \\
T B & & (f/g) & \to & A \\
& \searrow & \downarrow & \swArrow_\alpha &\downarrow^f \\
&& B & \xrightarrow{g} & C
} $$

where the empty quadrilaterals commute.  A similar argument shows that this map $T(f/g) \to (f/g)$ is the action map of a $T$-algebra structure on $(f/g)$, such that the projections $p:(f/g)\to A$ and $q:(f/g)\to B$ are strict $T$-morphisms and $\alpha$ is a 2-cell in the [[double category of algebras]] $T \mathbf{Alg}$.  (In fact, the latter assertion is precisely the equality of the above two pasting diagrams.)

The strictness of the projections $p,q$ is familiar from the behavior of [[rigged limits]] in $T Alg_l$ and $T Alg_c$.  However, it is unclear exactly what universal property this $T$-algebra $(f/g)$ has, although it seems likely to involve the double category $T \mathbf{Alg}$ somehow.

## Examples

* Some [[comma double categories]] are colax/lax comma objects in the double category of double categories (and lax and colax double functors).

[[!redirects oplax/lax comma object]]
[[!redirects oplax/lax comma objects]]
[[!redirects colax/lax comma object]]
[[!redirects colax/lax comma objects]]
[[!redirects lax/oplax comma object]]
[[!redirects lax/oplax comma objects]]
[[!redirects lax/colax comma object]]
[[!redirects lax/colax comma objects]]


Cole's theory of spectrum

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###Context###
#### Topos Theory
#### Higher Algebra
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[[!include topos theory - contents]]
[[!include higher algebra - contents]]
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# Contents
* table of contents
{:toc}

[[!redirects Cole spectrum]]

+-- {: .rightHandSide}
> Devinette: Trouver un point commun entre la Samaritaine et [[SGA4|SGA 4]].[^CM] {#quot}
=--

##Idea##

In the mid 1970s [[Julian Cole]] proposed a [[topos theory|topos-theoretic]] construction of [[spectrum (geometry)|spectra in geometry]] arising in the sense of [[spectrum of a commutative ring]] but for more general [[algebraic theories|(algebraic) theories]], as [[right adjoints]] to [[forgetful functors]] that generalized [[Monique Hakim|M. Hakim's]] approach to [[locally ringed topos|locally ringed toposes]].

Basic ingredients are pairs of [[geometric theories]] $\mathbb{S}$ and $\mathbb{T}$ over the same [[language]] such that $\mathbb{T}$ results from $\mathbb{S}$ by addition of further [[axioms]]. Then $\mathbb{T}-Mod_\mathcal{E}$ is a [[full subcategory]] of $\mathbb{S}-Mod_\mathcal{E}$ and the spectrum construction can be viewed as a sort of generalization of a right adjoint to the inclusion. The quotient relation between the two theories gives the construction a model-theoretic flavor.

###Definition###

Let $\mathbb{T}$ be [[geometric theory]]. The 2-category $\mathbb{T}-\mathfrak{Top}$ of _[[T-modelled toposes]]_ is given as follows:

* objects are pairs $(\mathcal{E},M)$ where $\mathcal{E}$ is a topos and $M$ a $\mathbb{T}$-model in $\mathcal{E}$,

* 1-cells $(\mathcal{F},L)\to (\mathcal{E},M)$ are pairs $(p,f)$ with $p:\mathcal{F}\to\mathcal{E}$ a geometric morphism and $f:p^*M\to L$ a $\mathbb{T}$-model homomorphism, and

* 2-cells $(p,f)\to(q,g)$ are [[natural transformations]] $\eta:p\to q$ such that $f=g\circ\eta_M$.

$\mathbb{T}-\mathfrak{Top}_N$ is the full sub-2-category on pairs $(\mathcal{E},M)$ such that $\mathcal{E}$ has a [[natural numbers object]].

###Definition###

Let $\mathbb{T}$ be a (geometric) quotient theory of $\mathbb{S}$. A class $\mathbb{A}$ of $\mathbb{T}$-model morphisms is called _admissible_ if

* $\mathbb{A}$ is closed under [[inverse image functor|inverse image functors]]: $p^*f\in \mathbb{A}$ for $f\in \mathbb{A}$.

* $\mathbb{A}$ contains all identity morphisms and given $g\in \mathbb{A}$ and composable $f$ : $f\in \mathbb{A}$ iff $gf\in \mathbb{A}$.

* Given an $\mathbb{S}$-model morphism $f:M\to L$ with $L$ a $\mathbb{T}$-model, there exists a factorization $M\overset{q}{\to}M_f\overset{\hat{f}}{\to}L$ with $M_f$ a $\mathbb{T}$-model and $\hat{f}\in \mathbb{A}$, such that any other such factorization $M\overset{r}{\to}P\overset{p}{\to}L$ factors with $gh=\hat{f}$ and $hq=r$ for a unique $h:M_f\to P$. Moreover, this factorization is preserved by inverse image functors.

The sub-category $\mathbb{A}-\mathfrak{Top}$ of $\mathbb{T}-\mathfrak{Top}$ for such an admissible class has 1-cells $(p,f)$ with $f\in \mathbb{A}$.

### Theorem

+--

*Let $\mathbb{S}$ and $\mathbb{T}$ be finitely presented geometric theories such that $\mathbb{T}$ is a quotient theory of $\mathbb{S}$, and let $\mathbb{A}$ be an admissible class of morphisms of $\mathbb{T}$-models. Then the inclusion functor $\mathbb{A}-\mathfrak{Top}_N\to \mathbb{S}-\mathfrak{Top}_N$ has a right adjoint $Spec:\mathbb{S}-\mathfrak{Top}_N\to \mathbb{A}-\mathfrak{Top}_N$.*

=--

###Example###

The classical example is given by the [[geometric theory|geometric theories]] of [[commutative ring|commutative rings]] and [[local ring|local rings]] with the factorization given by the class of local morphisms and appropriate rings of fractions as the local factors. The right adjoint maps a commutative ring $A$ basically to the pair consisting of the sheaf topos on the Zariski spectrum of $A$ and the structure sheaf of $A$ (cf. Johnstone 1977b).

###Remark###
In the context of his work with C. Lair on 'locally free diagrams' [[René Guitart|R. Guitart]] interprets the 'almost-freeness' of the spec construction as an 'almost-algebraicity' of topology (See Guitart 2008 and the references therein).

##Related concepts##

*[[spectrum of a commutative ring]]

*[[locally algebra-ed topos]]

*[[classifying topos]]

*[[geometric theory]]

*[[ringed topos]]

*[[Structured Spaces]]

*[[formally etale morphisms]]

##References##

The original article is reprinted as

* [[Julian Cole]], _The bicategory of topoi and spectra_, Reprints in Theory and Applications of Categories, No. 25 (2016) pp. 1-16 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/25/tr25abs.html))

Besides this, Johnstone's article ([1977a](#Johnstone77a)), the book of Johnstone ([1977b](#Johnstone77b)), on which the above exposition is based, is a good source for this material. Bunge (1981), Bunge and Reyes (1981) apply the results in a model-theoretic context. See also the short remark in Caramello (2014, p.55).

[[Michel Coste|M. Coste]]'s (1979) take on admissible maps flows in Coste-Michon (1981) via the 4-yoga (cf. [above](#quote)) into the more recent approach using open and étale maps pioneered by Joyal. Dubuc (2000) compares his axiomatic étale morphisms to Cole's class of admissible morphisms.

For a glimpse of [[André Joyal|A. Joyal]]'s original approach to the spectrum using distributive lattices see Joyal (1975), Español (1983,1986) or Coquand-Lombardi-Schuster (2007).  For higher categorical variations on the theme _spectrum_ (of a ring) see Lurie (2009).

* [[Marta Bunge|M. Bunge]], _Sheaves and Prime Model Extensions_ , J. Algebra **68** (1981) pp.79-96.

* [[Marta Bunge|M. Bunge]], [[Gonzalo E. Reyes|G. E. Reyes]] , _Boolean spectra and model completions_ , Fund.Math. **113** (1981) pp.165-173. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm113/fm113115.pdf))

* [[Olivia Caramello|O. Caramello]], _Topos-theoretic background_ , ms. 2014. ([pdf](http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf))

* [[Julian Cole|J. C. Cole]], _The Bicategory of Topoi, and Spectra_ , ms. ([pdf](http://www.oliviacaramello.com/Unification/ColeBicategoryTopoiSpectra.pdf))

* [[Thierry Coquand]], Henri Lombardi, Peter Schuster, _The projective spectrum as a distributive lattice_ , Cah. Top. Géom. Diff. Cat. **XLIIX** no.3 (2007) pp.220-228. ([numdam](http://www.numdam.org/item?id=CTGDC_2007__48_3_220_0))

* [[Michel Coste|M. Coste]], _Localisation, spectra and sheaf representation_ , pp.212-238 in Fourman, Mulvey & Scott (eds.), _Applications of Sheaves_ , Springer LNM **753** (1979).

* M. Coste, G. Michon, _Petits et Gros Topos en Géométrie Algébrique_ , Cah. Top. Géom. Diff. Cat. **XXII** no.1 (1981) pp.25-30. ([pdf](http://archive.numdam.org/article/CTGDC_1981__22_1_25_0.pdf)) {#CosteMichon81}

* [[Eduardo Dubuc|E. J. Dubuc]], _Axiomatic etal maps and a theory of spectrum_ , JPAA **149** (2000) pp.15-45.
 {#Dubuc}

* L. Español, _Le spectre d'un anneau dans l'algèbre constructive et applications à la dimension_ , Cah. Top. Géom. Diff. Cat. **XXIV** no.2 (1983) pp.133-144. ([numdam](http://www.numdam.org/numdam-bin/item?id=CTGDC_1983__24_2_133_0))

* L. Español, _El Espectro y la Dimension des Anillos en el Algebra sobre un Topos_ , Revista Colombiana de Matemáticas **XX** (1986) pp.105-128. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002541068))

* [[René Guitart|R. Guitart]], _Toute Théorie est Algébrique et Topologique_ , Cah.Top.Géom.Diff.Cat. **XLIX** no. 2 (2008) pp.83-128. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_2008__49_2/CTGDC_2008__49_2_83_0/CTGDC_2008__49_2_83_0.pdf))

* [[Monique Hakim|M. Hakim]], _Topos Annelés et Schémas Relatifs_ , Springer Heidelberg 1972.

* [[Peter Johnstone|P. T. Johnstone]], _Rings, fields and spectra_ , J. Algebra **49** (1977) pp.238-260. {#Johnstone77a}

* P. T. Johnstone, _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). pp.205-207 {#Johnstone77b}

* [[André Joyal|A. Joyal]], _Les Théorèmes de Chevalley-Tarski et Remarques sur l'Algèbre Constructive_ , Cah. Top. Géom. Diff. Cat. **XVI** (1975) pp.256-258. ([Proceedings Colloque Amiens 1975](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1975__16_3/CTGDC_1975__16_3_217_0/CTGDC_1975__16_3_217_0.pdf), 6.81 MB)

* [[Jacob Lurie|J. Lurie]], _Derived Algebraic Geometry V: [[Structured Spaces]]_ , ms. 2009. ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-V.pdf))

* R. O. Robson, _Model theory and spectra_ , JPAA **63** (1990) pp.301-327.

[^CM]: Coste&Michon (1981), p.27.

[[!redirects Cole spectral theory]]
[[!redirects Cole's Theory of Spectrum]]
[[!redirects Cole spectrum]]


Coleman-De Luccia instanton

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### Context
#### Vacua
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[[!include vacua -- contents]]
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#### Gravity
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## Idea

The local decay process of a [[false vacuum]] in the presence of [[gravity]] ([[general relativity]]), hence the case relevant in [[cosmology]].

## Related concepts

* [[bubble of nothing]]

* [[false vacuum eternal inflation]]

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## References

* {#Coleman77} [[Sidney Coleman]], _Fate of the false vacuum: Semiclassical theory_, Phys. Rev. D 15, 2929 – Published 15 May 1977 ([publisher](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.15.2929))

* [[Sidney Coleman]], Frank De Luccia, _Gravitational effects on and of vacuum decay_ Phys. Rev. D 21, 3305 – Published 15 June 1980 ([publisher](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.3305))

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Coleman-Mandula theorem

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# The Coleman--Mandula Theorem
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## Idea

The __Coleman--Mandula theorem__ is a [[no-go theorem]] motivated by the possibilities of [[Lie group]] symmetries in [[quantum field theory]] in [[Minkowski space-time]].

## Statement

Any [[Lie group]] containing the [[Poincaré group]] $P$ (in 4d) as a [[subgroup]] and containing a maximal [[internal symmetry|internal]] [[symmetry group]] $G$ must be a [[direct product]] of those. In addition, $G$ must be a [[semisimple Lie group]] with additional $U(1)$ ([[circle group]]) factors.

The generalization of this statement to [[super Lie algebras]] is known as the _[[Haag–Łopuszański–Sohnius theorem]]_.

## Remarks

Gel'fand and Likhtman showed that with a slight extension of the concept of Lie group, one can get that $P$ and $G$ combine in a nontrivial way. This happens for example in the [[supersymmetry|supersymmetric]] case.

## References

* [[Sidney Coleman]], [[Jeffrey Ellis Mandula]], _All Possible Symmetries of the S Matrix_, Physical Review __159__ (5): 1251--1256 (1967) ([doi:10.1103/PhysRev.159.1251](https://doi.org/10.1103/PhysRev.159.1251))

* [[I. M. Gel'fand]], E. S. Likhtman, JETP Letters __13__, 323 (1971)

Review includes

* [Wikipedia](http://en.wikipedia.org/wiki/Coleman%E2%80%93Mandula_theorem) (English)

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