core of a ring

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

#Contents#
* table of contents
{:toc}

## Idea

The **core** of a [[commutative ring]] $R$ consists of those [[elements]] for which [[homomorphisms]] out of that ring have no choice as to how to act: that is, those $r \in R$ such that for any [[pair]] of [[homomorphisms]] $f,g \colon R \to S$ we must have $f(r) = g(r)$.   By the [[universal property]] of the [[coproduct]] these are precisely the elements on which the [[coprojections]] $i_1, i_2 \colon R \to R \sqcup R$ are equal.  Since the [[coproduct]] of commutative rings is their [[tensor product]] (see [here](category+of+monoids#PushoutOfCommutativeMonoids)), these are precisely the elements such that

$$
  r \otimes 1
    \;=\;
  1 \otimes r
  \;\; \in \; R \otimes_{\mathbb{Z}} R
  \,.
$$

## Definition

### Core of a ring

\begin{definition}\label{CoreOfARing}
**(core of a ring)** \linebreak
For $R$ a [[unital ring|unital]] [[commutative ring]], its **[[core of a ring|core]]** is the following [[subobject|sub-ring]] of the [[tensor product of abelian groups]] $R \otimes_{{}_{\mathbb{Z}}} R$:

$$
  c R
  \;\coloneqq\;
  \Big\{
    \,
    r \in R
    \,\left\vert\,
    1 \otimes r \,=\, r \otimes 1
    \;\in\; R \otimes_{{}_{\mathbb{Z}}} R
    \right.
    \,
  \Big\}
$$
\end{definition}

([Bousfield & Kan 1972, Sec. 1](#BousfieldKan72))

\begin{remark}\label{CoreAsEqualizerAndAsRegularImage}
**(core as [[equalizer]] and as [[regular image]])** \linebreak
In [[category theory|category-theoretic]] terminology, Def. \ref{CoreOfARing} describes the [[equalizer]] ([Bousfield 1979, 6.4](#Bousfield79))

$$
  c R
    \xrightarrow{\;equ\;}
  R
    \overset{\phantom{AAAAAA}}\rightrightarrows
  R \otimes_{{}_{\mathbb{Z}}} R
  \,,
$$

where the top [[morphism]] is

$$
  R
    \;\simeq\;
  \mathbb{Z} \otimes R
    \xrightarrow{e \otimes id}
  R \otimes R
$$

and the bottom one is

$$
  R
    \;\simeq\;
  R \otimes \mathbb{Z}
    \xrightarrow{\;id \otimes e\;}
  R \otimes R
  \,,
$$

with

\[
  \label{UniqueRingHomomorphismFromTheIntegers}
  \array{
    \mathbb{Z}
    &\xrightarrow{ \;e\; } &
    R
    \\
    1 &\mapsto& 1
  }
\]

denoting the unique [[ring homomorphism]] form the [[commutative ring]] of [[integers]], which is the [[initial object]] in [[CommutativeRings]].

Observing (by [this Prop.](CRing#CoproductIsTensorProduct))
that the [[tensor product of abelian groups]] $R \otimes_{{}_{\mathbb{Z}}} R$ equipped with its canonically induced [[commutative ring]] [[mathematical structure|structure]] is the [[coproduct]] in [[CommutativeRings]]

$$
  R \otimes_{{}_{\mathbb{Z}}} R
    \;\simeq\;
  R \sqcup R
$$

this means equivalently that the core is the [[equalizer]] of the two [[coprojections]] into the [[coproduct]]:

$$
  c R
    \xrightarrow{\;equ\;}
  R
    \overset{\phantom{AAAAAA}}\rightrightarrows
  R \sqcup R
    \;\simeq\;
  R \underset{\mathbb{Z}}{\coprod} R
  \,,
$$

hence -- since $\mathbb{Z}$ is the [[initial object]] in [[CommutativeRings]] -- into the [[cofiber coproduct]] of $\mathbb{Z} \xrightarrow{e} R$ (eq:UniqueRingHomomorphismFromTheIntegers) with itself.

In this form, the core is manifestly ([here](image#AsEqualizer)) the *[[regular image]]* of the initial morphism $\mathbb{Z} \xrightarrow{e} R$:

$$
  c R
    \;\simeq\;
  Im_{reg}
  \left(
    \mathbb{Z} \xrightarrow{e} R
  \right)
    \xhookrightarrow{\phantom{AA}}
  R
  \,,
$$

hence the smallest [[regular monomorphism]] into $R$ in the [[category]] of [[CommutativeRings]].
\end{remark}

\begin{remark}\label{DualInterpretation}
**(geometric interpretation)** \linebreak
By [[duality between algebra and geometry]], we may think of the [[opposite category]] $CommutativeRings^{op}$ as that of [[affine scheme|affine]] [[arithmetic schemes]]. Here for $R \in CRing$ we write $Spec(R)$ for the same object, but regarded in $CRing^{op}$.

So the [[initial object]] $\mathbb{Z}$ in [[CRing]] becomes the [[terminal object]] [[Spec(Z)]] in $CRing^{op}$, and so for every $R$ there is a unique morphism

$$
  Spec(R) \longrightarrow Spec(\mathbb{Z})
$$

in $CRing^{op}$, exhibiting every affine [[arithmetic scheme]] $Spec(R)$ as equipped with a map to the base scheme [[Spec(Z)]].

Since the [[coproduct]] in [[CRing]] is the [[tensor product]] of rings ([prop.](CRing#CoproductIsTensorProduct)), this is the dually the [[Cartesian product]] in $CRing^{op}$ and hence

$$
  Spec(R \otimes R) \simeq Spec(R) \times Spec(R)
$$

exhibits $R \otimes R$ as the ring of functions on $Spec(R) \times Spec(R)$.

Hence the terminal morphism $Spec(R) \to Spec(\mathbb{Z})$ induces the corresponding [[Čech groupoid]] [[internal groupoid|internal]] to $CRing^{op}$

$$
  \array{
    Spec(R) \times Spec(R) \times Spec(R)
    \\
    \downarrow
    \\
    Spec(R) \times Spec(R)
    \\
    {}^{\mathllap{s}}\downarrow \uparrow \downarrow^{\mathrlap{t}}
    \\
    Spec(R)
  }
  \,.
$$

This exhibits $R \otimes R$ (the ring of functions on the scheme of morphisms of the [[Čech groupoid]]) as a [[commutative Hopf algebroid]] over $R$.

Moreover, the arithmetic scheme of [[isomorphism classes]] of this groupoid is the [[coequalizer]] of the [[source]] and [[target]] morphisms

$$
  Spec(R) \times Spec(R)
   \underoverset
     {\underset{s}{\longrightarrow}}
     {\overset{t}{\longrightarrow}}
     {\phantom{AA}}
   Spec(R)
     \overset{coeq}{\longrightarrow}
   Spec(c R)
  \,,
$$

also called the _[[coimage]]_ of $Spec(R) \to Spec(\mathbb{Z})$. Since [[limits]] in the [[opposite category]] $CRing^{op}$ are equivaletly colimits in $CRing$, this means that the ring of functions on the scheme of [[isomorphism classes]] of the [[Čech groupoid]] is precisely the core $c R$ or $R$ according to def. \ref{CoreOfARing}.

This is morally the reason why for $E$ a [[homotopy commutative ring spectrum]] the core $c \pi_0(E)$ of its underlying ordinary ring in degree 0 controls what the $E$-[[Adams spectral sequence]] converges to ([Bousfield 79, theorems 6.5, 6.6](#Bousfield79), see [here](Adams+spectral+sequence#ConvergenceStatements)), because the $E$-Adams spectral sequence computes [[E-nilpotent completion]] which is essentially the analog in [[higher algebra]] of the above story: namely the coimage ([(infinity,1)-image](infinity-image#ViaColimitOfCechNerve)) of $Spec(E) \to $ [[Spec(S)]] (see [here](Adams+spectral+sequence#DefinitionInHigherAlgebra)).

\end{remark}

### Solid rings

\begin{definition}\label{SolidRing}
**(solid rings)** \linebreak
A [[commutative ring]] $R$ which is [[isomorphism|isomorphic]] to its [[core of a ring|core]] (Def. \ref{CoreOfARing}), $R \,\simeq\, c R$ is called a **solid ring**.
\end{definition}
([Bousfield-Kan 72, §1, def. 2.1](#BousfieldKan72), [Bousfield 79, 6.4](#Bousfield79))

\begin{proposition}\label{SolidityMeansThatMultiplicationIsAnIsomorphism}
**([[solid ring|solidity]] means that [[multiplication]] is [[isomorphism]])** \linebreak
  A [[commutative ring]] $R$ is solid (Def. \ref{SolidRing}) iff its [[multiplication]] morphism is an [[isomorphism]]:
$$
  R\; \text{is solid}
  \;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;
  R \otimes_{{}_{\mathbb{Z}}} R
    \underoverset
      {\simeq}
      {\;(-) \cdot (-)\;}
      {\longrightarrow}
  R
  \,.
$$
\end{proposition}
([Bousfield & Kan 1972, 2.4](#BousfieldKan72) -- this is called a *T-ring* in [Bowshell & Schultz 1977, Def. 1.6](#BowshellSchultz77))
\begin{proof}
In one direction, assume that multiplication is an isomorphism. Since both
$r \mapsto 1 \otimes r$ and $r \mapsto r \otimes 1$ are [[right inverses]]

\begin{tikzcd}
    R
    \ar[
      rr,
      shift left=3pt,
      "{ r \,\mapsto\, 1 \otimes r }"{above}
    ]
    \ar[
      rr,
      shift right=3pt,
      "{ r \,\mapsto\, r \otimes 1 }"{below}
    ]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=-11pt]\tikztostart.south)
           --node[below]{
               \scalebox{.7}{$
                 \mathrm{Id}
               $}
             } ([yshift=-9pt]\tikztotarget.south)
           -- (\tikztotarget.south)}
    ]
    &&
    R \otimes_{{}_{\mathbb{Z}}} R
    \ar[
      rr,
      " (-) \cdot (-) "{above},
      "\sim"{below, yshift=1pt}
    ]
    &&
    R
    \mathrlap{\,,}
\end{tikzcd}

and since right inverses of isomorphisms are unique ([this Prop.](inverse#InversesAreUnique)), it follows that these two morphisms on the left are in fact equal, and therefore that their equalizer, hence the core of $R$ (by Rem. \ref{CoreAsEqualizerAndAsRegularImage}), is $R$.

In the other direction, assume that $1 \otimes r \,=\, r \otimes 1$ for all $r \in R$. Then
$$
  \begin{aligned}
    r_1 \,\otimes\, r_2
    & \;=\;
    \big( r_1 \,\otimes\, 1 \big)
      \cdot
    \big( 1 \otimes r_2 \big)
    \\
    & \;=\;
    \big( 1 \,\otimes\, r_1 \big)
      \cdot
    \big( 1 \otimes r_2 \big)
    \\
    & \;=\;
    1
      \,\otimes\,
    \big(
      r_1 \cdot r_2
    \big)
    \,,
  \end{aligned}
$$
and therefore $(-)\cdot(-)$ is a [[bijection]], hence an [[isomorphism]] of rings.
\end{proof}

## Properties

\begin{prop}\label{CoresAreSolid}
**(cores are solid)** \linebreak
The core (Def. \ref{CoreOfARing}) of any ring $R$ is solid (Def. \ref{SolidRing}):
$$
  c c R
    \;\simeq\;
  c R
  \,.
$$
\end{prop}
([Bousfield-Kan 72, prop. 2.2](#BousfieldKan72))

## Examples

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

The following is the complete list of solid rings (Def. \ref{SolidRing}) up to [[isomorphism]]:

1. The [[localization of a ring|localization]] of the ring of [[integers]] at a [[subset]] $Pr$ of [[prime numbers]]

   $$
     \mathbb{Z}\big[Pr^{-1}\big]
     \,;
   $$

1. the [[integers modulo n]]

   $$
     \mathbb{Z}/n\mathbb{Z}
   $$

   for $n \in \mathbb{N}$, $n \geq 2$;

1. the [[Cartesian product|product]] rings

   $$
     \mathbb{Z}[J^{-1}]
       \times
     \mathbb{Z}/n\mathbb{Z}
     \,,
   $$

   for $n \geq 2$ such that each [[prime factor]] of $n$ is contained in the set of primes $J$;

1. the ring cores of product rings

   $$
     c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)})
     \,,
   $$

   where $K \subset J$ are infinite sets of primes and $e(p)$ are positive natural numbers.

=--

([Bousfield-Kan 72, prop. 3.5](#BousfieldKan72), [Bousfield 79, p. 276](#Bousfield79))

In particular:

\begin{example}\label{RationalNumbersFormASolidRing}
**([[rational numbers]] form a [[solid ring]])** \linebreak
The ring $\mathbb{Q}$ of [[rational numbers]] is a [[solid ring]].
\end{example}
\begin{proof}
  Since every [[rational number]] may be written as
  $$
    r \,=\,q / p \,=\, \tfrac{1}{p} \cdot q \,\in\, \mathbb{Q}
  $$
  for some $p, q \,\in\, \mathbb{Z}$,
  we have
  $$
    \begin{aligned}
        1 \otimes r
      &
      \;=\;
      1
        \,\otimes\,
      \big( \tfrac{1}{p} \cdot q \big)
      \\
      & \;=\;
      \big( \tfrac{1}{p} \cdot p \big)
        \,\otimes\,
      \big( \tfrac{1}{p} \cdot q \big)
      \\
      & \;=\;
      \tfrac{1}{p}
        \,\otimes\,
      \big(
        p \cdot \tfrac{1}{p} \cdot q
      \big)
      \\
      & \;=\;
      \tfrac{1}{p}
        \,\otimes\,
      \big(
        q \cdot 1
      \big)
      \\
      & \;=\;
      \big(
        \tfrac{1}{p} \cdot q
      \big)
        \,\otimes\,
      1
      \\
      & \;=\;
      r \,\otimes\, 1
      \;\;\;\;\;\;
      \in
      \;
      \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q}
      \,.
    \end{aligned}
  $$
  Here the third and the fifth line use the [[equivalence relation]] defining the [[tensor product of abelian groups]] for the case of rings:
  $$
    r_1
      \,\otimes\,
    \big( n \cdot r_2 \big)
      \,=\,
    \big(
      r_1
      \,\cdot\,
      n
    \big)
      \,\otimes\,
    r_2
    \;\;\;
    \in
    \;
    \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q}
    \,,
    \;\;\;\;
    \text{for}\; n \,\in\, \mathbb{Z}
    \,.
  $$
\end{proof}

\begin{remark}\label{NoOtherCharZeroFieldIsSolid}
**(no other [[field]] of [[characteristic zero]] is a [[solid ring]])** \linebreak
  The same argument shows that no other [[field]] of [[characteristic zero]] $k$ is a solid ring, since, being a proper [[superset]] $\mathbb{K} \supset \mathbb{Q}$, it contains elements $k \in \mathbb{K}$ for which there are *no* pairs of [[integers]] $q$, $p$ such that $p \cdot k \,=\, q$.
In particular, the [[real numbers]] are not a [[solid ring]].

This is, ultimately, the reason why the [[derived adjunction|derived]] [[PL de Rham complex|PL de Rham]]-[[Quillen adjunction between simplicial sets and connective dgc-algebras]] is [[idempotent adjunction|idempotent]] only over $k = \mathbb{Q}$ (where it models [[rationalization]] of [[homotopy types]], see at *[[fundamental theorem of dg-algebraic rational homotopy theory]]*); away from this case other tools are needed; see also at *[[real homotopy theory]]*.
\end{remark}

## Related concepts

* [[nilpotent completion of spectra]]

* [[Adams spectral sequence]]

* [[idempotent monoid in a monoidal category]]

## References

The original articles:

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _The core of a ring_, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (doi:10.1016/0022-4049(72)90023-0)

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology 18 (1979), no. 4, 257--281. ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

The concept re-appears under the name "T-rings" in

* {#BowshellSchultz77} R. A. Bowshell and P. Schultz, *Unital rings whose additive endomorphisms commute*, Mathematische Annalen volume 228, pages 197–214 (1977) ([doi10.1007/BF01420290](https://doi.org/10.1007/BF01420290))

and under the name "$\mathbb{Z}$-epimorphs" in:

* {#DicksStephenson84} Warren Dicks, W. Stephenson, *Epimorphs and Dominions of Dedekind Domains*, Journal of the London Mathematical Society, Volume s2-29, Issue 2, April 1984, Pages 224–228 ([doi:10.1112/jlms/s2-29.2.224](https://doi.org/10.1112/jlms/s2-29.2.224))

Generalization to [[monoids in monoidal categories]]:

* {#Gutierrez13} [[Javier J. Gutiérrez]], *On solid and rigid monoids in monoidal categories*, Applied Categorical Structures 23, no. 2 (2015), 575-589 ([arXiv:1303.5265](https://arxiv.org/abs/1303.5265))

[[!redirects cores of rings]]

[[!redirects solid ring]]
[[!redirects solid rings]]


core-natural transformation

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

# Core-natural transformations
* table of contents
{: toc}

## Idea

A *canonical morphism* is a [[morphism]] which is equivariant under an [[automorphism group]], with [[actions]] mediated by two [[functors]].  It may be seen as a [[natural transformation]] in a particular context.  Conversely, canonical morphisms can be combined into a notion of transformation more general than a natural transformation, although this also may be seen as a natural transformation in a particular context.

This usage of the word 'canonical' is due to [[Jim Dolan]].  In general, this term is often used in mathematics to mean that the result of a construction may be specified using only the data at hand, without making arbitrary choices.  The idea behind using the word here is, roughly, that only canonical morphisms may be specified (in the situations in which they appear) without violating the [[principle of equivalence]].  (Both avoiding arbitrary choices and avoiding violation of the [[principle of equivalence]] are related to avoiding the [[axiom of choice]], but that does not seem to be directly relevant.)  However, there are certainly also uses of 'canonical' in mathematics that do not fall under this definition.

Arguably, 'natural' would be a better term for this intuition, but canonical morphisms are more general than the natural transformations that appear in the same contexts, so that word is taken.  Another possible term is 'core-natural' or 'groupoid-natural', since (as will be seen below) canonical morphisms may be interpreted as natural transformations between functors restricted to the [[core]] (underlying [[groupoid]]) of a given category.  The terms 'basis/coordinate--free/invariant' and 'generally covariant' also capture the same intuition, although these tend to be restricted to certain disciplines (linear algebra, geometry, physics); see also [[principle of equivalence]], [[definable set]] (in [[model theory]]), and [[parametric polymorphism]] (in [[type theory]]).

## Definitions

Given [[categories]] $C$ and $D$, [[functors]] $F, G\colon C \to D$, and an [[object]] $x$ of $C$, a __canonical morphism__ from $F(x)$ to $G(x)$ is a [[morphism]] $h\colon F(x) \to G(x)$ in $D$ such that the diagram

$$ \array {
   F(x)            & \overset{h}\to  & G(x) \\
   F(u) \downarrow &                 & \downarrow G(u) \\
   F(x)            & \underset{h}\to & G(x)
} $$

commutes for any [[automorphism]] $u$ of $x$ in $C$.

As $F$ and $G$ are only ever applied to [[isomorphisms]], this definition makes sense even when they are defined only on the [[core]] of $C$.  In fact, as they are applied only to $x$ and its automorphisms, the definition makes sense when they are defined only on $Aut_C(x)$, the [[automorphism group]] of $x$ in $C$.  In that case, $F$ and $G$ are [[representations]] of $Aut_C(x)$ in $D$, and a canonical morphism is precisely an [[intertwiner]] between these representations.

In the other direction, we can consider a family of canonical morphisms, one for each object of $C$, which are coherent in the sense that

$$ \array {
   F(x)            & \overset{h_x}\to  & G(x) \\
   F(u) \downarrow &                   & \downarrow G(u) \\
   F(y)            & \underset{h_y}\to & G(y)
} $$

commutes for every [[isomorphism]] $u\colon x \to y$ in $C$.  Such a family may be called a __canonical transformation__ from $F$ to $G$, although this should not be confused with a [[canonical coordinate transformation]].  Note that every natural isomorphism is canonical, but not conversely.  The main use of having a term like 'canonical' at all is to say that an expression for a morphism in $D$, involving a variable for an object in $C$, is 'canonical' in that variable, as a weaker condition than saying that the expression is 'natural' in that variable.

By the [[axiom of choice]], if there exists a canonical morphism from $F(x)$ to $G(x)$ for every object $x$, then there exists a canonical transformation from $F$ to $G$ (the converse is obvious).  Actually, this does not require the full axiom of choice; it uses only ... (that groupoid-relevant version, I need to find its name).

## Alternative characterisations
{#asNatural}

As remarked above, [[intertwiner|intertwiners]] between [[representations]] of [[groups]] may be seen as canonical morphisms between functors defined on (the [[delooping]] of) a group.  Conversely, we may *define* a canonical morphism from $F(x)$ to $G(x)$ to be an intertwiner between the restrictions of $F$ and $G$ to the [[automorphism group]] of $x$, thought of as representations of that group.

Simlarly, we may *define* a canonical transformation from $F$ to $G$ to be a [[natural transformation]] between the restrictions of $F$ and $G$ to the [[core]] $\tilde{C}$ of $C$.  This is the origin of the alternative term 'core-natural transformation'.

Finally, we note that just as every natural transformation between functors $F,G\colon C \to D$ defines a [[functor]] from $C$ to the [[arrow category]] $Arr D$ (and conversely), so a canonical transformation between such functors defines a functor from $\tilde{C}$ to $Arr D$ (and conversely if we allow the functors to be defined only on isomorphisms).

## Examples

Let $G$ be a [[group]], let $C$ be the [[delooping]] $\mathbf{B}G$ of $G$ (that is $G$ thought of as a $1$-object category), let $x$ be the object of $C$.  Then the functors $F,G\colon C \to D$ are simply [[representations]] of $G$ in $D$, and a canonical morphism from $F(x)$ to $G(x)$ is an [[intertwiner]] between these representations.

More generally, let $C$ be a [[groupoid]].  Then a canonical (or core-natural) transformation from $F$ to $G$ is simply a [[natural transformation]] from $F$ to $G$, which should make sense since $C$ is its own [[core]].  If $C$ is any category whatsoever, then a natural transformation from $F$ to $G$ is still a canonical transformation, although in general there are other canonical transformations.

We now consider an example that may serve to further motivate the term 'canonical'.  Given a $2$-element [[set]] $x$, there are $4$ [[functions]] from $x$ to itself, the [[identity function]], the non-identity [[involution]], and the two [[constant functions]].  However, there is no way to specify either constant function without using some property of the specific elements of the set in question; there is no 'canonical' way to define either of those.  Correspondingly, if we take $C$ and $D$ to be [[FinSet]], $F$ and $G$ to be the [[identity functor]] on $Fin Set$, and $x$ to be this $2$-element, then there are only $2$ canonical morphisms from $F(x) = 2$ to $G(x) = 2$: the identity function and the non-identity involution.

If we wish to extend this example to an entire canonical transformation between these $F,G \coloneqq id_{Fin Set}$, then this is actually the only choice that we can make.  This is because, if $x$ has $3$ or more elements, then we have no way to choose among the $2$ or more elements that are not equal to any given element, so only the identity function on $x$ is canonical.  Thus we have $2$ core-natural transformations from $id_{Fin Set}$ to itself, only one of which (the [[identity natural transformation]]) is natural.

On the other hand, if we let $C$ be the category $Fin Ord$ of finite [[well-ordered sets]], then every element of every such set is uniquely identifiable.  Accordingly, given any well-ordered set $x$, every function from (the underlying set of) $x$ to itself is a canonical morphism from $F(x)$ to $G(x)$, where now $F,G$ are both the [[forgetful functor]] from $Fin Ord$ to $Fin Set$.  Now there are infinitely many canonical transformations from this functor to itself, but still only the identity transformation is natural.

Now consider the operation of [[ordinal number|ordinal]] addition on $Fin Ord$.  We add two well-ordered sets $x$ and $y$ by taking their [[disjoint union]] as sets and placing the elements of $x$ before the elements of $y$.  This operation is associative up to a unique isomorphism, which is natural and so makes $Fin Ord$ into a [[monoidal category]].  The operation is also commutative up to a unique isomorphism, but the commutativity transformation is not natural.  Nevertheless, it is canonical (as it must be, being unique).

The examples above are all of canonical *isomorphisms*.  However, we can adjust the last example for a canonical transformation that is not an isomorphism.  Consider two functors from $Fin Ord \times Ord$ to $Ord$, given by ordinal addition in the two possible orders.  The canonical isomorphism in the last paragraph exends to a canonical transformation from $(x, y \mapsto x + y)$ to $(x, y \mapsto y + x)$.  As the canonical isomorphism above encapsulates the equation $x + y = y + x$ for finite ordinal numbers, so this canonical transformation encapsulates the equation $x + y \leq y + x$ for ordinal numbers when $x$ is finite.

## References

An analogous notion for higher functors makes a prominent appearance in [[Chris Schommer-Pries]]'s work on [[FQFT]] with defects/[[bi-brane]]s. See _slide 81_ (the penultimate page) of:

* [[Chris Schommer-Pries]], _Topological defects and classifying local topological field theories in low dimension_ ([pdf](http://sites.google.com/site/chrisschommerpriesmath/Home/Slides-MFO-6-11-09.pdf?attredirects=0))

See [[holographic principle of higher category theory]] for more on that.

On the terminology 'canonical', see:

* A 1993 [Usenet thread](http://groups.google.com/group/sci.math/browse_frm/thread/9f5ca96dc1a95ada/47a1bb21f6566547) may be the first public introduction of the contrast between 'natural' and 'canonical' by [[Jim Dolan]] (posting as 'Robert Scott'); see particularly posts 9&10.

* A 2010 [MathOverflow question](http://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical) about the meaning of 'canonical', with many different answers, including this one.

[[!redirects canonical map]]
[[!redirects canonical maps]]
[[!redirects canonical morphism]]
[[!redirects canonical morphisms]]

[[!redirects core-natural transformation]]
[[!redirects core-natural transformations]]
[[!redirects groupoid-natural transformation]]
[[!redirects groupoid-natural transformations]]


coreader comonad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given an object $W$ in a [[category]] $\mathcal{C}$, if all binary [[cartesian products]] with $W$ exist, then taking the cartesian product with $W$ assembles into a [[comonad]]

$$
  W \times (-)
  \;\colon\;
  \mathcal{C} \longrightarrow \mathcal{C}
$$

whose [[comultiplication]] is induced by the [[diagonal map]] on $W$ and whose [[counit of a comonad|counit]] is induced by the [[terminal object|terminal map]] $W \to \ast$.

When $W$ is furthermore equipped with the [[structure]] of a [[monoid]], then $W \times (-)$ also canonically inherits the structure of a [[monad]]. (Regarded as a [[monad in computer science]] this models the aggregation of a program's $W$-typed outputs, corresponding to a sort of side channel.) Equipped with this [[monad]]-structure, the operation $W \times (-)$ is known as the *[[writer monad]]*, see there for more.

On the other hand, if $W$ is further [[exponentiable object|exponentiable]] then $W \times (-)$ is [[left adjoint]] to the [[reader monad]] $W \to -$, so that it is known as the **reader comonad** (eg. in the [[Haskell]] documentation for *[Control.Comonad.Reader](https://hackage.haskell.org/package/category-extras-0.53.0/docs/Control-Comonad-Reader.html)*) or **coreader comonad** (e.g. [Ahman & Uustalu (2019)](#AhmanUustalu19)). It is also known as the **product comonad** (e.g. [Uustalu & Vene 2008, p. 270](#UustaluVene08)) and the **environment comonad** (eg. in the [[Haskell]] documentation for *[Control.Comonad.Env](https://hackage.haskell.org/package/comonad-5.0.8/docs/Control-Comonad-Env.html)*).

Coreader comonads in $\mathcal{C}^op$ are [[exception monads]] in $\mathcal{C}$, because products turn into [[coproducts]].

## Properties

### As an Indexed Comonad

If all binary cartesian products in $C$ exist, then the coreader comonad extends to a $C$-indexed comonad on $C$, i.e. a functor $C \to Comonad$. If $C$ has a [[terminal object]] $1$, then the environment comonad $1 \times -$ is isomorphic to the identity comonad.

### Generalization to Tensor Products

If $C$ is a [[monoidal category]] rather than having all binary products, then tensoring with an object $\Gamma \otimes -$ can be given an analogous comonad structure when $\Gamma$ has a [[comonoid]] structure. In a [[cartesian monoidal category]] every object has a unique comonoid structure, which is what induces the coreader comonad. Even without assuming objects carry comonoid structures, the action sending $\Gamma$ to the functor $\Gamma \otimes -$ defines a strong [[monoidal functor]] from $C$ to the monoidal category of endofunctors $[C,C]$, a degenerate kind of [[graded monad|graded comonad]].

### Relation to reader monad and state monad

In a [[cartesian closed category]]/[[type theory]] $\mathcal{C}$, the coreader comonad $W\times (-)$ is [[left adjoint]] to the [[reader monad]] $[W,-]$.

The composite of coreader comonad followed by reader monad is the [[state monad]].

### In terms of dependent type theory

If the type system is even a [[locally Cartesian closed category]]/[[dependent type theory]] then for each type $W$ there is the [[base change]] [[adjoint triple]]

$$
  \mathcal{C}_{/W}
    \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}}
  \mathcal{C}
$$

In terms of this then the coreader comonad is equivalently the composite

$$
  \sum_W W^\ast = W\times (-) \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}
$$

of [[context extension]] followed by [[dependent sum]].

One may also think of this as being the [[integral transform]] through the span

$$
  \ast \leftarrow W \rightarrow \ast
$$

(with trivial kernel) or as the [[polynomial functor]] associated with the span

$$
  \ast \leftarrow W \stackrel{id}{\rightarrow} W \rightarrow \ast
  \,.
$$

## Coalgebras

A [[coalgebra of a comonad|coalgebra]] of the coreader comonad $W \times -$ is equivalent to simply a morphism $A \to W$ for $A$ the carrier of the co-algebra. Furthermore, the category of coalgebras is the isomorphic to the [[slice category]] over $W$. This shows that if all cartesian products with $W$ exist, then the forgetful functor $\Sigma_W \colon C / W \to C$ is [[comonadic]].

## Related concepts

[[!include reader-writer (co)monads -- table]]

* [[maybe monad]]

* [[continuation monad]]

* [[state monad]]

* [[exception monad]]

## References

Eg.

* {#AhmanUustalu19} [[Danel Ahman]], [[Tarmo Uustalu]], p. 3 of: *Decomposing Comonad Morphisms*, CALCO 2019, Leibniz International Proceedings in Informatics (LIPIcs) **139** (2019) [[doi:10.4230/LIPIcs.CALCO.2019.14](https://doi.org/10.4230/LIPIcs.CALCO.2019.14)]

* [[Tarmo Uustalu]], p. 13 of: *Monads and Interaction Lecture 3*, lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021) [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs3.pdf), [[Uustalu-Monads3.pdf:file]]]

* {#UustaluVene08} [[Tarmo Uustalu]], [[Varmo Vene]], *Comonadic Notions of Computation*, Electronic Notes in Theoretical Computer Science **203** 5 (2008) 263-284 [[doi:10.1016/j.entcs.2008.05.029](https://doi.org/10.1016/j.entcs.2008.05.029)]

In [[Haskell]]:

* *[Control.Comonad.Reader](https://hackage.haskell.org/package/category-extras-0.53.0/docs/Control-Comonad-Reader.html)*

See also:

* {#Wadler92} [[Philip Wadler]], _[The essence of functional programming](https://page.mi.fu-berlin.de/scravy/realworldhaskell/materialien/the-essence-of-functional-programming.pdf)_, 1992.

* {#Verdier14} [[Olivier Verdier]], _[The Reader and Writer Monads and Comonads](http://www.olivierverdier.com/posts/2014/12/31/reader-writer-monad-comonad/)_, 2014

[[!redirects reader comonad]]
[[!redirects reader comonads]]

[[!redirects coreader comonad]]
[[!redirects coreader comonads]]

[[!redirects co-reader comonad]]
[[!redirects co-reader comonads]]

[[!redirects product comonad]]
[[!redirects environment comonad]]

[[!redirects writer comonad]]
[[!redirects writer comonads]]


corecursion

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

# Corecursion
* table of contents
{: toc}

## Idea

Corecursion exploits the existence of a morphism from a [[coalgebra for an endofunctor]] to a [[terminal coalgebra]] for the same endofunctor to define an operation.  It is [[duality|dual]] to [[recursion]]. See also [[coinduction]].

## Examples

\begin{example}
For the endofunctor $H(X) = 1 + X$ on [[Set]], the terminal coalgebra is $\bar{\mathbb{N}}$, the [[extended natural number system]]. Define a function $add\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}} \times \bar{\mathbb{N}}$:

$$
   add(n, m) =
   \begin{cases}
   (pred(n), m) & if\; n \gt 0; \\
   (0, pred(m)) & if\; n = 0,\; m \gt 0; \\
   * & if\; m = n = 0,
   \end{cases}
$$

where $pred(x)$ is as defined at [[extended natural number]].

Then $(\bar{\mathbb{N}} \times \bar{\mathbb{N}}, add)$ is an $H$-coalgebra. The unique coalgebra morphism ${+}\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to \bar{\mathbb{N}}$ (to the terminal coalgebra $\bar{\mathbb{N}}$) is addition on the extended natural numbers.  From this definition, we may read off these basic facts about $+$:

* $n + m \gt 0$ with $pred(n + m) = pred(n) + m$ if $n \gt 0$,
* $n + m \gt 0$ with $pred(n + m) = 0 + pred(m)$ if $n = 0$ and $m \gt 0$,
* $n + m = 0$ if $n = 0$ and $m = 0$.

(From the last two, it's immediate to prove by [[coinduction]] that $0 + m = m$ for all $m$.)
\end{example}

## Reference

* [[Jiří Adámek]], _Introduction to Coalgebra_, Theory and Applications of Categories **14** 8 (2005) 157-199 [[tac:14-8](http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/14/8/14-08.pdf)]

* Lawrence Moss, Norman Danner, _On the Foundations of Corecursion_ ([journal](http://jigpal.oxfordjournals.org/content/5/2/231), [pdf](ftp://ftp.cs.indiana.edu/pub/techreports/TR444.pdf))

[[!redirects corecursion]]
[[!redirects corecursive definition]]
[[!redirects corecursive definitions]]


coreduced object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a context of [[synthetic differential geometry]]/[[differential cohesion]] a _coreduced object_ is one all whose [[infinitesimal space|infinitesimal]] paths are constant.
Compare the [[discrete object|discrete objects]], in which all paths are constant, meaning all discrete objects are also coreduced.

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]].

A **coreduced object** or **coreduced type** is one in the [[full subcategory]] defined by the [[infinitesimal shape modality]] $\Im$ or equivalently the [[infinitesimal flat modality]] $\&$.

Note that an object $X$ being coreduced is the same as it being [[formally etale]].

## Examples

* A coreduced [[scheme]] is also called a [[de Rham space]]. The [[cohomology]] over such is [[crystalline cohomology]], the [[quasicoherent sheaves]] over these define [[D-geometry]].
* In a [[differential geometry|differential geometric]] setting, [[formally etale morphism|formally etale maps]] are the [[local diffeomorphism|local diffeomorphisms]], and the coreduced objects are those for which the terminal map $X \to 1$ is a local diffeomorphism.

## Related concepts

[[!include cohesion - table]]

* [[formally smooth morphism]], [[formally etale morphism]], [[formally unramified morphism]]

[[!redirects coreduced objects]]

[[!redirects coreduced type]]
[[!redirects coreduced types]]

[[!redirects coreduced space]]
[[!redirects coreduced spaces]]


coreflective subcategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __coreflective subcategory__ is a [[full subcategory]] whose inclusion functor has a [[right adjoint]] $R$ (a [[cofree functor]]):

$$
  C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D
  \,.
$$

The dual concept is that of a [[reflective subcategory]]. See there for more details.

## Characterizations

+-- {: .num_prop #CharacterizationByLocalization}
###### Proposition
**(equivalent characterizations)**

Given any pair of [[adjoint functors]]

$$
  (L \dashv R)
  \;:\;
  B
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot}
  A
$$

the following are equivalent:

1. The [[left adjoint]] $L$ is [[full and faithful functor|fully faithful]]. (In this case $A$ is equivalent to its [[essential image]] in $B$ under $L$, a full [[coreflective subcategory]] of $B$.)

2. The [[unit of an adjunction|unit]] $\eta : 1_A \to R L$ of the [[adjunction]] is a [[natural isomorphism]] of functors.

3. The [[comonad]] $(L R, L\eta R,\epsilon)$ associated with the adjunction is [[idempotent comonad|idempotent]], the left adjoint $L$ is [[conservative functor|conservative]], and the right adjoint $R$ is [[essentially surjective functor|essentially surjective on objects]].

4. If $S$ is the set of morphisms $s$ in $B$ such that $R(s)$ is an [[isomorphism]] in $A$, then $R \colon B \to A$ realizes $B$ as the (nonstrict) [[colocalization]] of $B$ with respect to the class $S$.

5. The [[right adjoint]] $R$ is [[codense functor|codense]].

=--

For proofs, see the corresponding characterisations for [[reflective subcategories]].

## Properties

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

[[Vopěnka's principle]] is equivalent to the statement:

For $C$ a [[locally presentable category]],  every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[colimit]]s is a [[coreflective subcategory]].

=--

This is ([AdamekRosicky, theorem 6.28](#AdamekRosicky)).

## Examples

* the inclusion of [[Kelley space]]s into [[Top]], where the right adjoint "kelleyfies"

* the inclusion of [[torsion theory|torsion]] [[abelian groups]] into [[Ab]], where the right adjoint takes the [[torsion subgroup]].

* the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.

* [[Lie integration]], which constructs a [[simply connected space|simply connected]] [[Lie group]] from a finite-dimensional real [[Lie algebra]]. The coreflector is Lie differentiation (taking a Lie group to its associated Lie algebra), and the [[counit]] is the natural map to a given Lie group $G$ from the universal [[covering space]] of the [[connected space|connected component]] at the identity of $G$.

* In a [[recollement]] situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory $i_*: A' \hookrightarrow A$ with reflector $i^*$ and coreflector $i^!$. The functor $j^*$ is both a reflector for the reflective subcategory $j_*: A'' \hookrightarrow A$, and a coreflector for the coreflective subcategory $j_!: A'' \hookrightarrow A$.

## Related concepts

* [[Quillen coreflection]]

* [[reflective subcategory]]

* [[bireflective subcategory]]

## References

* {#AdamekRosicky} [[Jiri Adamek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, London Mathematical Society Lecture Note Series 189

* Robert El Bashir, Jiri Velebil, _Simultaneously Reflective And Coreflective Subcategories of Presheaves_ ([TAC](http://www.tac.mta.ca/tac/volumes/10/16/10-16abs.html))

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On reflective and coreflective hulls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]] **56** (2015) 162–208 [[pdf](https://www.math.mcgill.ca/barr/papers/biref.pdf), [[BarrKennisonRaphael-Reflective.pdf:file]]]

[[!redirects coreflective subcategories]]

[[!redirects coreflective embedding]]
[[!redirects coreflective embeddings]]

[[!redirects co-reflective subcategory]]
[[!redirects co-reflective subcategories]]

[[!redirects coreflector]]
[[!redirects coreflectors]]
[[!redirects coreflection]]
[[!redirects coreflections]]

[[!redirects coreflective localization]]
[[!redirects coreflective localizations]]


coregular action

Let $k$ be a field, $H$ a $k$-bialgebra and $H^*$ a [[dual bialgebra]] with duality $\langle, \rangle : H\otimes H^*\to k$. The __left coregular action__ is a [[Hopf action]] of $H$ on $H^*$ given by

$$
(h,\phi) \mapsto R_h(\phi) := \phi_{(1)} \langle h, \phi_{(2)}\rangle
$$

The corresponding representation $R_h: H\to End H^*$ is called the __left coregular representation__. It is used in the definition of [[Heisenberg double]].

[[!redirects coregular representation]]


corepresentable functor

A [[functor]] $C \to Set$ is **corepresentable** if it is ([[isomorphism|isomorphic]] to) a functor of the form $Hom_C(c,-)$ for some [[object]] $c\in C$.

This is equivalently a [[representable functor]] defined on the [[opposite category]] $C^{op}$.  Often no terminological distinction is made between representable and corepresentable ones (both being called simply "representable"), since a functor $C\to Set$ can only be "corepresentable" while a functor $C^{op}\to Set$ can only be "representable".

Particularly in the study of moduli problems, there is also the notion of corepresentable contravariant functors. A functor $F : C^op \to Set$ is **corepresentable** if and only if there exists an object $X$ in $C$ and a morphism $F \to h_X$ such that for any object $T$ in $C$, the canonical map $Hom_C(X,T) \to Hom_{\PSh(C)}(h_X, h_T) \to Hom_{PSh(C)}(F, h_T)$ is bijective.

[[!redirects corepresentable functors]]


corepresentation

A left (right) __corepresentation__ is a synonym for a left (right) [[coaction]] of a coalgebra (comonoid) in one of the monoidal categories which have linear, and possibly, functional connotation; e.g. in the context of topological vector spaces.

There is however also another meaning of a corepresentation for a [[Leibniz algebra]].

Both a representation and a corepresentation of a right Leibniz $k$-algebra $\mathfrak{g}$ involve a $k$-module $M$ and two $k$-linear maps "actions" $M\otimes\mathfrak{g}\to M$ and $\mathfrak{g}\otimes M\to M$ with 3 axioms.

For representations:

$$[m, [x, y]] = [[m, x], y] - [[m, y], x]$$
$$[x, [a, y]] = [[x, m], y] - [[x, y], m]$$
$$[x, [y, m]] = [[x, y], m] - [[x, m], y]$$

for $x,y\in\mathfrak{g}$ and for $m\in M$.

For corepresentations:

$$[[x, y], m] = [x, [y, m]] - [y, [x, m]] $$
$$[y, [a, x]] = [[y, m], x] - [m, [x, y]] $$
$$[[m, x], y] = [m, [x, y]] - [[y, m], x].$$

If the two "actions" are symmetric, i.e. $[x,m] + [m,x] = 0$ for all $m\in M$, $x\in\mathfrak{g}$ then all the 6 axioms of representation or corepresentation are equivalent. If $M$ is underlying a Leibniz algebra then an action of $\mathfrak{g}$ on $M$ is by definition symmetric, hence all the 6 equivalent conditions hold.


corestriction

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

#Contents#
* table of contents
{:toc}

## Definition

The notion of _co-restriction_ of a [[function]], or more generally of a [[morphism]], is either the [[formal duality|formal dual]] of that of _[[restriction]]_, or something similar. For the latter, recall:

Given any [[subset]] $S\subset A$, we may consider the corresponding inclusion $i_S \colon S \hookrightarrow A$ as a [[function]]. More generally, for a [[subobject]] $S\subset A$ (an [[equivalence class]] of [[monomorphisms]] in some ambient [[category]]) we may consider any representative monomorphism $i_S \colon S\hookrightarrow A$.

Then for any [[morphism]] $f \colon A\to B$, the _[[restriction]]_ $f|_S \colon S \to B$ of $f$ onto $S$ is the [[composition|precomposition]] $f|_S \coloneqq f \circ i_S$ of $f$ by $i_S$.
(For a different representative of the subobject, $i_{\tilde{S}} \colon \tilde{S}\to A$ there is a unique [[isomorphism]] $b \colon S\to\tilde{S}$ such that $i_{\tilde{S}}\circ b = i_S$, hence $f_{\tilde{S}} = f\circ b$.)

This way, restriction of a [[morphism]] amounts to replacing its [[domain]] by one of its [[subobjects]].

### As factorization through the image
  {#AsFactorizationThroughTheImage}

Similarly, the standard notion of _corestriction_ (e.g. [Böhm 08, Def. 3.1](#Bohm08)) of a [[morphism]] $f \colon A \longrightarrow B$ is to instead replace the [[codomain|*co*domain]] $B$ by a [[subobject]]:

For a [[monomorphism]] $i_T \colon T \hookrightarrow B$, the _corestriction_ $f|^T \colon A \to T$ of $f$ onto $S$ is, if it exists, the unique morphism $f|^T \colon A \to S$ such that there is a decomposition $f = i_T \circ f|^T$.

The corestriction exists if and only if $T$ contains the [[image]] of $f$, that is $i_{Im(f)}$ factors through $i_{T}$. In particular, by the [[universal property]] of the image, the corestriction onto the [[image]] $S = Im(f)$ always exists and this is sometimes understood as the corestriction of $f$, by default.

### As the formal dual of restriction

Alternatively, one may consider the notion of corestriction to be the [[formal dual]] to the notion of a restriction: if $p^U \colon B\to U$ is an [[epimorphism]], then the co-restriction of a morphism $f \colon A\to B$, in this dual sense, is the [[composition|postcomposition]] $p^U \circ f \colon \colon A\to U$ of $f$ by $p^U$, which surely always exists.

This is the notion of corestriction considered e.g. in [Andreotti 57, page 14](#Andreotti57), where the factorization through the image ([above](#AsFactorizationThroughTheImage)) is instead called _coastriction_.

## References

The notion of corestriction is well known, while rarely made explicit in print. One may find it e.g. in

* {#Bohm08} [[Gabriella Böhm]], Definition 3.1 and Remarks 3.2 in: _Hopf algebroids_, in: _Handbook of Algebra_, Volume 6, 2009, Pages 173-235 ([arXiv:0805.3806](https://arxiv.org/abs/0805.3806), doi:10.1016/S1570-7954(08)00205-2)

See also:

* {#Andreotti57} A. Andreotti, page 14 (paragraph 2-14) in: _Généralités sur les catégories abéliennes_ (suite) Séminaire A. Grothendieck, Tome 1 (1957) Exposé no. 2 ([numdam:SG_1957__1__A2_0](http://www.numdam.org/item/SG_1957__1__A2_0))

[[!redirects corestrictions]]

[[!redirects co-restriction]]
[[!redirects co-restrictions]]


Corina Cirstea

Corina Cirstea is a Computer Scientist and Modal Logician who is based in Southampton.

* [Home Page](http://www.ecs.soton.ac.uk/people/cc2)

category:people


Corina Keller

* [InSpire page](https://inspirehep.net/authors/1892802)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=242093)

## Selected writings

On [[factorization homology]] for [[surfaces]] equipped with [[flat connections]] for a [[finite group]]:

* [[Corina Keller]], [[Lukas Müller]], *Finite symmetries of quantum character stacks*, Theory and Applications of Categories **39** 3 (2023) 51-97  [[arXiv:2107.12348](https://arxiv.org/abs/2107.12348), [tac:39-03](http://www.tac.mta.ca/tac/volumes/39/3/39-03abs.html)]

On [[factorization homology]] for [[surfaces]] with [[defects]], dynamical quantum groups, and Chern--Simons theory with sources:

* [[Corina Keller]], *Generalized Character Varieties and Quantization via Factorization homology*, PhD Thesis, Université de Montpellier [[tel-04062148](https://theses.hal.science/tel-04062148), [pdf] (https://imag.umontpellier.fr/~calaque/students-Corina_Keller_PhD_Thesis.pdf)]

Review:

* [[Corina Keller]], *Twisted Character Varieties and Quantization via Factorization Homology*, talk at PIRSA (2022) [video: [doi:10.48660/21100028](https://doi.org/10.48660/21100028)]

category: people


coring

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of coring is a generalization of a $k$-[[coalgebra]]. While for
a coalgebra $k$ must be a __commutative__ ring (often a [[field]]), a coring is defined over a general noncommutative ring $k$ or even an associative algebra $A$.

Whereas a coalgebra structure is defined on a $k$-module (if $k$ is a field, it is a [[vector space]]) -- which may be regarded as a central $k$-[[bimodule]] -- a coring structure is defined on a general bimodule over a general [[ring]].

## Definition

An __$A$-coring__ is a [[comonoid]] in the [[monoidal category]] of [[bimodule | bimodules]] over a fixed (typically noncommutative) unital [[ring]] $A$.

This generalizes the notion of $A$-[[coalgebra | coalgebras]] which are defined only if $A$ is commutative and where the bimodules in question are [[central bimodule|central]].

## Base ring extension

More generally, fix a ground commutative ring $R$. Corings will be now over $R$-algebras. So a coring will mean a pair $(A,C)$ where $A$ is an $R$-algebra and $C$ an $A$-coring.

Let $\alpha:A\to B$ be a morphism of rings and $C$ an $A$-coring. Then the $B$-$B$-bimodule $B\otimes_A C\otimes_A B$ has an induced structure of a $B$-coring
with comultiplication

$$
B\otimes_A C\otimes_A B \stackrel{B\otimes \Delta_C\otimes B}\longrightarrow
 B\otimes_A C\otimes_A C\otimes_A B
\stackrel{B\otimes C\otimes 1_B\otimes C\otimes B}\longrightarrow
B\otimes_A C\otimes_A B\otimes_A C\otimes_A B \cong
(B\otimes_A C\otimes_A B)\otimes_B (B \otimes_A C\otimes_A B)
$$
and the counit
$$
B\otimes_A C\otimes_A B \stackrel{B\otimes\epsilon_C\otimes B}\longrightarrow B\otimes_A A\otimes_A B \stackrel{B\otimes\phi\otimes B}\longrightarrow
B\otimes_A B\otimes_A B \stackrel{mult}\longrightarrow B
$$

## Morphisms of corings over varying bases

A morphism $(A,C)\to (B,D)$ is a pair $(\alpha,\gamma)$ where

(i) $\alpha : A\to B$ is an $R$-algebra morphism; by restriction this makes $D$ an $A$-$A$-bimodule by restriction. Denote also by $p:D\otimes_A D\to D\otimes_B D$ the canonical projection of bimodules induced by $\alpha$.
\begin{xymatrix}
D\otimes A\otimes D\ar@<.5ex>[r]^{act\otimes C}\ar@<-.5ex>[r]_{C\otimes act}\ar[d]^{D\otimes\alpha\otimes D}& D\otimes D\ar[d]\ar[r] & D\otimes_A D\ar@{.>}[d]^{p}\\
D\otimes B\otimes D\ar@<.5ex>[r]^{act\otimes D}\ar@<-.5ex>[r]_{D\otimes act} & D\otimes D\ar[r] & D\otimes_B D
\end{xymatrix}
(ii) $\gamma : {}_A C_A\to {}_A D_A$ is a map of $A$-$A$-bimodules, that is
\begin{xymatrix}
A\otimes C\otimes A\ar[r]^{\alpha\otimes\gamma\otimes\alpha}\ar[d] & B\otimes D\otimes B\ar[d]\\
C\ar[r]^{\gamma} & D
\end{xymatrix}
commutes where the vertical arrows are the combined bimodule actions

(iii) $\gamma$ commutes with counit $\alpha \circ \epsilon_C = \epsilon_D\circ \gamma$

(iv) $p\circ (\gamma\otimes_A\gamma)\circ \Delta_C = \Delta_D\circ \gamma$, or diagramatically,
\begin{xymatrix}
C\ar[d]^{\gamma}\ar[r]^{\Delta_C} & C\otimes_A C\ar[r]^{\gamma\otimes_A\gamma} & D\otimes_A D\ar[d]_{p}\\
D\ar[rr]^{\Delta_D} && D\otimes_B D
\end{xymatrix}
Map $b\otimes c\otimes b'\mapsto b\gamma(c)b'$, $B\otimes_A C\otimes_A B\to D$ from the base extension of $C$ to $D$ is by construction a map of $B$-bimodules (externally we just use the $B$-actions on $B$ and on $D$ which are compatible by action axioms) and the conditions (iii),(iv) express that this map of $B$-bimodules is a morphism of $B$-corings.

Morphism $(\alpha,\gamma)$ factorizes into a morphism of $A$-corings ${}_A C_A\to {}_A B\otimes_A C\otimes_A B_A$, $c\mapsto 1\otimes c\otimes 1$ into the base extension coring (determined by $\alpha$, $A$, $B$ and $C$), followed by a morphism of $B$-corings ${}_B B\otimes_A C\otimes_A B_B\to {}_B D_B$.

Every morphism $(\alpha,\gamma)$ as above induces an __induction functor__ ${}^C\mathcal{M}\to{}^D\mathcal{M}$, among the categories of (say, left) comodules, $M\mapsto B\otimes_A M$, $f\mapsto id_B\otimes f$ with, which for $C$-coaction $\rho^M: M\to C\otimes_A M, m\mapsto \sum m_{(-1)}\otimes m_{(0)}$ gives $D$-comodule structure
$$
B\otimes_A M \to D\otimes_B B\otimes_A M\cong D\otimes_A M,\,\,\,
b\otimes m \mapsto  b\gamma(m_{(-1)}) \otimes m_{(0)}.
$$
Similarly, one defines the induction functor for right comodules, and in particular coaction $M'\otimes_A B\to M'\otimes_A B\otimes_B D\cong M'\otimes_A D$,
$m'\otimes b\mapsto m_{(0)}\otimes\gamma(m_{(1)})b$. In particular, one can start with $\Delta_C$ and induce $C\otimes_A B\to C\otimes_A D$ via $c\otimes b\mapsto c_{(1)}\otimes \gamma(c_{(2)})b$.

Under the assumption that the morphism $(\alpha,\gamma)$ is a pure morphism of corings (left hand version), which is for example satisfied automatically when $C$ is flat as a right $A$-module, the induction functor has a right adjoint, the coinduction functor. It is given by a cotensor product which is a comodule under the purity condition. Thus if $N$ is a left $D$-comodule, consider the cotensor product written in two ways as the equalizer
\begin{xymatrix}
(C\otimes_A B)\square^D N\ar[r]&
C\otimes_A B\otimes_B N \ar@<.5ex>[r]^{C\otimes_A B\otimes_B\rho^N}\ar@<-.5ex>[r]_{c\otimes b\otimes n\mapsto c_{(1)}\otimes 1\otimes\gamma(c_{(2)})b\otimes n} &
C\otimes_A B\otimes_B D\otimes_B N
\end{xymatrix}
\begin{xymatrix}
(C\otimes_A B)\square^D N\ar[r]&
C\otimes_A N \ar@<.5ex>[rr]^{C\otimes_A\rho^N}\ar@<-.5ex>[rr]_{(C\otimes_A p)\circ(C\otimes_A\gamma\otimes_A N)\circ(\Delta\otimes_A N)} &&
C\otimes_A D\otimes_B N
\end{xymatrix}
where $p:D\otimes_A N\to D\otimes_B N$ is the canonical projection.

### Comodules over a coring

Given an $A$-coring $C = ({}_A C_A, \Delta,\epsilon)$, a left $C$-comodule $M$ is a left $A$-module with a map of left $A$-modules $\nu: M\to M\otimes_A {}_A C_A$ (left $C$-coaction), such that
the composition $M\stackrel{\rho}\to M\otimes_A C \stackrel{\rho\otimes C}\to (M\otimes_A C)\otimes_A C$ after rebracketing becomes identical to the composition
$M\stackrel{\rho}\to M\otimes_A C \stackrel{M\otimes \Delta_C}\to M\otimes_A (C \otimes_A C)$ and the counitality axiom holds.

A morphism of left $C$-comodules $f:(M,\rho^M)\to(N,\rho^N)$ is a morphism of underlying $A$-modules $f:M\to N$ such that $(M\otimes_A f)\circ\rho^M = \rho^N\circ f$. Thus there is a category of left $C$-comodules equipped with a forgetful functor to the category of left $A$-modules.

The functor $F: M\mapsto M\otimes_A C$ is canonically a comonad on ${}_A Mod$ with  comultiplication $\Delta^F = Id\otimes_A\Delta_C : F\to F F$ and the [[Eilenberg-Moore category]] of the comonad $F$ is isomorphic to the category of $C$-comodules.

Analogously, one considers right $C$-comodules as right $A$-comodules with right $C$-coactions.

## Examples

### Canonical (Sweedler) coring

The classical example of a coring is the canonical or [[Sweedler coring]] corresponding to an extension $R\hookrightarrow S$ of unital rings. The category of [[descent]] data for this ring extension is equivalent to the category of [[comodule | comodules]] over the canonical coring.

Corings are in general useful for the treatment of [[descent in noncommutative algebraic geometry]].

### Coring of a projective module

Suppose we are given ring $R,S$ and an $S$-$R$-bimodule, finitely generated as right projective $R$-module $M = {}_S M_R$. Clearly, $M^* = Hom_R(M,R)$ is an $R$-$S$-bimodule.
Let $M$ be given as a direct summand of a free module $F = \oplus_i R_i$; the decomposition $F = M\oplus L$ induces projection $p:F\to M$, $\sum_i r_i\mapsto \sum_i r_i x_i$, a section $s: M\hookrightarrow F$, $s(m) = \sum_i x_i^*(m)\in\oplus_i R_i$, and [[dual basis]] $x_1,\ldots,x_n\in M$, $x_1^*,\ldots,x_n^*\in Hom_R(M,R)$ characterized by $m = \sum_i x_i^*(m) x_i$ for all $m\in M$. Then define an $R$-coring $C$ over $R$ as $R$-bimodule $M^*\otimes_S M$ with comultiplication
$$
\Delta : M^*\otimes_S M\to (M^*\otimes_S M)\otimes_R(M^*\otimes_S M)
$$
$$
\Delta(f\otimes m) = \sum_{i=1}^n f\otimes x_i^*\otimes x_i\otimes m
$$
and $\epsilon:M^*\otimes_S M, f\otimes m\mapsto f(m)\in R$.

### Matrix coring

Another major class of examples are the so-called [[matrix coring | matrix corings]].

## References

The notion of an $A$-coring is introduced by M. Sweedler and recently lived through a renaissance in works of [[T. Brzeziński]], R. Wisbauer, [[G. Böhm]], L. Kaoutit, Gómez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about $A$-[[cocategory|cocategories]].

There is already a monograph:

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.

Special topics:

* [[T. Brzeziński]], _Descent cohomology and corings_,  Comm. Algebra 36:1894-1900, 2008, [arxiv:math.RA/0601491](http://arxiv.org/abs/math.RA/0601491)

* L. El Kaoutit, J. Gomez-Torrecillas, _On the set of grouplikes of a coring_, [arxiv/0901.4291](http://arxiv.org/abs/0901.4291)

* [[T. Brzeziński]], _Flat connections and (co)modules_, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 [arxiv:math.QA/0608170](http://arxiv.org/abs/math.QA/0608170)

* [[T. Brzeziński]], _The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties_, Algebras and Representation Theory __5__ (2002) 389-410, [math.QA/0002105](http://arxiv.org/abs/math.QA/0002105)

* Lars Kadison, _Depth two and Galois coring_, [math.RA/0408155](http://arxiv.org/abs/math.RA/0408155)

* {#BergmanHausknecht96} [[George M. Bergman]], Adam O. Hausknecht, _Cogroups and co-rings in categories of associative rings_, A.M.S. Math. Surveys and Monographs __45__ (1996) [ISBN 0-8218-0495-2, [ams:surv-45](https://bookstore.ams.org/surv-45), [MR 97k:16001](http://www.ams.org/mathscinet-getitem?mr=97k:16001), [errata and updates](http://math.berkeley.edu/~gbergman/papers/updates/coalg.html)]

  > (discussion [[internalization|internal to]] the [[category]] of [[associative algebras]])

* [[T. Brzeziński]], L. Kadison, [[R. Wisbauer]], _On coseparable and biseparable corings_,  Hopf algebras in noncommutative geometry and physics,  71--87, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005.
* T. Brzeziński, L. El Kaoutit, J. Gómez-Torrecillas, _The bicategories of corings_, J. Pure & Appl. Alg. __205__:3 (2006) 510-541 [doi:10.1016/j.jpaa.2005.07.013](https://doi.org/10.1016/j.jpaa.2005.07.013) [math.RA/0408042](https://arxiv.org/abs/math/0408042)

There is a generalization of corings:

* Jawad Y. Abuhlail, _Semicorings and semicomodules_, [arxiv/1303.3924](http://arxiv.org/abs/1303.3924)

category: algebra
[[!redirects corings]]
[[!redirects semicoring]]


coring spectrum

#Contents#
* table of contents
{:toc}

## Idea

A _coring spectrum_ is a [[comonoid object]] in the [[symmetric monoidal (infinity,1)-category of spectra]].
The concept of a _coring spectrum_  is to that of a  _[[ring spectrum]]_ like a [[coalgebra]] is to an [[associative algebra]].

## Examples
 {#Examples}

\begin{example}\label{SuspensionSpectraAsCoringSpectra}
**([[suspension spectra]] carry canonical [[structure]] of [[coring spectra]])**
\linebreak
Every [[infinity-groupoid|$\infty$-groupoid]] ([[homotopy type]] of a [[topological space]]) $X$ is canonically a [[comonoid object]] in the [[Cartesian monoidal (infinity,1)-category]] [[∞Grpd]] ([here](cartesian+monoidal+infinity,1-category#CoalgebraObjects)). Accordingly, since forming [[suspension spectra]] is [[strong monoidal functor|strong monoidal]] (see [there](suspension+spectrum#StrongMonoidalness)), its suspension spectrum $\Sigma^\infty X_+$ is a coring spectrum (via the [smash-monoidal diagonals](suspension+spectrum#SmashMonoidalDiagonals)).

For more on this coring structure on suspension spectra see also ([here](suspension+spectrum#SmashMonoidalDiagonals)) at *[[suspension spectrum]]* and see discussion of *[[A-theory]]* as in [Hess & Shipley 2014](#HessShipley14).
\end{example}

## Related concepts

* [[comodule spectrum]]

## References

The canonical coring-spectrum structure on [[suspension spectra]] is used in

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _Waldhausen K-theory of spaces via comodules_, Advances in Mathematics 290 (2016): 1079-1137 [[arXiv:1402.4719](https://arxiv.org/abs/1402.4719), [doi:10.1016/j.aim.2015.12.019](https://doi.org/10.1016/j.aim.2015.12.019)]

(for discussion of *[[A-theory]]*).

[[!redirects coring spectra]]

[[!redirects coalgebra spectrum]]
[[!redirects coalgebra spectra]]


Corinne Manogue

* [Wikipedia entry](https://en.wikipedia.org/wiki/Corinne_Manogue)

## Selected writings

On [[octonions]], [[quaternions]], [[normed division algebras]] etc.:

* [[Tevian Dray]], [[Corinne Manogue]], _The Geomety of Octonions_, World Scientific 2015 ([doi:10.1142/8456](https://doi.org/10.1142/8456))

On _[[supersymmetry and division algebras]]_:

* {#ManogueSudbery89} [[Corinne Manogue]], [[Anthony Sudbery]], _General solutions of covariant superstring equations of motion_, Phys. Rev. D 12 (1989), 4073–4077 ([doi:10.1103/PhysRevD.40.4073](https://doi.org/10.1103/PhysRevD.40.4073))

* [[Tevian Dray]], J. Janesky, [[Corinne Manogue]], _Octonionic hermitian matrices with non-real eigenvalues_, Adv. Appl. Clifford Algebras 10 (2000), 193–216 ([arXiv:math/0006069](https://arxiv.org/abs/math/0006069))

On [[E7]] via [[division algebra]]:

* [[Tevian Dray]], [[Corinne Manogue]], [[Robert Wilson]], *A New Division Algebra Representation of $E_7$* [[arxiv:2401.10534](https://arxiv.org/abs/2401.10534)]

## Related entries

* [[octonions]]

* [[exceptional Jordan algebra]]

* [[E6]]

category: people


Cornelia Druţu

* [personal page](http://people.maths.ox.ac.uk/drutu/)

* [institute page](https://www.maths.ox.ac.uk/people/cornelia.drutu)

## Selected writings

On [[geometric group theory]]:

* [[Cornelia Druţu]], [[Michael Kapovich]] (appendix by [[Bogdan Nica]]), *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/) [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

category: people

[[!redirects Cornelia Drutu]]


Cornelius Mertzlufft-Paufler

* [website](http://www.mertzlufft-paufler.de/)

#related $n$Lab entries#

* [[multisymplectic geometry]]

category: people

[[!redirects Cornelis Mertzlufft-Paufler]]


corner

#Contents#
* table of contents
{:toc}

## Idea

A kink in a [[boundary]].

See for instance at _[[manifold with corners]]_.

## Related concepts

* [[boundary]]

[[!redirects corners]]


Corrado de Concini

* [Wikipedia entry](https://en.wikipedia.org/wiki/Corrado_de_Concini)

## selected writings

* {#deConciniProcesi95} [[Corrado de Concini]], [[Claudio Procesi]], _Wonderful models of subspace arrangements_, Selecta Mathematica, New Series Vol. 1, No. 3, 1995 ([pdf](https://link.springer.com/content/pdf/10.1007/BF01589496.pdf))

## related $n$Lab entries

* [[wonderful compactification]]

* [[Fulton-MacPherson compactification]]

category: people


Corrado Segre

__Corrado Segre__ was an [[algebraic geometry|algebraic geometer]], of the classical Italian school.

* wikipedia in [English](http://en.wikipedia.org/wiki/Corrado_Segre), [Italian](http://it.wikipedia.org/wiki/Corrado_Segre)

category: people


correlator

[[!redirects correlators]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[Euclidean field theory]] a _correlator_ is an [[expectation value]] of [[observables]] in a given [[state]]. For the product of $n$ local field observables $\mathbf{\Phi}(x)$ this is also called an _[[n-point function]]_, see there for more.

Euclidean $n$-point functions are typically [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]]. Their [[restriction of distributions]] to the [[complement]] of the fat diagonal hence yields a [[non-singular distribution]] exhibiting the [[correlator as a differential form on a configuration space of points]].

Under [[Wick rotation]] (if applicable, see _[[Osterwalder-Schrader theorem]]_) this translates correlators to [[n-point functions]] in [[relativistic field theory]].

In [[functorial quantum field theory]] a correlator is simply the value of the [[functor]] on a given (class of) [[cobordisms]].

## Related concepts

* [[n-point function]]

* [[correlator as differential form on configuration space of points]]

* [[Ward identity]]

* [[entanglement]]

## References

See most any text on [[quantum field theory]]/[[statistical mechanics]].

Discussion specifically of [[non-perturbative effect|non-perturbative]] [[monopole]] correlators:

* {#Iqbal14} [[Nabil Iqbal]], _Monopole correlations in holographically flavored liquids_, Phys. Rev. D 91, 106001 (2015) ([arXiv:1409.5467](https://arxiv.org/abs/1409.5467))

[[!include holographic principle -- table]]

[[!redirects correlation function]]
[[!redirects correlation functions]]

[[!redirects correlation]]
[[!redirects correlations]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[Euclidean field theory]], an alternative to regarding [[propagators]]/[[correlators]] as [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]], is to [[pullback of distributions|pull-back]] these distributions to [[smooth functions]]/[[differential forms]] on ([[Fulton-MacPherson compactifications]] of) [[configuration spaces of points]] and regard them in this incarnation, in particular discuss their [[renormalization]] from this perspective.

Analogous to the perspective of [[wavefront sets]] for distributions, this perspective amounts to recording around each potentially singular point an [[n-sphere|(d-1)-sphere]] worth of extra [[direction of a vector|directional]] information carried by the [[correlator]]/[[Feynman amplitude]] in the vicinity of the point.

This approach goes back to [Axelrod-Singer 93](#AxelrodSinger93) in the discussion of [[perturbative quantum field theory|perturbative]] [[quantization of Chern-Simons theory]]. Here the [[graph complex]] of [Kontsevich 94](#Kontsevich94) (full details due to [Lambrechts-Volić 14](#LambrechtsVolic14)) shows that the [[de Rham algebra]] of the [[configuration space of points]] is actually [[quasi-isomorphism|quasi-isomorphic]] to all possible [[Feynman amplitudes]] for free [[Chern-Simons theory|Chern-Simons]]/[[AKSZ theory]].

A general and systematic discussion of [[perturbative quantum field theory]] and its [[renormalization]] from this perspective is offered in [Berghoff 14a](#Berghoff14a), [Berghoff 14b](#Berghoff14b) (albeit presently only for [[Euclidean quantum field theory]], not for [[relativistic quantum field theory]]).

## Examples

### Higher Chern-Simons theory

This approach to [[pQFT]] was originally considered specifically for the [[Chern-Simons propagator]] in  [[quantization of 3d Chern-Simons theory]] in [Axelrod-Singer 93](#AxelrodSinger93), see also [Bott-Cattaneo 97, Remark 3.6](#BottCattaneo97) and [Cattaneo-Mnev 10, Remark 11](#CattaneoMnev10). The analysis applies verbatim to [[higher Chern-Simons theory]] such as notably the [[AKSZ sigma-models]], too, since the [[Feynman propagator]]  depends only on the [[free field theory]]-[[equations of motion]], which is $d A = 0$ in all these cases.

Here the [[Chern-Simons propagator]] regarded as a non-singular [[differential form]] on the [[Fulton-MacPherson compactification|compactification]] of the [[configuration space of points]] serves to exhibit its [[Feynman amplitudes]] as providing [[graph complex]]-models for the [[de Rham cohomology]] of these [[Fulton-MacPherson compactification|compactified]] [[configuration spaces of points]], a point due to [Kontsevich 94](#Kontsevich94), [Kontsevich 93, 5](#Kontsevich93):

\[
  \label{TheQuasiIsomorphism}
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma
    }
  }{
    Graphs_n(\Sigma)
  }
  \underoverset{
    \simeq_{\mathrlap{qi}}
  }
  {
    \color{blue}
    \array{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{de Rham algebra}
      \\
      \text{of semi-algebraic differential forms}
      \\
      \text{on the FM-compactification}
      \\
      \text{of the configuration space of n points}
      \\
      \text{in}\; \Sigma
    }
  }{
  \Omega^\bullet_{PA}
  \big(
    Conf_n\big(  \Sigma \big)
  \big)
  }
  \,.
\]

## Related concepts

* [[quantization of 3d Chern-Simons theory]]

## References

The approach was originally considered specifically for [[Chern-Simons theory]] in

* {#AxelrodSinger93} [[Scott Axelrod]], [[Isadore Singer]],  _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

which was re-amplified  in

* Daniel Altschuler, Laurent Freidel, Section 3 of: _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. 187 (1997) 261-287 ([arxiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010))

* {#BottCattaneo97} [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in _Integral invariants of 3-manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

* {#CattaneoMnev10} [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in _Remarks on Chern-Simons invariants_, Commun. Math. Phys. 293: 803-836, 2010 ([arXiv:0811.2045](https://arxiv.org/abs/0811.2045))

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], appendix B of _Perturbative quantum gauge theories on manifolds with boundary_, Communications in Mathematical Physics, January 2018, Volume 357, Issue 2, pp 631–730  ([arXiv:1507.01221](https://arxiv.org/abs/1507.01221), [doi:10.1007/s00220-017-3031-6](https://doi.org/10.1007/s00220-017-3031-6))

and highlighted as a means to obtain [[graph complex]]-models for the [[de Rham cohomology]] of [[configuration spaces of points]]/[[knot spaces]] in

* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

* {#Kontsevich94} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

with full details and proofs in

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society no. 1079, 2014  ([arxiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

see also

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

A systematic development of [[Euclidean field theory|Euclidean]] [[perturbative quantum field theory]] with [[n-point functions]] considered as [[smooth functions]] on [[Fulton-MacPherson compactifications]]/[[wonderful compactifications]] of [[configuration spaces of points]] and more generally of subspace arrangements is due to

* {#BergbauerBrunettiKreimer09} [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], _Renormalization and resolution of singularities_, ESI preprint 2010 ([arXiv:0908.0633](https://arxiv.org/abs/0908.0633), [ESI:2244](https://mat.univie.ac.at/~esiprpr/esi2244.pdf))

* {#Bergbauer09} [[Christoph Bergbauer]], _Renormalization and resolution of singularities_, talks as IHES and Boston, 2009 ([pdf](http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf))

* {#Berghoff14a} [[Marko Berghoff]], _Wonderful renormalization_, 2014 ([pdf](http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf), [doi:10.18452/17160](https://doi.org/10.18452/17160))

* {#Berghoff14b} [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_,  Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))

Analogous discussion for configuration spaces of points regarded as [[Hilbert schemes of points]] and with their [[ordinary cohomology]] replaced by [[K-theory]]:

* Jian Zhou, _K-Theory of Hilbert Schemes as a Formal Quantum Field Theory_ ([arXiv:1803.06080](https://arxiv.org/abs/1803.06080))

Discussion specifically in [[topological quantum field theory]] with an eye towards [[supersymmetry|supersymmetric]] field theory, in terms of the [[ordinary homology]] of [[configuration spaces of points]]:

* [[Christopher Beem]], [[David Ben-Zvi]], [[Mathew Bullimore]], [[Tudor Dimofte]], [[Andrew Neitzke]], _Secondary products in supersymmetric field theory_ ([arXiv:1809.00009](https://arxiv.org/abs/1809.00009))

[[!redirects correlator as differential form on configuration space of points]]
[[!redirects correlators as differential forms on configuration space of points]]

[[!redirects correlator as differential form on a configuration space of points]]
[[!redirects correlator as a differential form on a configuration space of points]]
[[!redirects correlators as differential forms on a configuration space of points]]
[[!redirects correlators as differential forms on configuration spaces of points]]

[[!redirects Feynman amplitudes as differential forms on configuration spaces of points]]

[[!redirects Feynman amplitude on compactified configuration space of points]]
[[!redirects Feynman amplitude on compactified configuration spaces of points]]
[[!redirects Feynman amplitudes on compactified configuration spaces of points]]

[[!redirects propagators regarded as differential forms on configuration spaces]]


correspondence > history

< [[correspondence]]
[[!redirects correspondence -- history]]


correspondence between measure and valuation theory

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

[[valuation (measure theory)|Valuations]] and [[measures]] have very similar constructions and applications, with [[$\tau$-additive measures]] somewhat in between.
Here we review which concepts correspond to each other.
Note that on many topological spaces of interest, such as [[metric spaces]], most of the constructions below [[valuation (measure theory)#extending_valuations_to_measures|coincide]].

## Table

  Concept↓\Theory→
  [[measure|Measures (traditional)]]
  [[τ-additive measures]]
  [[valuation (measure theory)|Valuations]]

  [[space|Spaces]]
  [[measurable space|Measurable spaces]]
  [[topological space|Topological spaces]]
  [[locale|Locales]]

  [[morphism|Maps]]
  [[measurable function|Measurable maps]]
  [[continuous map|Continuous maps]]
  [[continuous map|Continuous maps (of locales)]]

  Events
  [[measurable subset|Measurable sets]]
  [[open set|Open sets]]
  [[open set|Open sets (elements of the frame)]]

  Values
  Non-negative [[real number|reals]]
  Non-negative [[real number|reals]], with the lower [[semicontinuous topology]]
  Non-negative [[one-sided real number|lower reals]]

  Functionals
  [[measure|Measures]]
  [[tau-additive measure|τ-additive Borel measures]]
  [[valuation (measure theory)|Continuous valuations]]

  Finitary linearity
  Additivity
  Additivity
  Modularity

  Infinitary linearity
  $\sigma$-additivity
  $\tau$-additivity
  [[Scott continuity|(Scott) continuity]]

  [[intensive or extensive quantity|Integrands]]
  [[measurable function|Measurable]] real-valued functions
  [[semicontinuous map|Lower semicontinuous]] real functions
  [[continuous map|Continuous functions]] into the [[one-sided real number|lower reals]]

  Approximation of integrands
  Pointwise-increasing [[sequence|sequences]] (of measurable real functions)
  Pointwise-increasing [[net|nets]] (of lower semicontinuous real functions)
  Pointwise-increasing [[net|nets]] (of continuous functions into the lower reals)

  Integral
  [[Lebesgue integral]]
  [[Lebesgue integral]]
  [[valuation (measure theory)#integration|Lower integral]]

  [[monad|Monad]]*
  [[Giry monad]]
  [[extended probabilistic powerdomain##the_measure_monad_on_top|measure monad on Top]]
  [[valuation monad on locales]]

$*$ See also [[monads of probability, measures, and valuations]].

## References

* V. Bogachev, _Measure Theory_, vol. 2 (2007).

* {#monad} Reinhold Heckmann, _Spaces of valuations_, Papers on General Topology and Ap-plications, 1996. [Link here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf).

* Mauricio Alvarez-Manilla, Achin Jung, Klaus Keimel, _The probabilistic powerdomain for stably compact spaces_, Theoretical Computer Science 328, 2004. [Link here](https://www.sciencedirect.com/science/article/pii/S0304397504004074).

* {#kirch} Olaf Kirch, _Bereiche und Bewertungen_ (in German), Master Thesis, Technische Hochschule Darmstadt, 1993. [Link here](http://fb04286.mathematik.tu-darmstadt.de/fbereiche/logik/research/Domains/papers/kirch/diplom.ps.gz).

* {#jung} Achim Jung, _Stably compact spaces and the probabilistic powerspace construction_, ENTCS 87, 2004. [Link here](https://www.sciencedirect.com/science/article/pii/S1571066104051345)

* [[Thierry Coquand]] and [[Bas Spitters]], _Integrals and Valuations_, 2009. [Link here](http://logicandanalysis.org/index.php/jla/article/view/14/12).

* {#vmonad} [[Steve Vickers]], _A monad of valuation locales_, 2011. [Link here](https://www.cs.bham.ac.uk/~sjv/Riesz.pdf).

* {#vintegral} [[Steve Vickers]], _A localic theory of lower and upper integrals_, 2008. [Link here](https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.200710028).

[[!redirects correspondence between measures and valuations]]
[[!redirects correspondence between measure theory and valuations]]
[[!redirects correspondence between measure theory and valuation theory]]
[[!redirects correspondence between valuations and measures]]
[[!redirects correspondence between valuations and measure theory]]
[[!redirects correspondence between valuation theory and measure theory]]


correspondence theory of truth

The **correspondence theory of truth** is generally applied to any view explicitly embracing the idea that the truth of propositions consists in their relation to reality, i.e., that truth is a relational property been judgments and some portion of reality.

According to some commentators, one version of the correspondence theory is the theory of  _[[truth]]_ in [[object languages]] with respect to [[metalanguages]] due to [[Alfred Tarski]]. However, the bearing of this on a correspondence theory of truth is debated ([SEP: The Correspondence Theory of Truth](https://plato.stanford.edu/entries/truth-correspondence/#7.2)).

## References

* Herbert Keuth, _Tarski's definition of truth and the correspondence theory_, Philosophy of Science Vol. 45, No. 3 (Sep., 1978), pp. 420-430

* Gila Sher, _What is Tarski's theory of truth?_ ([pdf](http://philosophyfaculty.ucsd.edu/faculty/gsher/WTTT.pdf))

* Stanford Encyclopedia of Philosophy, _[The Correspondence Theory of Truth](http://plato.stanford.edu/entries/truth-correspondence/)_


correspondence type

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### Context
#### Type theory
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\tableofcontents

## Idea

In the same way that one could define [[equivalence types]] as types of [[one-to-one correspondences]] and [[function types]] as types of [[anafunctions]], one could define correspondence types as types of [[correspondences]].

Correspondences are [[type families]] $x\colon A, y \colon B \vdash R(x, y) \; \mathrm{type}$. From every correspondence, one could derive the [[multivalued partial function]]
$$x:A, p:\sum_{y:B} R(x, y) \vdash f(x, p):B$$
and for every type family $x:A \vdash P(x)$ and every [[multivalued partial function]] $x:A, p:P(x) \vdash f(x, p):B$, one could define the [[correspondence]] $x:A, y:B \vdash R(x, y)$ as
$$R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$$

## Rules for correspondence types

The rules for correspondence types are as follows:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{Corr}(A, B) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:\mathrm{Corr}(A, B), x:A, y:B \vdash \mathcal{C}_{A, B}(f, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x) \vdash f(x, p):B}{\Gamma \vdash (x, p) \mapsto f(x, p):\mathrm{Corr}(A, B)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x) \vdash f(x, p):B}{\Gamma, x:A, p:P(x) \vdash \alpha(x, p):\mathcal{C}_{A, B}((x, p) \mapsto f(x, p), x, f(x, p))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type}}{\Gamma, f:\mathrm{Corr}(A, B), x:A, p:P(x) \vdash \mathrm{ev}(f, x, p):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type}}{\Gamma, f:\mathrm{Corr}(A, B), x:A, p:P(x) \vdash \beta(f, x, p):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, x, p))}$$

## See also

* [[correspondence]]
* [[function type]], [[dependent function type]]
* [[equivalence type]]
* [[partial function type]]

[[!redirects correspondence type]]
[[!redirects correspondence types]]


cosemisimple coalgebra

Let $C$ be a $k$-[[coalgebra]] and $\rho:V\to V\otimes C$ its right [[corepresentation]]. Recall that a sub-$k$-module $W\subset V$ is __$\rho$-[[invariant subspace|invariant]]__ if $Im \rho|_W\subset W\otimes C$; if $C$ is [[flat coalgebra|flat]] over $k$ then $\rho|_W$ is a $C$-subcorepresentation and $(W,\rho|_W)$ a $C$-subcomodule of $(V,\rho_V)$. If $V=C$ and $\rho=\Delta_C$ then a $C$-subcomodule is the same as a subcoalgebra.

A $k$-coalgebra $C$ is __cosimple__ if it has no subcoalgebras except for $C$ and $0$ (with $C \neq 0$); in other words, it is the only [[simple object]] in $Comod^C$. Emphasising 'cosimple' instead of 'simple' is convenient because, for [[Hopf algebra]]s, both semisimplicity and cosemisimplicity make sense. It is a basic fact (not paralleled in module theory) that if $k$ is a [[field]], for every $k$-coalgebra $C$ and every $C$-comodule $(V,\rho)$, every element $v\in V$ is contained in some finite-dimensional $C$-subcomodule, and in particular every simple comodule is finite-dimensional and every cosimple coalgebra is finite-dimensional. A $k$-coalgebra $C$ is __cosemisimple__ if it is a direct sum of simple $k$-subcoalgebras. This is equivalent to saying that every $C$-comodule is a direct sum of simple subcomodules. A common criterion of cosemisimplicity is the existence of (say) left integrals on $H$ (left-invariant normalized functionals on $H$).

For example (over a field $k$) any [[group algebra]] $k[G]$ is cosemisimple as a coalgebra, while the universal [[enveloping algebra]] $U(g)$ of any nontrivial Lie $k$-algebra $g\neq 0$ is not cosemisimple. The function algebra $\mathcal{O}(G)$ of an affine algebraic $k$-group is cosemisimple iff $G$ is linearly reductive; over a transcendental parameter $q$ of deformation, this is preserved for quantized function algebras (cf. [[quantum group]]).

[[!redirects cosemisimple coalgebra]]


coseparable coring

Coseparability of [[coring]]s is a dual notion to [[separable ring|separability of rings]] (where one requires that the multiplication map is split).

An $A$-[[coring]] $(C,\Delta,\epsilon)$ is __coseparable__ if the comultiplication $\Delta:C\to C\otimes_A C$ splits as a $C$-$C$-bicomodule morphism. In other words, there is a morphism of $A$-$A$-bimodules $p: C\otimes_A C\to C$ such that

$$\Delta\circ p = (C \otimes_A p)\circ(\Delta\otimes_A C)
= (p\otimes_A C)\circ(C\otimes_A\Delta)$$

$$p\circ\Delta = C$$

where $C = \mathrm{Id}_C$.


coset

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###Context###
#### Group Theory
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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[group]] $G$ and a [[subgroup]] $H$, then their _coset object_ is the [[quotient]] $G/H$, hence the set of [[equivalence classes]] of elements of $G$ where two are regarded as equivalent if they differ by right multiplication with an element in $H$.

If $G$ is a [[topological group]], then the quotient is a [[topological space]] and usually called the _coset space_. This is in particular a [[homogeneous space]], see there for more.

## Definition

### Internal to a general category

In a category $C$, for $G$ a [[group object]] and $H \hookrightarrow G$ a [[subgroup|subgroup object]], the left/right _object of cosets_ is the [[orbit|object of orbits]] of $G$ under left/right multiplication by $H$.

Explicitly, the left coset space $G/H$ [[coequalizes]] the parallel morphisms
$$
  H \times G \underoverset{\mu}{proj_G}\rightrightarrows G
$$
where $\mu$ is (the inclusion $H\times G \hookrightarrow G\times G$ composed with) the group multiplication.

Simiarly, the right coset space $H\backslash G$ [[coequalizes]] the parallel morphisms
$$
  G \times H \underoverset{proj_G}{\mu}\rightrightarrows G
$$

### Internal to $Set$

Specializing the above definition to the case where $C$ is the well-pointed topos $Set$, given an element $g$ of $G$, its orbit $g H$ is an element of $G/H$ and is called a _left coset_.

Using [[comprehension]], we can write

$$
  G/H = \{g H | g \in G\}
$$

Similarly there is a coset on the right $H \backslash G$.

### For Lie groups and Klein geometry

If $H \hookrightarrow G$ is an inclusion of [[Lie groups]] then the quotient $G/H$ is also called a _[[Klein geometry]]_.

### For $\infty$-groups
 {#ForInfinityGroups}

More generally, given an [[(∞,1)-topos]] $\mathbf{H}$ and a [[homomorphism]] of  [[∞-group]] objects $H \to G$, hence equivalently a morphism of their [[deloopings]]
$\mathbf{B}H \to \mathbf{B}G$, then the [[homotopy quotient]] $G/H$ is given by the [[homotopy fiber]] of this map

$$
  \array{
    G/H &\longrightarrow& \mathbf{B}H
    \\
    && \downarrow
    \\
    && \mathbf{B}G
  }
  \,.
$$

See at _[[∞-action]]_ for more on this definition. See at _[[higher Klein geometry]]_ and _[[higher Cartan geometry]]_ for the corresponding concepts of [[higher geometry]].

## Properties

### For normal subgroups

The coset inherits the structure of a group if $H$ is a [[normal subgroup]].

Unless $G$ is abelian, considering both left and right coset spaces provide different information.

### Topology of the quotient map
 {#QuotientMaps}

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

For $X$ a [[smooth manifold]] and $G$ a [[compact Lie group]] equipped with a [[free action|free]] smooth [[action]] on $X$, then the [[quotient]] [[projection]]

$$
  X \longrightarrow X/G
$$

is a $G$-[[principal bundle]] (hence in particular a [[Serre fibration]]).

=--

This is originally due to ([Gleason 50](#Gleason50)). See e.g. ([Cohen, theorem 1.3](#Cohen))

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

For $G$ a [[Lie group]] and $H \subset G$ a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]]

$$
  G \longrightarrow G/H
$$

is an $H$-[[principal bundle]] (hence in particular a [[Serre fibration]]).

=--

This is originally due to ([Samelson 41](#Samelson41)).

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

For $G$ a [[compact Lie group]] and $K \subset H \subset G$ [[closed subspace|closed]] [[subgroups]], then the [[projection]] map

$$
  p \;\colon\; G/K \longrightarrow G/H
$$

is a locally trivial $H/K$-[[fiber bundle]] (hence in particular a [[Serre fibration]]).

=--

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

Observe that the projection map in question is equivalently

$$
  G \times_H (H/K) \longrightarrow G/H
  \,,
$$

(where on the left we form the [[Cartesian product]] and then divide out the [[diagonal action]] by $H$). This exhibits it as the $H/K$-[[fiber bundle]] [[associated bundle|associated]] to the $H$-[[principal bundle]] of corollary \ref{QuotientProjectionForCompactLieSubgroupIsPrincipal}.

=--

\begin{prop}\label{CosetSpaceCoprojectionsWithLocalSections}
**([[coset space coprojections with local sections]])**\linebreak
Let $G$ be a [[topological group]] and $H \subset G$ a [[subgroup]].

Sufficient conditions for the [[coset space]] [[coprojection]] $G \overset{q}{\to} G/H$ to admit [[local sections]], in that there is an [[open cover]] $\underset{i \in I}{\sqcup}U_i \to G/H$ and a [[continuous section]] $\sigma_{\mathcal{U}}$ of the [[pullback]] of $q$ to the cover,

$$
  \array{
    &&
    G_{\vert \mathcal{U}}
    &\longrightarrow&
    G
    \\
    &
    {}^{\mathllap{
      \exists \sigma
    }}
    \nearrow
    &
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \mathllap{ \exists \; }
    \underset{i \in I}{\sqcup} U_i
    &=&
    \underset{i \in I}{\sqcup} U_i
    &\longrightarrow&
    G/H
    \mathrlap{\,,}
  }
$$

include the following:

* $G$ is any [[topological group]]

  and $H$ is a [[compact Lie group]]

  (in particular for $H$ a [[closed subgroup]] if $G$ itself is a [[compact Lie group]], since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]).

  ([Gleason 50, Thm. 4.1](#Gleason50))

or:

* The underlying [[topological space]] of $G$ is

  1. [[locally compact topological space|locally compact]];

  1. [[separable metric space]];

  1. of [[finite number|finite]] [[dimension of a separable metric space]]

  (e.g. if $G$ is a [[Lie group]])

  and $H \subset G$ is a [[closed subgroup]].

  ([Mostert 53, Theorem 3](#Mostert53))

\end{prop}

### As a homotopy fiber

\begin{remark}\label{InGeometricHomotopyTheory}
In [[geometric homotopy theory]] (in an [[(∞,1)-topos]]), for $H \longrightarrow G$ any homomorphisms of [[∞-group]] objects, then the natural projection $G \longrightarrow G/H$, generally realizes $G$ as an $H$-[[principal ∞-bundle]] over $G/H$. This is exhibited by a [[homotopy pullback]] of the form

$$
  \array{
   G & \longrightarrow &*
   \\
    \downarrow && \downarrow
   \\
   G/H &\longrightarrow& \mathbf{B}H
  }
  \,.
$$

where $\mathbf{B}H$ is the [[delooping|delooping groupoid]] of $H$. This also equivalently exhibits the [[∞-action]] of $H$ on $G$ (see there for more).

By the reverse [[pasting law]] for [[homotopy pullbacks]] ([here](pasting+law+for+pullbacks#ReversePastingLawForInfinityGroupoids), using that $\ast \to \mathbf{B}H$ is an [effective epimorphism](effective+epimorphism+in+an+infinity1-category) by definition of [[delooping]]) then we get the [[homotopy pullback]]

$$
  \array{
   G/H & \longrightarrow &\mathbf{B}H
   \\
    \downarrow && \downarrow
   \\
   * & \longrightarrow & \mathbf{B}G
  }
$$

which exhibits the coset as the [[homotopy fiber]] of $\mathbf{B}H \to \mathbf{B}G$.

See also [SS21, Ex. 3.2.35](#SatiSchreiber21).
\end{remark}

## Examples

### $n$-Spheres

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

The [[n-spheres]] are coset spaces of [[orthogonal groups]]:

$$
  S^n \simeq O(n+1)/O(n)
  \,.
$$

The odd-dimensional spheres are also coset spaces of [[unitary groups]]:

$$
  S^{2n+1}
  \simeq
  U(n+1)/U(n)
$$

=--

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

Regarding the first statement:

Fix a [[unit vector]] in $\mathbb{R}^{n+1}$. Then its [[orbit]] under the defining $O(n+1)$-[[action]] on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector [[stabilizer group|stabilizes]] it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.

The second statement follows by the same kind of reasoning:

Clearly $U(n+1)$ [[transitive action|acts transitively]] on the [[unit sphere]] $S^{2n+1}$ in $\mathbb{C}^{n+1}$. It remains to see that its [[stabilizer subgroup]] of any point on this sphere is $U(n)$. If we take the point with [[coordinates]] $(1,0, 0, \cdots,0)$ and regard elements of $U(n+1)$ as [[matrices]], then the stabilizer subgroup consists of matrices of the block diagonal form

$$
  \left(
    \array{
      1 & \vec 0
      \\
      \vec 0 & A
    }
  \right)
$$

where $A \in U(n)$.

=--

There are also various exceptional realizations of spheres as coset spaces. For instance:

[[!include coset space structure on n-spheres -- table]]

\linebreak

### Sequences of coset spaces
 {#QuotientMapsOfCosetSpaces}

Consider $K \hookrightarrow H \hookrightarrow G$ two consecutive group inclusions with their induced coset [[quotient]] [[projections]]

$$
  \array{
    H/K & \longrightarrow&  G/K
    \\
    && \downarrow
    \\
    && G/H
  }
  \,.
$$

When $G/K \to G/H$ is a [[Serre fibration]], for instance in the situation of prop. \ref{ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup} (so that this is indeed a [[homotopy fiber sequence]] with respect to the [[classical model structure on topological spaces]]) then it induces the corresponding [[long exact sequence of homotopy groups]]

$$
  \cdots
  \to
  \pi_{n+1}(G/H)
  \longrightarrow
  \pi_n(H/K) \longrightarrow \pi_n(G/K) \longrightarrow \pi_n(G/H)
  \longrightarrow
  \pi_{n-1}(H/K)
  \to \cdots
  \,.
$$

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

Consider a sequence of inclusions of [[orthogonal groups]] of the form

$$
  O(n) \hookrightarrow O(n+1) \hookrightarrow O(n+k)
  \,.
$$

Then by example \ref{nSphereAsCosetSpace} we have that $O(n+1)/O(n) \simeq S^n$ is the [[n-sphere]] and by corollary \ref{QuotientProjectionForCompactLieSubgroupIsPrincipal} the quotient map is a [[Serre fibration]]. Hence there is a [[long exact sequence of homotopy groups]] of the form

$$
  \cdots
  \to
  \pi_q(S^n)
  \longrightarrow \pi_q(O(n+k)/O(n)) \longrightarrow \pi_q(O(n+k)/O(n+1))
  \longrightarrow
  \pi_{q-1}(S^n)
  \to \cdots
  \,.
$$

Now for $q \lt n$ then $\pi_q(S^n) = 0$ and hence in this range we have [[isomorphisms]]

$$
  \pi_{\bullet \lt n}(O(n+k)/O(n))
     \stackrel{\simeq}{\longrightarrow}
  \pi_{\bullet \lt n}(O(n+k)/O(n+1))
 \,.
$$

=--

## Related concepts

* [[coset space]]

* [[coadjoint orbit]]

* [[index of a subgroup]]

* [[class equation]]

* [[flag variety]]

* [[Klein geometry]]

* [[WZW model]]

* [[double coset]]

## References

* {#Samelson41} [[Hans Samelson]], _Beiträge zur Topologie der Gruppenmannigfaltigkeiten_, Ann. of Math. 2, 42, (1941), 1091 - 1137. ([jstor:1970463](https://www.jstor.org/stable/1970463), [doi:10.2307/1970463](https://doi.org/10.2307/1970463))

* {#Gleason50} [[Andrew Gleason]], _Spaces with a compact Lie group of transformations_, Proc. of A.M.S 1, (1950), 35 - 43 ([jstor:2032430](https://www.jstor.org/stable/2032430), [doi:10.2307/2032430](https://doi.org/10.2307/2032430))

* {#Steenrod51} [[Norman Steenrod]], Section I.7 of: _The topology of fibre bundles_, Princeton Mathematical Series 14, Princeton Univ. Press, 1951.

* {#Mostert53} [[Paul Mostert]], _Local Cross Sections in Locally Compact Groups_, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 ([jstor:2032540](https://www.jstor.org/stable/2032540), [doi:10.2307/2032540](https://doi.org/10.2307/2032540))

* {#Bredon72} [[Glen Bredon]], Section I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#Cohen} R. Cohen, _Topology of fiber bundles_, Lecture notes ([pdf](http://math.stanford.edu/~ralph/fiber.pdf))

On coset spaces with the same [[rational cohomology]] as a [[product space|product]] of [[n-spheres]]:

* [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_,  Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false))

Discussion in [[(infinity,1)-topos theory|$\infty$-topos theory]]:

* {#SatiSchreiber21} [[Hisham Sati]], [[Urs Schreiber]], Ex. 3.2.35 ([p. 104](https://arxiv.org/pdf/2112.13654.pdf#page=104)) of: *[[schreiber:Equivariant principal infinity-bundles]]* [[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)]

[[!redirects coset]]
[[!redirects cosets]]
[[!redirects left coset]]
[[!redirects right coset]]
[[!redirects left cosets]]
[[!redirects right cosets]]

[[!redirects coset space]]
[[!redirects coset spaces]]