colimit

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#### Category theory
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[[!include category theory - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
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## Idea

The concept of _colimit_ is that [[duality|dual]] to a [[limit]]:

a colimit of a [[diagram]] in a [[category]] is, if it exists, the [[representable functor|co-classifying space]] for morphisms _out_ of that diagram.

The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the diagram.

We have

* the notion of _colimit_ generalizes the notion of [[direct sum]];

* the notion of _[[weighted colimit]]_ generalizes the notion of _weighted (direct) sum_.

Sometimes colimits (or some colimits) are called _[[inductive limits]]_ or _[[direct limits]]_; see the discussion of terminology at [[limit]].

A [[weighted colimit]] in $C$ is a
[[weighted limit]] in $C^{op}$.

## Definition

A [[colimit]] in a [[category]] $C$ is the same as a [[limit]] in the [[opposite category]], $C^{op}$.

More in detail, for $F : D^{op} \to C^{op}$ a functor, its [[limit]] $\lim F$ is the colimit of
$F^{op} : D \to C$.

## Examples

Here are some important examples of colimits:

* A colimit of the [[diagram|empty diagram]] is an [[initial object]].
* A colimit of a diagram consisting of two (or more) objects and no nontrivial morphisms is their [[coproduct]].
* A colimit of a [[span]] is a [[pushout]].
* A colimit of two (or more) [[parallel morphisms]] is a [[coequalizer]].
* A colimit of a diagram whose domain is a [[sifted category]] is a [[sifted colimit]].
* A colimit of a diagram whose domain is a [[filtered category]] is a [[filtered colimit]].
* A colimit of a connected diagram is a [[connected colimit]].
* A colimit of a nonempty diagram is a [[nonempty colimit]].

See also *[[limits and colimits by example]]*.

## Properties

The properties of colimits are of course [[formal duality|dual]] to those of [[limits]]. It is still worthwhile to make some of them explicit:

\begin{proposition}
\label{ColimitsInTermsOfCoequalizers}
  All colimits may be expressed via [[coequalizers]] of maps between [[coproducts]].
\end{proposition}
This was historically first observed by [Maranda 1962, Thm. 1](#Maranda62). See the [[formal dual|dual]] discussion ([here](limit#ConstructionFromProductsAndEqualizers)) of [[limits]] via [[products]] and [[equalizers]].

\begin{proposition}
**([[hom-functor preserves limits|contravariant Hom sends colimits to limits]])**
\linebreak
For $C$ a [[locally small]] category,
for $F : D \to C$ a functor, for $c \in C$ an object and writing $C(F(-), c) : D \to Set$, we have
$$
  C(colim F, c) \simeq lim C(F(-), c)
  \,.
$$
\end{proposition}

Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the fact that the covariant Hom respects [[limit]]s (as described there) in $C^{op}$ in terms of $C$:

$$
  \begin{aligned}
    C(colim F, c) & \simeq C^{op}(c, colim F)
    \\
    & \simeq C^{op}(c, lim F^{op})
    \\
    & \simeq lim C^{op}(c, F^{op}(-))
    \\
    & \simeq lim C(F(-), c)
  \end{aligned}
$$

Notice that this actually says that $C(-,-) : C^{op} \times C \to Set$ is a [[continuous functor]] in both variables: in the first it sends limits in $C^{op}$ and hence equivalently colimits in $C$ to limits in $Set$.

\begin{proposition}
**([[adjoints preserve (co-)limits|left adjoint functors preserve colimits]])**
\linebreak
Let $L : C \to C'$ be a functor that is [[left adjoint]] to some functor $R : C' \to C$. Let $D$ be a [[small category]] such that $C$ admits limits of shape $D$. Then $L$ commutes with $D$-shaped colimits in $C$ in that

for $F : D \to C$ some diagram, we have
$$
  L(colim F) \simeq colim (L \circ F)
  \,.
$$
\end{proposition}

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

Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every $c' \in C'$
$$
\begin{aligned}
  C'(L (colim F), c) & \simeq C(colim F, R(c'))
  \\
  & \simeq lim C(F(-), R(c'))
  \\
  & \simeq lim C'(L \circ F(-), c')
  \\
  & \simeq C'(colim (L \circ F), c')
  \,.
\end{aligned}
  \,.
$$

Since this holds naturally for every $c'$, the [[Yoneda lemma|Yoneda lemma, corollary II]] on uniqueness of representing objects implies that $R (lim F) \simeq lim (R \circ F)$.
=--

## Related concepts

* [[filtered colimit]]

  * [[directed colimit]]

    * [[sequential colimit]]

* [[sifted colimit]]

* [[direct sum]]

* [[homotopy colimit]], [[(∞,1)-colimit]]

* [[lax colimit]]

## References

[[limit|Limits]] and colimits were defined in [[Daniel M. Kan]] in Chapter II of the paper that also defined [[adjoint functors]] and [[Kan extensions]]:

* [[Daniel M. Kan]], _Adjoint functors_, Transactions of the American Mathematical Society 87:2 (1958), 294–294 ([doi:10.1090/s0002-9947-1958-0131451-0](https://doi.org/10.1090/s0002-9947-1958-0131451-0)).

The observation that colimits may be constructed from [[coequalizers]] and set-indexed [[coproducts]]:

* {#Maranda62} [[Jean-Marie Maranda]], Thm. 1 in: *Some remarks on limits in categories*, Canadian Mathematical Bulletin **5** 2 (1962) 133-146 [[doi:10.4153/CMB-1962-015-0](https://doi.org/10.4153/CMB-1962-015-0)]

Beware that these early articles refer to colimits as _[[direct limits]]_.

Textbook account:

* [[Saunders MacLane]], §III.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects colimits]]
[[!redirects colimit functor]]
[[!redirects colimit functors]]


colimit in a dg-category

## Idea

One way to define [[homotopy colimits]] and [[homotopy limits]] in a [[dg-category]] is to compute them in the [[(infinity,1)-category]] of [[dg-modules]].

## References

See section 2.1 in

* [[B. Toen]], _Derived Azumaya algebras and generators for twisted derived categories_, [arXiv:1002.2599](http://arxiv.org/abs/1002.2599).

[[!redirects colimits in a dg-category]]

[[!redirects homotopy colimit in a dg-category]]
[[!redirects homotopy colimits in a dg-category]]

[[!redirects limit in a dg-category]]
[[!redirects limits in a dg-category]]

[[!redirects homotopy limit in a dg-category]]
[[!redirects homotopy limits in a dg-category]]


colimits in categories of algebras

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###Context###
#### Limits and colimits
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[[!include infinity-limits - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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[[!redirects cocompleteness of categories of algebras]]

#Contents#
* tic
{: toc}

## Introduction

Let $T$ be a [[monad]] on a [[category]] $C$, and let $C^T$ denote the [[Eilenberg-Moore category]] of $T$, i.e., the category of $T$-algebras. Let

$$U: C^T \to C$$

be the usual underlying or [[forgetful functor]], with [[left adjoint]] $F: C \to C^T$, [[unit of an adjunction|unit]] $\eta: 1_C \to U F$, and counit $\varepsilon: F U \to 1_{C^T}$. It is well-known that $U$ [[reflected limit|reflects limits]], so that if $C$ is [[complete category|complete]], then $C^T$ is also complete and $U$ is [[continuous functor|continuous]].

The situation with regard to colimits is more complicated. It is not generally true that if $C$ is cocomplete, then $C^T$ is also. (See [Adámek & Koubek](#AK), III.10, for an example when $C =$ [[Pos]].) In this article we collect various results that guarantee existence of colimits of algebras.

## Reflexive coequalizers and cocompleteness

A simple but basic fact is the following. Suppose $J$ is a [[small category]], and suppose that the functors $T$ and $T \circ T$ preserve colimits over $J$, that is, suppose that for every $F: J \to C$ the canonical map

$$colim_J T \circ F \to T(colim_J F)$$

is an [[isomorphism]], and similarly for $T \circ T$.

+-- {: .num_prop}
###### Proposition
Under these hypotheses, $U: C^T \to C$ [[creates colimits]] over $J$.
=--

For a proof, see Proposition 4.3.2 of [[Handbook of Categorical Algebra]] (Volume 2).

Here are some sample applications of this proposition which arise frequently in practice. Let $J$ be the generic _reflexive fork_, having exactly two objects $0, 1$, generated by three non-identity arrows

$$0 \to 1 \stackrel{\longrightarrow}{\longrightarrow} 0,$$

and subject to the condition that the two composites from $0$ to $0$ are the identity. A colimit over $J$ is called a [[reflexive coequalizer]]. It frequently happens that a monad $T: C \to C$ preserves reflexive coequalizers; in this case, if $C$ has reflexive coequalizers, then so does $C^T$.

The following very useful observation was first made by Linton.

+-- {: .num_theorem}
######Theorem
If $C$ is cocomplete and $C^T$ has reflexive coequalizers, then $C^T$ is cocomplete.
=--

+-- {: .proof}
######Proof
First observe that if $(c, \xi: T c \to c)$ is a $T$-algebra, then $\xi$ is the coequalizer of the reflexive fork

$$\array{
F U c & \stackrel{F \eta U c}{\longrightarrow} & F U F U c & \stackrel{\overset{\varepsilon F U c}{\longrightarrow}}{\underset{F \xi}{\longrightarrow}} & F U c
}$$

To show $C^T$ has coproducts, let $(c_i, \xi_i)$ be a collection of algebras. Then $F(\sum_i U c_i)$ is the coproduct $\sum_i F U c_i$ in $C^T$ (since $F$ preserves coproducts and $C$ has them). We have a reflexive fork

$$\array{
\sum_i F U c_i & \stackrel{\sum_i F \eta U c_i}{\to} & \sum_i F U F U c_i & \stackrel{\overset{\sum_i \varepsilon F U c_i}{\longrightarrow}}{\underset{\sum_i F \xi_i}{\longrightarrow}} & \sum_i F U c_i
}$$

and it is not difficult to show that the coequalizer in $C^T$ of this diagram is the coproduct $\sum_i c_i$.

Finally, general coequalizers in $C^T$ are constructed from coproducts and reflexive coequalizers: given a parallel pair $f, g: c \stackrel{\longrightarrow}{\longrightarrow} d$ in $C^T$, the coequalizer of $f$ and $g$ is the colimit of the reflexive fork

$$\array{
d & \to & c + d & \stackrel{\overset{(f, 1_d)}{\to}}{\underset{(g, 1_d)}{\to}} & d}$$

where the first arrow is the coproduct coprojection.
=--

+-- {: .num_cor}
###### Corollary
If $T$ is a monad on a complete and cocomplete category $C$ that preserves reflexive coequalizers, then $C^T$ is complete and cocomplete.
=--

The hypotheses of the preceding corollary hold when $C$ is a complete, cocomplete, cartesian closed category and $T$ is the monad corresponding to a [[Lawvere theory|finitary algebraic theory]]. (The key observation being that the finitary power functors $x \mapsto x^n$ preserve reflexive coequalizers if $C$ is cartesian closed.)

+-- {: .num_cor}
######Corollary
Assuming the [[axiom of choice]], if $T$ is a monad on [[Set]], then $Set^T$ is cocomplete. Similarly upon replacing $Set$ by a slice $Set/X \simeq Set^X$, or by [[Vect]].
=--

+-- {: .proof}
######Proof
It is enough to show that $Set^T$ has coequalizers. Suppose given a pair of algebra maps $f, g: A \stackrel{\to}{\to} B$ whose coequalizer we wish to construct. Let $R$ be the $T$-algebra relation
$$R = \langle f, g \rangle: A \to B \times B$$
and then let $E$ be the smallest $T$-congruence (equivalence relation that is a $T$-subalgebra map $E \hookrightarrow B \times B$) through which $R$ factors. (This is the intersection of all $T$-congruences through which $R$ factors, and may be calculated in $Set$, where it is reflected in $T$-$Alg$ since $U: Set^T \to Set$ reflects arbitrary intersections.) The coequalizer as calculated in $Set$,
$$\array{
U E & \stackrel{\overset{U \pi_1}{\longrightarrow}}{\underset{U \pi_2}{\longrightarrow}} & U B & \stackrel{p}{\longrightarrow} & Q
}$$
is a split coequalizer, because every quotient of an equivalence relation in $Set$ is a split coequalizer. (This requires the [[axiom of choice]]. A splitting is given by any splitting $i: Q \to U B$ of $p$, which picks a representative in each equivalence class, together with $\langle i p, 1 \rangle: U B \to U E$.) The proof is completed by the following lemma.
=--

+-- {: .num_lemma #split}
###### Lemma
In a category $\mathbf{C}$, given a pair $\pi_1, \pi_2: E \rightrightarrows B$ in $\mathbf{C}^T$ such that $U\pi_1, U\pi_2: U E \rightrightarrows U B$ has a split coequalizer $U B \to Q$ in $\mathbf{C}$, the pair $\pi_1, \pi_2$ has a coequalizer in $\mathbf{C}^T$ (reflected by the split coequalizer).
=--

+-- {: .proof}
###### Proof
(Cf. [[monadicity theorem]].) A split coequalizer is an absolute colimit, which the functor $T$ preserves. Hence the top row in

$$\array{
T U E & \stackrel{\overset{T U\pi_1}{\to}}{\underset{T U\pi_2}{\to}} & T U B & \stackrel{T p}{\longrightarrow} & T Q \\
\downarrow & & \downarrow & & \downarrow \\
U E & \stackrel{\overset{U\pi_1}{\longrightarrow}}{\underset{U\pi_2}{\longrightarrow}} & U B & \stackrel{p}{\longrightarrow} & Q
}$$

(the first two vertical arrows being algebra structure maps) is a coequalizer in $\mathbf{C}^T$. The last vertical arrow making the diagram commute gives $Q$ a $T$-algebra structure, and the split coequalizer in the bottom row is thereby reflected in $\mathbf{C}^T$ (i.e., lifts to a coequalizer in $\mathbf{C}^T$, albeit not necessarily to one that is itself split).
=--

### Categories of algebras are Barr exact

+-- {: .num_theorem #exact}
###### Theorem
If $\mathbf{C}$ is a [[regular category]] or [[exact category]] in which regular epimorphisms [[split epimorphism|split]], and $T$ is any monad on $\mathbf{C}$, then $\mathbf{C}^T$ is a regular category (or exact category, respectively).
=--

+-- {: .proof}
###### Proof
For regularity, we first construct coequalizers of kernel pairs in $\mathbf{C}^T$. So suppose $\pi_1, \pi_2: E \rightrightarrows B$ is the kernel pair of some $f: B \to C$ in $\mathbf{C}^T$. The kernel pair $U\pi_1, U\pi_2: U E \to U B$ of $U f$ in $\mathbf{C}$ has a coequalizer $q: U B \to Q$ in $\mathbf{C}$, and of course $U\pi_1, U\pi_2$ is the kernel pair of $q$ as well. It follows from the splitting hypothesis that the fork

$$\array{
U E & \stackrel{\overset{U\pi_1}{\longrightarrow}}{\underset{U\pi_2}{\longrightarrow}} & U B & \stackrel{q}{\longrightarrow} & Q}$$

splits in $\mathbf{C}$, hence by Lemma \ref{split} this diagram lifts to a coequalizer for $\pi_1, \pi_2: E \rightrightarrows B$ in $\mathbf{C}^T$. Thus kernel pairs in $\mathbf{C}^T$ have coequalizers.

That regular epis in $\mathbf{C}^T$ are stable under pullback follows a similar line of reasoning: let $p: B \to P$ be a regular epi in $\mathbf{C}^T$. It is the coequalizer of its kernel pair $\pi_1, \pi_2: E \rightrightarrows B$. We just calculated that the coequalizer $q: U B \to Q$ of $U\pi_1, U\pi_2$ in $\mathbf{C}$ lifts to $\mathbf{C}^T$, so that $Q$ is identified with $U P$ and $q$ with $U p$. Thus $U p$ is a regular epi in $\mathbf{C}$. Now if $f: A \to P$ is a map in $\mathbf{C}^T$, and $b = f^\ast p$ is the pullback of $p$ along $f$ (with kernel pair $\ker(b)$), then $U b$ is the pullback of $U p$ along $U f$ since $U$ preserves pullbacks, and so $U b$ is a regular epi since $\mathbf{C}$ is regular. This $U b$ is the coequalizer of its kernel pair, and splits, so by Lemma \ref{split}, its lift $b$ is the coequalizer of $\ker(b)$. Thus regular epis in $\mathbf{C}^T$ are stable under pullback.

For Barr-exactness, suppose $\pi_1, \pi_2: E \rightrightarrows B$ is an equivalence relation (or [[congruence]]) in $\mathbf{C}^T$. Then $U\pi_1, U\pi_2: U E \rightrightarrows U B$ is an equivalence relation in $\mathbf{C}$, and hence a kernel pair since $\mathbf{C}$ is exact. It is the kernel pair of its coequalizer $q$ in $\mathbf{C}$. By Lemma \ref{split}, the split coequalizer

$$\array{
U E & \stackrel{\overset{U\pi_1}{\longrightarrow}}{\underset{U\pi_2}{\longrightarrow}} & U B & \stackrel{q}{\to} & Q
}$$

lifts to a coequalizer diagram in $\mathbf{C}^T$, and since kernel pairs are preserved and reflected by $U: \mathbf{C}^T \to \mathbf{C}$, we conclude that $\pi_1, \pi_2$ is the kernel pair of the lifted regular epi over $q$.
=--

+-- {: .num_cor}
###### Corollary
If $T$ is a monad on a [[slice category]] $Set/X$, then the category of $T$-algebras is (Barr-)exact. If $T$ is a monad on $Vect$, then $Vect^T$ is exact.
=--

### For functors preserving filtered colimits

Returning now to existence of general coequalizers, here is a more difficult and arcane result given in [Toposes, Triples, and Theories](#BarrWells) (theorem 3.9, p. 267):

+-- {: .num_prop}
###### Proposition
If $C$ has coequalizers and equalizers of arbitrary sets of parallel morphisms, and if a monad $T: C \to C$ preserves colimits of countable chains $\omega \to C$, then $C^T$ has coequalizers.
=--

+-- {: .num_cor}
###### Corollary
If $C$ is complete and cocomplete and $T: C \to C$ preserves [[filtered colimits]], or even just colimits of $\omega$-chains, then $C^T$ is complete and cocomplete.
=--

### For locally presentable categories

If $C$ is a [[locally presentable category]] and $T$ is an [[accessible monad]] (aka a bounded monad, aka a monad with rank) on $C$, then $C^T$ is also locally presentable and in particular cocomplete. Details may be found in [[Locally presentable and accessible categories]].

## Algebraic functors have left adjoints

+-- {: .num_theorem}
###### Theorem
Suppose that $\theta: S \to T$ is a morphism of monads on $C$, and suppose that $C^T$ has coequalizers. Then the induced **algebraic functor**

$$C^\theta: C^T \to C^S$$

(pulling back a $T$-algebra $(c, \xi: T c \to c)$ to the $S$-algebra $(c, S c \stackrel{\theta c}{\longrightarrow} T c \stackrel{\xi}{\longrightarrow} c)$, thus remembering only underlying $S$-algebra structure) has a left adjoint.
=--

+-- {: .proof}
###### Proof
Since the following diagram is commutative:
$$
\begin{array}{cccc}C^T & \overset{C^{\theta}}{\to}  & C^S    \\
\mathllap{U^T} \downarrow     &                   & \downarrow \mathrlap{U^S} \\
C                    & =                 & C
\end{array}
$$
(using an obvious notation), it follows immediately from a corollary
to the [[adjoint lifting theorem]] that if $C^T$ has coequalizers of
reflexive pairs, then $C^{\theta}$ has a left adjoint
and is, in fact, monadic.

This actually completes the proof, but here is a concrete description of the left adjoint to $C^\theta$: it sends an $S$-algebra $(c, \xi: S c \to c)$ to the (reflexive) coequalizer of the pair

$$\label{coeq}\array{
 & & T T c & & \\
 & \mathllap{T \theta c} \nearrow & & \searrow \mathrlap{\mu c} \\
T S c & & \stackrel{\; \; \; \; \; \; \; \; T \xi \; \; \; \; \; \; \; \; }{\longrightarrow} & & T c
}$$

where $\mu: T T \to T$ is the monad multiplication. (If $u: 1_C \to S$ is the unit of $S$, then $T u c: T c \to T S c$ is a common right inverse of both arrows of the pair.) This coequalizer is analogous to the construction of the left adjoint $B \otimes_A -$ to the "restriction" functor $Ab^f: Ab^B \to Ab^A$ between module categories (restricting scalar multiplication on a $B$-module along a ring map $f: A \to B$); given an $A$-module $(M, \alpha_M: A \otimes M \to M)$, the $B$-module $B \otimes_A M$ is the coequalizer in $Ab^B$ of

$$\array{
 & & B \otimes B \otimes M & & \\
 & \mathllap{B \otimes f \otimes M} \nearrow & & \searrow \mathrlap{mult_B \otimes M} \\
B \otimes A \otimes M & & \stackrel{\; \; \; \; \; \; \; \; B \otimes \alpha_M \; \; \; \; \; \; \; \; }{\longrightarrow} & & B \otimes M,
}$$

and so the coequalizer of (eq:coeq) will be denoted $T \circ_S c$ to underline the analogy.

To see that $T \circ_S -$ is the left adjoint, let $(d, \alpha: T d \to d)$ be a $T$-algebra. Any map $f: c \to d$ in $C$ induces a unique $T$-algebra map $\phi: T c \to d$:

$$\phi = (T c \stackrel{T f}{\to} T d \stackrel{\alpha}{\to} d)$$

and the claim is that $f: c \to d$ is an $S$-algebra map $c \to C^\theta(d)$ if and only if $\phi$ coequalizes the pair in (eq:coeq), i.e., if $\phi$ factors (uniquely) through a $T$-algebra map $T \circ_S c \to d$.

Indeed, assume $f$ is an $S$-algebra map, so we have a commutative diagram

$$\label{alg}\array{
S c & \stackrel{\xi}{\to} & c \\
\mathllap{S f} \downarrow & & \downarrow \mathrlap{f} \\
S d & \underset{\alpha \circ \theta d}{\to} & d.
}$$

That $\phi = \alpha \circ T f$ coequalizes the pair of (eq:coeq) follows by expanding the diagram

$$\label{coeq2} \array{
T S c & \stackrel{T \theta c}{\to} & T T c & & & & \\
 & \mathllap{T \xi} \searrow & \downarrow \mathrlap{\mu c} & & & & \\
 & & T c & \underset{T f}{\to} & T d & \underset{\alpha}{\to} & d
}$$

to

$$\label{commute}\array{
 & & T S d & & & & \\
 & \mathllap{T S f} \nearrow & \; \; (nat) & \searrow \mathrlap{T \theta d} & & & \\
T S c & \stackrel{T \theta c}{\to} & T T c & \stackrel{T T f}{\to} & T T d & \stackrel{T \alpha}{\to} & T d \\
 & \mathllap{T \xi} \searrow & \downarrow \mathrlap{\mu c} & \; \; (nat) & \downarrow \mathrlap{\mu d} & \; \; (alg) & \downarrow \mathrlap{\alpha} \\
 & & T c & \underset{T f}{\to} & T d & \underset{\alpha}{\to} & d
}$$

where using (eq:alg), the path along the top may be replaced by $T f \circ T \xi: T S c \to T d$, reducing the desired coequalizing of (eq:coeq2) to the tautology $\alpha \circ T f \circ T \xi = \alpha \circ T f \circ T \xi$.

Conversely, assuming the coequalizing of (eq:coeq2), the perimeter of (eq:commute) commutes, and on top of that we stack naturality diagrams for the monad unit $\eta$ of $T$:

$$\label{commute2}\array{
 & & S d & & & & \\
 & _\mathllap{S f} \nearrow & _\mathllap{\eta S d} \downarrow & \searrow _\mathrlap{\theta d} & & & \\
S c & (nat) & T S d & (nat) & T d & \stackrel{\alpha}{\to} & d \\
_\mathllap{\eta S c} \downarrow & _\mathllap{T S f} \nearrow & \; \; (nat) & \searrow _\mathrlap{T \theta d} & \downarrow_\mathrlap{\eta T d} & (nat) & \downarrow _\mathrlap{\eta d}\\
T S c & \stackrel{T \theta c}{\to} & T T c & \stackrel{T T f}{\to} & T T d & \stackrel{T \alpha}{\to} & T d \\
 & \mathllap{T \xi} \searrow & \downarrow \mathrlap{\mu c} & \; \; (nat) & \downarrow \mathrlap{\mu d} & \; \; (alg) & \downarrow \mathrlap{\alpha} \\
 & & T c & \underset{T f}{\to} & T d & \underset{\alpha}{\to} & d
}$$

The vertical composite on the right is $1_d$ by a unit equation for a $T$-algebra, and thus we may simplify the perimeter. Retaining the (simplified) perimeter of (eq:commute2), and inserting some naturality squares and a unit diagram inside, we arrive at the commutative diagram

$$\array{
S c & \stackrel{S f}{\to} & S d & \stackrel{\theta d}{\to} & T d & & \\
_\mathllap{\eta S c} \downarrow & \searrow\; _\mathllap{\xi} & & & & \searrow \mathrlap{\alpha} & \\
T S c & (nat) & c & \stackrel{f}{\to} & d & \stackrel{1_d}{\to} & d \\
 & \mathllap{T \xi} \searrow & \downarrow _\mathrlap{\eta c} & (nat) & \downarrow _\mathrlap{\eta d} & (unit) & \downarrow \mathrlap{1_d} \\
 & & T c & \underset{T f}{\to} & T d & \underset{\alpha}{\to} & d
}$$

where the commutativity of the unlabeled polygonal region is just the commutativity of (eq:alg). This completes the proof of the claim.
=--

## References

* [[Michael Barr]] and [[Charles Wells]], _Toposes, Triples, and Theories_, Reprints in Theory and Applications of Categories (2005), 1-289. ([online pdf](http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf))
{#BarrWells}

* [[Jiří Adámek]] and Václav Koubek, _Are colimits of algebras simple to construct?_,  Journal of Algebra, Volume 66, Issue 1 (September 1980), 226-250. ([link](http://www.sciencedirect.com/science/article/pii/0021869380901222))
 {#AK}


colimits of equifibered transformations -- section

### Colimits of equifibered transformations
  {#ColimitsOfEquifiberedTransformations}

+-- {: .num_prop #CartesianTransformationsOfColimitsInAnInfinityTopos}
###### Proposition
**([[equifibered natural transformations]] of [[(∞,1)-colimits]] in an [[(∞,1)-topos]])**

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. For $\mathcal{I}$ a [[small (∞,1)-category]], write $\mathcal{I}^\rhd$ for the result of adjoining a [[terminal object]] (the shape of [[cocones]] under $\mathcal{I}$-shaped diagrams), and let

$$
  X^\rhd \overset{f^\rhd}{\Rightarrow} Y^\rhd \;\colon\; \mathcal{I}^\rhd  \longrightarrow \mathbf{H}
$$

be a [[natural transformation]] between two $\mathcal{I}^\rhd$-shaped [[diagrams]] ([[(∞,1)-functors|∞-functors]]), with
$$
  X \overset{f}{\Rightarrow} Y \;\colon\; \mathcal{I}  \longrightarrow \mathbf{H}
$$

denoting its restriction away from the cocone tip.

If

* $Y^\rhd$ is an [[(∞,1)-colimit]] diagram,

and

* $f$ is a [[cartesian natural transformation]],

then the following are equivalent:

1. $X^\rhd$ is an [[(∞,1)-colimit]] diagram,

1. $f^\rhd$ is a [[cartesian natural transformation]].

=--

([Rezk 10, 6.5](infinity,1-topos#Rezk10), [[Higher Topos Theory|Lurie, Theorem 6.1.3.9 (4)]])

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

Let $\mathcal{I} = \Delta^{op}$ be the [[opposite (∞,1)-category|opposite]] of the [[simplex category]], so that $\mathcal{I}^{\rhd} = \Delta_+^{op}$ is the opposite of the [[augmented simplex category]].

Let

$$
  X_\bullet, Y_\bullet \colon \Delta^{op} \longrightarrow \mathbf{H}
$$

be [[groupoid objects in an (∞,1)-category|groupoid objects]] and write

$$
  \array{
     X_0
     \\
     \big\downarrow
     \\
     \mathcal{X}
  }
  \phantom{AAAAA}
  ,
  \phantom{AAAAA}
  \array{
     Y_0
     \\
     \big\downarrow
     \\
     \mathcal{Y}
  }
$$

for the corresponding [[effective epimorphisms]] into their [[(∞,1)-colimits]].

Then Prop. \ref{CartesianTransformationsOfColimitsInAnInfinityTopos} implies that the following are equivalent:

1. a morphism of [[groupoid objects in an (∞,1)-category|groupoid objects]] $X_\bullet \overset{f_\bullet}{\Rightarrow} Y_\bullet$ is a [[cartesian natural transformation]];

1. the corresponding transformation of [[effective epimorphisms]]

   $$
     \array{
        X_0 &\overset{f_0}{\longrightarrow}& Y_0
        \\
        \big\downarrow &\swArrow& \big\downarrow
        \\
        \mathcal{X}
        &\underset{
          \underset{\longrightarrow}{\lim}f
        }{\longrightarrow}&
        \mathcal{Y}
     }
   $$

   is an [[(∞,1)-pullback]] square.

=--


colimits of normal spaces

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

# Contents
* table of contents
{:toc}

This page will collect some technical material concerning [[colimits]] of [[normal spaces]] as computed in [[Top]]. Here "normal" means $T_4$, i.e., we assume as part of normality that points are closed.

In particular it shows that [[CW-complexes are paracompact Hausdorff spaces|CW-complexes are normal spaces]] (theorem \ref{CWComplexesAreNormal} below).

A basic technique is the exploitation of the Tietze extension condition which characterizes normal spaces. Similar techniques may be used to prove a parallel set of results for [[paracompact spaces]]; see [[colimits of paracompact spaces]].

## Basic results

We start with some easy but useful results, using only our "bare hands" (i.e., using the definition of normality and applying simple reasoning).

The first very easy observation is that normal spaces are closed under [[coproducts]] in [[Top]] (so-called "[[disjoint union spaces]]"). The proof may be safely left to the reader.

There is no hope that normal spaces are closed under [[coequalizers]] or [[pushouts]]. A first example that comes to mind is the [[line with double origin]], which is the topological [[pushout]] of the diagram

$$\mathbb{R} \leftarrow \mathbb{R} \setminus \{0\} \to \mathbb{R}$$

where both maps are the inclusion map. This space isn't even [[Hausdorff space|Hausdorff]], as every [[neighborhood]] of the point $0_1$ (the image of the origin under the first pushout [[coprojection]]) intersects every neighborhood of the point $0_2$ (the image of the origin under the second coprojection).

However, there are reasonable conditions under which pushouts will be normal. (Throughout we assume all spaces are $T_1$, i.e., that [[singletons]] are [[closed set|closed]] ([here](separation+axioms#T1InTermsOfClosureOfPoints)), so that "normal" means $T_4$).

+-- {: .num_prop #normal}
###### Proposition
Let $X$ be normal, and suppose $q: X \to Y$ in $Top$ is a [[closed maps|closed]] [[surjective function|surjection]]. Then $Y$ is normal.
=--

+-- {: .proof}
###### Proof
First we claim singletons of $Y$ are closed: for each $y \in Y$ there exists $x \in X$ such that $y = q(x)$. Since $\{x\}$ is closed in $X$ and $q$ is closed, $\{y\} = q(\{x\})$ is closed in $Y$.

Let $C, D$ be disjoint closed sets of $Y$, so that $\neg C, \neg D$ form an open cover of $Y$. Then $A = q^{-1}(\neg C), B = q^{-1}(\neg D)$ form an open cover of $X$. Normality of $X$ (in "[[De Morgan duality|De Morganized]]" form) implies there are closed subsets $E \subseteq A, F \subseteq B$ which together also cover $X$. Then $Y = q(X) = q(E \cup F) = q(E) \cup q(F)$, so the closed sets $q(E), q(F)$ cover $Y$. And we have $q(E) \subseteq q(A) = \neg C$ and similarly $q(F) \subseteq \neg D$, which is equivalent to $C \subseteq \neg q(E)$ and $D \subseteq \neg q(F)$, where $\neg q(E), \neg q(F)$ are disjoint open sets of $Y$, and we are done.
=--

Here is one way in which such closed surjections arise: we know that [[compact Hausdorff spaces]] are normal spaces, and this is an especially nice class because the category of compact Hausdorff spaces is a [[pretopos]]. This is the categorical backdrop for the following observation. (We don't need this lemma in any result below; we include it merely as a handy lemma to have around.)

+-- {: .num_lemma #pretopos}
###### Lemma
Let $X$ be a compact Hausdorff space. Then an equivalence relation $\sim \hookrightarrow X \times X$ is closed as a subset of $X \times X$ iff the quotient map $q: X \to X/\sim$ is a closed map. (In which case $X/\sim$ is compact Hausdorff.)
=--

+-- {: .proof}
###### Proof
If $q: X \to X/\sim$ is a closed map, making $X/\sim$ compact Hausdorff, then the diagonal $\Delta \hookrightarrow X\sim \times X/\sim$ is a [[embedding of topological spaces|closed embedding]], so that its pullback along $q \times q$

$$\array{
\sim & \hookrightarrow & X \times X \\
\downarrow & & \downarrow \mathrlap{q \times q} \\
\Delta & \hookrightarrow & X/{\sim} \times X/{\sim}
}$$

also defines a closed subset of $X \times X$.

In the other direction, suppose $\sim$ is closed, and let $A \subseteq X$ be a closed subset of $X$. Consider its $\sim$-*saturation* $\bar{A}$, namely

$$\{x \in X: (\exists_{a \in A})\; x\sim a\} = \pi_1((\sim)\cap (X \times A)).$$

This is a closed set because the first projection $\pi_1$ is a closed map (by compactness of $X$). Moreover, $\bar{A} = q^{-1}(q(A))$, essentially by definition ($x \in q^{-1}(q(A))$ means $q(x) = q(a)$ for some $a \in A$). Since
$q^{-1}(q(A))$ is closed, $q(A)$ is closed by definition of quotient topology.
=--

+-- {: .num_remark}
###### Remark
A [[closed map|closed]] [[surjection]] is a quotient map (a [[regular epimorphism|regular epi]] in [[Top]]), and a closed [[injection]] is an [[embedding of topological spaces|embedding]] (a [[regular monomorphism|regular mono]] in $Top$).
=--

+-- {: .num_lemma #poclosed}
###### Lemma
In $Top$, the pushout of a closed embedding along any continuous map is again a closed embedding.
=--

+-- {: .proof}
###### Proof
We reproduce the proof given [here](/nlab/show/subspace+topology#pushout).

Since $U = \hom(1, -): Top \to Set$ is [[faithful functor|faithful]], we have that monos are reflected by $U$; also monos and pushouts are preserved by $U$ since $U$ has both a [[left adjoint]] and a [[right adjoint]]. In $Set$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $Top$. Furthermore, such a pushout diagram in $Set$ is also a pullback, so that we have the [[Beck-Chevalley condition|Beck-Chevalley equality]] $\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B)$ (where $\exists_i \colon P(A) \to P(B)$ is the [[direct image]] map between [[power sets]], and $f^\ast: P(C) \to P(A)$ is the [[inverse image]] map).

To prove that $j$ is a subspace, let $U \subseteq C$ be any open set. Then there exists open $V \subseteq B$ such that $i^\ast(V) = f^\ast(U)$ because $i$ is a subspace inclusion. If $\chi_U \colon C \to \mathbf{2}$ and $\chi_V \colon B \to \mathbf{2}$ are the maps to [[Sierpinski space]] that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map $\chi_W \colon D \to \mathbf{2}$ which extends the pair of maps $\chi_U, \chi_V$. It follows that $j^{-1}(W) = U$, so that $j$ is a subspace inclusion.

If moreover $i$ is an open inclusion, then for any open $U \subseteq C$ we have that $j^\ast(\exists_j(U)) = U$ (since $j$ is monic) and (by Beck-Chevalley) $g^\ast(\exists_j(U)) = \exists_i(f^\ast(U))$ is open in $B$. By the definition of the topology on $D$, it follows that $\exists_j(U)$ is open, so that $j$ is an open inclusion. The same proof, replacing the word "open" with the word "closed" throughout, shows that the pushout of a closed inclusion $i$ is a closed inclusion $j$.
=--

## Tietze characterization

A powerful tool for proving theorems about topological colimits of normal spaces is the characterization of normal spaces via the Tietze extension theorem:

+-- {: .num_theorem}
###### Theorem
A space $X$ is normal if and only if, for each closed subset $C \subseteq X$ and map $f: C \to \mathbb{R}$, there is an extension map $g: X \to \mathbb{R}$, i.e., a map whose restriction $g|_C$ coincides with $f$.
=--

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

1. There are variations on the theorem in which stronger separation properties (such as perfect normality, or $T_6$) may be reformulated in terms of extension conditions.

1. We remark that the "if" part of the proof is very easy. If $A, B$ are closed disjoint subsets of $X$, then the closed subspace $C = A \cup B$ is the coproduct on $A, B$ in $Top$, and we may define a map $f: A \cup B \to \mathbb{R}$ to be the constant $0$ on $A$ and the constant $1$ on $B$. Let $g: X \to \mathbb{R}$ be any extension of $f$; then $U = g^{-1}(\{x: x \lt 1/3\})$ and $V = g^{-1}(\{x: x \gt 2/3\})$ are separating open sets.
=--

The following is a sample application.

+-- {: .num_theorem #attach}
###### Theorem
If $X, Y, Z$ are normal spaces and $h: X \to Z$ is a closed embedding and $f: X \to Y$ is a continuous map, then in the pushout diagram in $Top$

$$\array{
X & \stackrel{h}{\to} & Z \\
\mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\
Y & \underset{k}{\to} & W,
}$$

the space $W$ is normal (and $k: Y \to W$ is a closed embedding, by the preceding Lemma).
=--

+-- {: .proof}
###### Proof
By the Tietze characterization, it suffices to show that any map $\phi: C \to \mathbb{R}$ on a closed subspace $C$ of $W$ can be extended to a map $\psi$ on all of $W$. Pulling back $C \hookrightarrow W$ along $k$, we have a closed subspace $k^{-1}(C) \hookrightarrow Y$ and a composite map $k^{-1}(C) \stackrel{k|}{\to} C \stackrel{\phi}{\to} \mathbb{R}$; call it $\alpha: k^{-1}(C) \to \mathbb{R}$. By normality of $Y$, the map $\alpha: k^{-1}(C) \to \mathbb{R}$ extends to a map $\beta: Y \to \mathbb{R}$. We obtain a map $\beta f: X \to \mathbb{R}$.

Similarly, pulling $C \hookrightarrow W$ back along $g$, we have a composite map $g^{-1}(C) \stackrel{g|}{\to} C \stackrel{\phi}{\to} \mathbb{R}$. We may now define a map

$$\gamma: h(X) \cup g^{-1}(C) \to \mathbb{R}$$

by $\gamma(h(x)) = \beta f(x)$ for $h(x) \in h(X)$, and $\gamma(z) = \phi(g(z))$ for $z \in g^{-1}(C)$. It is not hard to check that $\gamma$ is well-defined (by commutativity of the pullback square) and is continuous (using the fact that $h|: X \to h(X)$ is a homeomorphism, since $h$ is an embedding), and that $h(X) \cup g^{-1}(C)$ is a closed subset of $Z$ (since $h$ is closed). By normality of $Z$, we may extend $\gamma$ to a map $\delta: Z \to \mathbb{R}$, and we have just observed that $\delta h = \beta f$, so the pair $(\beta, \delta): Y + Z \to \mathbb{R}$ induces a unique map $\psi: W \to \mathbb{R}$ such that $\psi k = \beta$ and $\psi g = \delta$. Finally, the restriction of $\psi$ to $C$ is $\phi$, as required.
=--

## Sequences of normal spaces

+-- {: .num_prop #sequence}
###### Proposition
If $(i_n: X_n \to X_{n+1})_{n \in \mathbb{N}}$ is a countable sequence of closed embeddings between normal spaces, then the colimit $X = colim_n X_n$ is also normal.
=--

+-- {: .proof}
###### Proof
Let $A, B$ be disjoint closed subsets of $X$, and for all $n$ put $A_n = X_n \cap A$, $B_n = X_n \cap B$. Working recursively, suppose given disjoint open sets $U_n, V_n$ such that $A_n \subseteq U_n$ and $B_n \subseteq V_n$. Since normality guarantees that we can refine further if necessary, i.e., find an open $O_n$ such that $A_n \subseteq O_n$ and $\widebar{O_n} \subseteq U_n$, we may assume that the closures $\widebar{U_n}$, $\widebar{V_n}$ are disjoint in $X_n$, as are their images in $X_{n+1}$ under the closed embedding $i_n$. The closed sets $i_n(\widebar{U_n}) \cup A_{n+1}$ and $i_n(\widebar{V_n}) \cup B_{n+1}$ are disjoint in $X_{n+1}$:

* $A_{n+1} \cap B_{n+1} = \emptyset$ since $A \cap B = \emptyset$,

* $i_n(\widebar{U_n}) \cap i_n(\widebar{V_n}) = i_n((\widebar{U_n} \cap \widebar{V_n}) = i_n(\emptyset) = \emptyset$ where the direct image operator $i_n(-)$ preserves binary intersections since $i_n$ is monic,

* $i_n(\widebar{U_n}) \cap B_{n+1} = i_n(\widebar{U_n} \cap i_n^{-1}(B_{n+1})) = i_n(\widebar{U_n} \cap B_n) \subseteq i_n((\widebar{U_n} \cap \widebar{V_n}) = \emptyset$ (using Frobenius reciprocity), and similarly,

* $A_{n+1} \cap i_n(\widebar{V_n}) = \emptyset$.

Use normality of $X_{n+1}$ to select disjoint open sets $U_{n+1}, V_{n+1}$ such that $i_n(\widebar{U_n}) \cup A_{n+1} \subseteq U_{n+1}$ and $i_n(\widebar{V_n}) \cup B_{n+1} \subseteq V_{n+1}$, thus completing the recursive construction.

It is clear that $i_n(U_n) \subseteq U_{n+1}$ and the union $U = colim_n U_n$ defines an open set of $X$ (by definition of colimit topology), as does $V = colim_n V_n$. The sets $U, V$ include $A, B$ respectively and are disjoint since any element they have in common must belong to $U_n$ and $V_n$ for sufficiently large $n$, which is impossible. This completes the proof.
=--

+-- {: .num_remark}
###### Remark
An alternative proof using the Tietze characterization is easily given: if $C \subseteq colim_n X_n$ is closed and $f: C \to \mathbb{R}$ is continuous, then putting $C_n = X_n \cap C$ and $f_n = f|_{C_n}$, we define a suitable extension of each $f_n: C_n \to \mathbb{R}$ to a map $g_n: X_n \to \mathbb{R}$ by induction. Supposing at stage $n$ the map $g_n$ is given, we have a well-defined map $h_{n+1}: C_{n+1} \cup i_n(X_n) \to \mathbb{R}$ defined as $f_{n+1}$ on $C_{n+1}$ and as the composite $i_n(X_n) \cong X_n \stackrel{g_n}{\to} \mathbb{R}$ on $i_n(X_n)$. Then using normality of $X_{n+1}$, extend $h_{n+1}$ to a map $g_{n+1}: X_{n+1} \to \mathbb{R}$. Clearly $g_{n+1}$ extends $f_{n+1}$ and we have the compatibility equations $g_n = g_{n+1} i_n$, so the $g_n$ paste together to form a map $g: colim_n X_n \to \mathbb{R}$ which extends $f$.

A little use of notation allows a short exposition of this proof.
Let $Z$ be the topological space corresponding to the partial order $a\lt b\gt c\lt d\gt e$ (where $a,c,d$ are closed points, $b,d$ are open, and closed subsets are $\{a,b,c\},\{c,d,e\},\{a\},\{c\},\{e\},\emptyset$), and let
$g$ be the map gluing together points $b,c,d$. By Theorem 2.1,
$f$ has the left lifting property wrt $g$ whenever $f$ is a closed embedding into a normal space. As each $i_n: X_n \to X_{n+1}$, ${n \in \mathbb{N}}$ has the left lifting property wrt $g$, their transfinite composition $\emptyset\to colim_n X_n$ has the same lifting property,
which means that $X$ is normal.

=--

+-- {: .num_theorem #CWComplexesAreNormal}
###### Theorem
A [[CW-complex]] is a [[normal space]].
=--

+-- {: .proof}
###### Proof
A CW-complex $X$ is formed by an inductive process where the $n$-skeleton $X_n$ is formed as an [[attachment space]] formed from normal spaces. That is, we start with the normal space $X_{-1} = \emptyset$, and given the normal space $X_{n-1}$ and an attaching map $f: S_{n-1} = \sum_{i \in I} S_i^{n-1} \to X_{n-1}$, we push out the closed embedding $S_{n-1} = \sum_{i \in I} S_i^{n-1} \hookrightarrow \sum_{i \in I} D_i^n = D_n$ along the attaching map $f$ to get a closed embedding

$$i_n: X_{n-1} \to X_n (= X_{n-1} \cup_{S_{n-1}} D_n)$$

and deduce $X_n$ is normal by Theorem \ref{attach}. Then $X = colim_n X_n$ is normal by applying Proposition \ref{sequence}.
=--

## Related entries

* [[colimits of paracompact Hausdorff spaces]]


colimits of paracompact Hausdorff spaces

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

# Contents
* table of contents
{:toc}

## Introduction

In this article we collect some results on [[colimits]] of [[paracompact Hausdorff spaces]] as computed in the category of [[topological spaces]] [[Top]], and particular conditions under which the colimit is again paracompact and Hausdorff.

The account is designed to be parallel to that given in [[colimits of normal spaces]], centering particularly on how paracompactness may be reformulated in terms of an extension or selection property. The relevant result, due to Ernest Michael, is particularly well-adapted to the study of colimits.

In the sequel, a *paracompactum* (pl. *paracompacta*) is a paracompact Hausdorff space, just as [[compactum]] is a compact Hausdorff space.

## Basic results

+-- {: .num_prop #coproduct}
###### Proposition
The [[coproduct]] in $Top$ of a small family of paracompacta is a paracompactum.
=--

The proof is very easy. See [here](https://ncatlab.org/nlab/show/paracompact+topological+space#ParacompactnessPreservedByDisjointUnion).

+-- {: .num_prop}
###### Proposition
A [[closed subspace]] $A$ of a paracompactum $X$ is also a paracompactum.
=--

+-- {: .proof}
###### Proof
Let $U_\alpha$ be an [[open cover]] of $A$; note $\neg A \cup U_\alpha$ is the maximal open $V_\alpha$ in $X$ such that $U_\alpha = V_\alpha \cap A$. The $V_\alpha$ cover $X$. If $\mathcal{V}$ is a locally finite open refinement of this cover (via paracompactness of $X$), then the family $\{V \cap A: V \in \mathcal{V}\}$ is a locally finite open refinement of the family of sets $U_\alpha$. Finally, $A$ is Hausdorff because Hausdorffness is a [[hereditary property]].
=--

+-- {: .num_lemma #poclosed}
###### Lemma
In $Top$, the pushout $j$ of a (closed/open) embedding $i$ along any continuous map $f$,

$$\array{
A & \stackrel{i}{\hookrightarrow} & B \\
\mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\
C & \underset{j}{\hookrightarrow} & D,
}$$

is again a (closed/open) embedding.
=--

+-- {: .proof}
###### Proof
For the sake of convenience, we reproduce the proof given [here](/nlab/show/subspace+topology#pushout).

Since $U = \hom(1, -): Top \to Set$ is [[faithful functor|faithful]], we have that monos are reflected by $U$; also monos and pushouts are preserved by $U$ since $U$ has both a [[left adjoint]] and a [[right adjoint]]. In $Set$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $Top$. Furthermore, such a pushout diagram in $Set$ is also a pullback, so that we have the [[Beck-Chevalley condition|Beck-Chevalley equality]] $\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B)$ (where $\exists_i \colon P(A) \to P(B)$ is the [[direct image]] map between [[power sets]], and $f^\ast: P(C) \to P(A)$ is the [[inverse image]] map).

To prove that $j$ is a subspace, let $U \subseteq C$ be any open set. Then there exists open $V \subseteq B$ such that $i^\ast(V) = f^\ast(U)$ because $i$ is a subspace inclusion. If $\chi_U \colon C \to \mathbf{2}$ and $\chi_V \colon B \to \mathbf{2}$ are the maps to [[Sierpinski space]] that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map $\chi_W \colon D \to \mathbf{2}$ which extends the pair of maps $\chi_U, \chi_V$. It follows that $j^{-1}(W) = U$, so that $j$ is a subspace inclusion.

If moreover $i$ is an open inclusion, then for any open $U \subseteq C$ we have that $j^\ast(\exists_j(U)) = U$ (since $j$ is monic) and (by Beck-Chevalley) $g^\ast(\exists_j(U)) = \exists_i(f^\ast(U))$ is open in $B$. By the definition of the topology on $D$, it follows that $\exists_j(U)$ is open, so that $j$ is an open inclusion. The same proof, replacing the word "open" with the word "closed" throughout, shows that the pushout of a closed inclusion $i$ is a closed inclusion $j$.
=--

## Michael's selection characterization of paracompactness

(See also [[Michael's theorems]].)

It is well-known (see [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity|here]]) that a $T_1$ space $X$ is a paracompactum iff every open cover of $X$ [admits a subordinate partition of unity](/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity). Michael's innovation was in working out how the partition of unity characterization may be recast in terms of existence of continuous sections of suitable projection maps.

From here on out, all spaces will be assumed to be $T_1$ ([[singletons]] are [[closed subset|closed]]); see [[separation property]].

Recall that a [[relation]] in a [[category]] is a jointly monic [[span]]

$$\array{
 & & R & & \\
 & \mathllap{p} \swarrow & & \searrow \mathrlap{q} & \\
X & & & & Y
}$$

and that the relation is [[entire relation|entire]] if $p$ is an [[epimorphism]]. We will be working in the category $Top$, where epimorphisms are the same as surjective continuous functions. A *selection* for such a relation is by definition a [[section]] of $p$, where of course in $Top$ this means a continuous section. (In the category $Set$, the [[axiom of choice]] may be equivalently stated as saying that every entire relation admits a selection, a formulation which is sometimes credited to [[Charles Sanders Peirce|Peirce]].)

If $X$ is a space, let $\mathcal{O} X$ denote its topology, considered as a subset of the power set $P X$. Adapting the terminology of Michael, we will say a relation $X \stackrel{p}{\leftarrow} R \stackrel{q}{\to} Y$ is *lsc* ("lower semi-continuous") if the composite

$$P Y \stackrel{q^\ast}{\to} P R \stackrel{\exists_p}{\to} P X$$

restricts to a map $\exists_p q^\ast: \mathcal{O} Y \to \mathcal{O} X$. (In more down-to-earth terms, if we consider $R$ as a subspace of $X \times Y$, this says the [[image]] $\pi_X(R \cap (X \times V))$ under the projection $\pi_X: X \times Y \to X$ is open in $X$ whenever $V$ is open in $Y$.)

Let $B$ be a [[Banach space]] (for an example relevant to paracompactness concerns, $B$ could be $L^1(S)$, the Banach space freely generated by a set $S$ that indexes an open cover $\{U_s\}_{s \in S}$ of a given space $X$). We will say an (entire) relation $(R, p, q)$ from $X$ to $B$ in $Top$ is *closed and convex* if for each $x \in X$ the fiber $R_x = \exists_q p^\ast(x)$ is a closed [[convex set|convex]] subset of $B$.

+-- {: .num_theorem #Michael}
###### Theorem
**(E. Michael's selection theorem)**
A $T_1$ space $X$ is a paracompactum iff the condition

* for every [[Banach space]] $B$, every entire, lsc, closed and convex relation $(R, p, q)$ from $X$ to $B$ admits a selection

is satisfied.
=--

For now we omit giving the proof (given in [Michael](#Michael56)).

Before applying this theorem, we need a few lemmas that play supporting roles.

+-- {: .num_lemma #pullback}
###### Lemma
If $(R, p, q): X \nrightarrow B$ is entire, lsc, closed and convex, and $f: A \to X$ is a continuous map, then the pullback relation $(f^\ast R, r, q g): A \nrightarrow Y$ (as in the pullback diagram)

$$\array{
 & & f^\ast R & \stackrel{g}{\to} & R & & \\
 & \mathllap{r} \swarrow & (pb) & \swarrow \mathrlap{p} & & \searrow \mathrlap{q} & \\
A & \underset{f}{\to} & X & & & & B
}$$

is also entire, lsc, closed and convex.
=--

+-- {: .proof}
###### Proof
Clearly $r$ is surjective (being a pullback of a surjection), so the relation $f^\ast R$ is entire, and for $a \in A$ we have $(f^\ast R)_a = R_{f(a)}$ so that $f^\ast R$ is closed and convex if $R$ is. We also have

$$\exists_r (q g)^\ast = \exists_r g^\ast q^\ast = f^\ast \exists_p q^\ast$$

where the second equation follows from a Beck-Chevalley equation (induced by the pullback square). The map $\exists_p q^\ast$ preserves openness since $R$ is lsc, and $f^\ast$ preserves openness by continuity, so $\exists_r (q g)^\ast$ preserves openness, which proves $f^\ast R$ is lsc.
=--

The next result is an important observation about extending continuous selections.

+-- {: .num_lemma #extension}
###### Lemma
If $X$ satisfies the condition $\bullet$ of Theorem \ref{Michael} and $f: A \hookrightarrow X$ is a closed embedding, then for every $(R, p, q): X \to B$, every selection $\sigma: A \to f^\ast R$ of $(f^\ast R, r, q g)$ can be extended to a selection of $(R, p, q)$.
=--

+-- {: .proof}
###### Proof
Replace the relation $R$ by the relation $S$ from $X$ to $B$ where $S_{f(x)} = \{\sigma(x)\}$ if $x \in A$, and $S_x = R_x$ if $x \notin f(A)$. Clearly every fiber $S_x$ is closed, convex, and inhabited, so $S$ is closed and convex and entire. We check that $S$ is lsc. Let $V$ be open in $Y$; we must verify that every point $x$ of $\pi_X(S \cap (X \times V))$ is an interior point of that set. If $x \in \neg A$, this is clear since

$$\neg A \cap \pi_X(S \cap (X \times V)) = \neg A \cap \pi_X(R \cap (X \times V)$$

is open. If $x \in A$, then by continuity of $\sigma$ there is open neighborhood $U$ of $x$ such that $\sigma(x') \in V$ for all $x' \in U \cap A$, and then one may easily check that

$$U \cap \pi_X(R \cap (X \times V)) \subseteq \pi_X(S \cap (X \times V))$$

so that the left side is an open neighborhood of $x$ contained within the right side. Thus $S$ is lsc.

Since hypothesis $\bullet$ holds for $X$, we conclude that $S$ admits a selection $\tau$; by construction of $S$ we must have that $\tau |_A = \sigma$, and this completes the proof.
=--

## Applications to colimits of paracompacta

Now we put Michael's selection theorem to use in developing colimits of paracompacta.

+-- {: .num_theorem #ClosedEmbeddingsOfParacompactaClosedUnderPushout}
###### Theorem
If $X, Y, Z$ are paracompacta and $h: X \to Z$ is a closed embedding and $f: X \to Y$ is a continuous map, then in the pushout diagram in $Top$

$$\array{
X & \stackrel{h}{\to} & Z \\
\mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\
Y & \underset{k}{\to} & W,
}$$

the space $W$ is a paracompactum (and $k: Y \to W$ is a closed embedding, by Lemma \ref{poclosed}).
=--

(Compare a similar result for normal spaces, [here](https://ncatlab.org/nlab/show/colimits+of+normal+spaces#attach).)

+-- {: .proof}
###### Proof
Let $W \stackrel{p}{\leftarrow} R \stackrel{q}{\to} B$ be an entire, lsc, closed and convex relation to a Banach space $B$; by Michael's selection criterion, it suffices to show that $W$ is $T_1$ and $p$ admits a section. That $W$ is $T_1$ is easy: a point $x \in W$ either belongs to the closed subspace $Y \hookrightarrow W$ or it doesn't. If it does, then because $Y$ is Hausdorff, it is a closed point in a closed subspace, so it is a closed point in $W$. If it doesn't, then $k^{-1}(x) = \emptyset$ and $g^{-1}(x)$ is a singleton in $Z$ and thus closed in $Z$ since $Z$ is Hausdorff, i.e., the inverse image of $x$ under the quotient map $(k, g): Y + Z \to W$ is the closed subset $\emptyset + g^{-1}(x)$, so $x$ must be closed in $W$ by definition of quotient topology.

Thus we have only to prove $p$ admits a section. Using Lemma \ref{pullback}, we pull back $(R, p, q)$ to entire, lsc, closed and convex relations to $B$ from $Y$ and from $Z$:

$$\array{
X & \stackrel{h}{\to} & Z & & \\
\mathllap{f} \downarrow & & \downarrow \mathrlap{g} & \nwarrow & \\
Y & \underset{k}{\to} & W & & g^\ast R \\
 & \nwarrow & & \nwarrow \mathrlap{p} & \downarrow \\
 & & k^\ast R & \to & R.
}$$

By paracompactness of $Y$ and the selection criterion, there is a section $s$ of $k^\ast R \to Y$. The composite $s f: X \to k^\ast R$ in turn induces a selection $t: X \to f^\ast k^\ast R \cong h^\ast g^\ast R$. By Lemma \ref{extension}, the selection $t$ may be extended to a selection $u: Z \to g^\ast R$. The two maps

$$Y \stackrel{s}{\to} k^\ast R \to R, \qquad Z \stackrel{u}{\to} g^\ast R \to R$$

may be pasted together (i.e., their respective composites with $f$ and $h$ agree) to give a map out of the pushout, $i: W \to R$, which is easily checked to be a section of $p$.
=--

+-- {: .num_prop #ColimitsOfSequencesOfParacompacta}
###### Proposition
If $(i_n: X_n \to X_{n+1})_{n \in \mathbb{N}}$ is a countable sequence of closed embeddings between paracompacta, then the colimit $X = colim_n X_n$ is also a paracompactum.
=--

+-- {: .proof}
###### Proof
Clearly $X$ is $T_1$: for any $x \in X$, the intersection $\{x\} \cap X_n$ is closed in $X_n$ for all $n$, so $x$ must a closed point of $X$.

Let $j_n: X_n \to X$ denote a component of the colimit cocone. Let $(R, p, q)$ from $X$ to a Banach space $B$ be an entire, lsc, closed and convex relation. Pulling this back to $X_0$, by paracompactness we have a selection $X_0 \to j_0^\ast R$. Given a selection $X_n \to j_n^\ast R$, we may extend along $i_n$ to a selection $X_{n+1} \to j_{n+1}^\ast R$ using paracompactness and Lemma \ref{extension}. These selections, being compatible with the inclusions $i_n$, paste together to give a selection $X \to R$.
=--

In particular this may be used to see that _[[CW-complexes are paracompact Hausdorff spaces]]_.

## Related entries

* [[colimits of normal spaces]]

## References

* {#Michael56} [[Ernest Michael]], *Continuous Selections I*, The Annals of Mathematics, 2nd Series, Vol. 63 No. 2. (March 1956), 361-382. [stable URL](http://www.jstor.org/stable/1969615?seq=1#page_scan_tab_contents), ([pdf](http://www.renyi.hu/~descript/papers/Michael_1.pdf))

[[!redirects colimits of paracompact spaces]]


Colin Guillarmou

* [personal page](https://www.imo.universite-paris-saclay.fr/~guillarmou/)

## Selected writings

On the rigorous construction of [[Liouville theory]] via the [[conformal bootstrap]]:

* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))

and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88):

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

category: people


Colin McLarty

* [website](http://www.colinmclarty.com/)

## Selected writings

On [[The Rising Sea]]-analogy by [[Alexander Grothendieck]]:

* {#MacLarty03} [[Colin McLarty]], _The Rising Sea: Grothendieck on simplicity and generality I_, 2003 ([pdf](http://www.cwru.edu/artsci/phil/RisingSea.pdf))

On the history of [[topos theory]]:

* [[Colin McLarty]], _The Uses and Abuses of the History of Topos Theory_ , Brit. J. Phil. Sci., 41 (1990) ([JSTOR](http://www.jstor.org/stable/687825)) [PDF](http://bjps.oxfordjournals.org/content/41/3/351.full.pdf)

* {#McLarty18} [[Colin McLarty]], _Grothendieck's 1973 topos lectures_, Séminaire Lectures grothendieckiennes, 3 May (2018) ([YouTube video](https://www.youtube.com/watch?v=5AR55ZsHmKI))

Exposition of [[sheaf toposes]]:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

On [[category theory]] and [[elementary topos|elementary]] [[topos]] [[sheaf and topos theory|theory]]:

* [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

On [[Saunders MacLane]]:

* [[Colin McLarty]], _The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics_,Brit. J. Phil. Sci. 2007 ([pdf](http://www.cwru.edu/artsci/phil/BJPSMacLane.pdf))

category: people


Colin Rourke

* [webpage](http://msp.warwick.ac.uk/~cpr/)

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

## Selected writings

On [[semi-simplicial sets]] and their [[geometric realization]]:

*  {#RourkeSanderson71} [[Colin Rourke]],  [[Brian Sanderson]], _Δ-Sets I: Homotopy Theory_. The Quarterly Journal of Mathematics 22: 321–338 (1971) ([doi:10.1093/qmath/22.3.321](https://doi.org/10.1093/qmath/22.3.321), [pdf](http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf))

Discussion of [[equivariant homotopy theory|equivariant]] [[configuration spaces of points]] in [[Euclidean G-spaces]] and the [[Cohomotopy charge map]]-equivalence:

* {#RourkeSanderson00} [[Colin Rourke]], [[Brian Sanderson]], _Equivariant Configuration Spaces_, J. London Math. Soc. 62 (2000) 544-552 ([arXiv:math/9712216](https://arxiv.org/abs/math/9712216))

category: people


Colin Tan

I am a postdoc working in type-theoretic approaches to linked data at the [Singapore University of Technology and Design](https://sutd.edu.sg). My PhD thesis approached a problem with origins in real algebraic geometry using analytic methods from complex geometry.

category: people


Colin Zwanziger

I am a graduate student in the philosophy department at Carnegie Mellon University interested in categorical logic, applications of category theory to [[linguistics]], philosophy of math/language...

[Website](https://www.colinzwanziger.com)

## Selected writings and talks

* [[Colin Zwanziger]], *Hofmann-Streicher Universes via Coalgebra*, [[Homotopy Type Theory Electronic Seminar Talks]], 3 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zwanziger-2022-11-03-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=UG0xwTNUplo))

category:people


collar neighbourhood theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _collar neighbourhood theorem_ due to [Brown 62](#Brown62), hence also known as _Brown's collaring theorem_, says that the [[boundary]] of any [[manifold with boundary]] always admits a "collar", namely an [[open neighbourhood]] which is the [[Cartesian product]] of the boundary with a [[half-open interval]].

This theorem is central notably for the definition and behaviour of [[categories of cobordisms]].

## Statement

Let $X$ be a [[topological manifold]] or [[smooth manifold]] [[manifold with boundary|with boundary]] $\partial X$. Then the [[boundary]] [[topological subspace|subspace]] inclusion has an [[open neighbourhood]] which is [[homeomorphism|homeomorphic]] or [[diffeomorphic]], respectively, to a [[collar]], i.e. to the [[product space|Cartesian product]] manifold with boundary $\partial X \times [0,1)$ of $\partial X$ with the [[half-open interval]]:

$$
  \partial X
  \overset{
     (id, 0)
  }{\hookrightarrow}
  (\partial X) \times [0,1)
  \overset{et}{\hookrightarrow}
  X
  \,.
$$

## Related concepts

* [[cobordism]]

## References

Original references are:

* {#Brown62} [[Morton Brown]], _Locally flat imbeddings of topological manifolds_, Annals of Mathematics, Vol. 75 (1962), p. 331-341 ([jstor:1970177](https://www.jstor.org/stable/1970177))

* [[Robert Connelly]], _A new proof of Brown's collaring theorem_, Proceedings of the American Mathematical Society 27 (1971), 180 – 182 ([jstor:2037284](https://www.jstor.org/stable/2037284))

Quick review and sketch of the proof is in

* p. 5 of _Manifolds with boundary_ ([pdf](http://math.ucr.edu/~res/math260s10/manwithbdy.pdf), [[ManifoldsWithBoundary.pdf:file]])

[[!redirects collar]]
[[!redirects collars]]


collection

Mathematicians sometimes use the word **collection** to denote a bunch of "things" without prejudice as to whether those things form a [[set]], a [[proper class]], or some other formal notion of collection such as a [[type]].  For example, the word is often used this way in the definition of a [[category]].


collective field theory

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

__Collective field theory__ is a generalization of the method of treating plasma oscillations from

* D. Bohm, D. Pines, Phys. Rev. __92__ (1953) 609;

to more general quantum systems. It has many applications in [[large N limit]].

* A. Jevicki, B. Sakita, _The quantum collective field method and its application to the planar limit_, Nucl. Physics B __165__, 3 (1980) 511-527, doi
* A. Jevicki, B. Sakita, _Collective field approach to the large-N limit: Euclidian field theories_, Nucl. Physics B __185__, 1 (1981) 89-100, doi, [kek scan](http://www-lib.kek.jp/cgi-bin/img_index?8103461)
* A. Jevicki, B.Sakita, Phys. Rev. D 22, 467 (1980).
* A. Jevicki, _Non-perturbative collective field theory_, Nucl. Physics B __376__, 1 (1992) 75-98, doi
* A. Jevicki, _Higher Spin AdS-Duality from CFT:a constructive approach_, conference slides 2011, [pdf](http://bajnok.web.elte.hu/JHW/Talks/Jevicki.pdf)
* Antal Jevicki, João P. Rodrigues, _Supersymmetric collective field theory_, Physics Letters B __268__, Issue 1 (1991), 53-58, doi
* B. Sakita, _Field theory of strings as a collective field theory of $U(N)$ gauge fields_, Phys. Rev. D 21, 1067–1073 (1980) [doi](http://dx.doi.org/10.1103/PhysRevD.21.1067)
* M. Mondello, E. Onofri, _Planar limit of the singlet spectrum for $SU(N)$-invariant quantum hamiltonians by the quantum collective field method_, Physics Letters B __98__, Issue 4 (1981), 277-279, doi


Colleen Delaney

* [webpage](https://web.math.ucsb.edu/~cdelaney/research.html)

## Selected writings

On [[braid group representations]] (specifically the [[Burau representation]] and its relation to the [[Alexander polynomial]]) and their application to [[topological quantum computation]]:

* [[Colleen Delaney]], [[Eric C. Rowell]], [[Zhenghan Wang]], *Local unitary representations of the braid group and their applications to quantum computing*, Revista Colombiana de Matemáticas(2017), 50 (2):211 ([arXiv:1604.06429](https://arxiv.org/abs/1604.06429), [doi:10.15446/recolma.v50n2.62211](http://dx.doi.org/10.15446/recolma.v50n2.62211))

On [[modular tensor categories]] and [[anyon|anyonic]] [[topological order]]:

* [[Colleen Delaney]], *A categorical perspective on symmetry, topological order, and quantum information* (2019) $[$[pdf](https://crdelane.pages.iu.edu/dissertationch1-5.pdf), [uc:5z384290](https://escholarship.org/uc/item/5z384290)$]$

On [[modular tensor categories]] and [[braid group representations]]:

* [[Colleen Delaney]], *Lecture notes on modular tensor categories and braid group representations* (2019) $[$[pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf), [[DelaneyModularTensorCategories.pdf:file]]$]$

category: people


collineation

A __collineation__ is a bijection from a projective space to a projective space of the same finite dimension which preserves the inclusions of subspaces. Equivalently, it is often defined by a seemingly weaker condition, which is also used in the infinite-dimensional case: a collineation is a bijection of projective spaces which sends each triple of collinear points into a triple of collinear points.

A __projective collineation__ is a collineation such that its restriction to every projective hyperplane is a projective isomorphism.

A __perspective collineation__ is a (projective) collineation from a projective plane to itself such that the restriction to one, and hence to any projective line is a perspectivity onto its image line.

Similarly, a __correlation__ is a bijection from a projective space to a projective space of the same finite dimension which reverses the inclusions of subspaces.

The terms _collineation_ and _correlation_ have been introduced into projective geometry
by Moebius in 1827.

## Literature

* Jacquelline Lelong-Ferrand, Les fondements de la géométrie
* Coxeter, Real projective geometry

category: geometry

[[!redirects projective collineation]]
[[!redirects perspective collineation]]


collision entropy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

(...)

The collision entropy of a [[probability distribution]] on a [[finite set]] is the [[Rényi entropy]] at order 2:

$$
  S_2(p)
    \;=\;
  -\log
  \left(
    \sum_{i=1}^n
    p_i^2
  \right)
  \,,
$$

hence is the negative [[logarithm]] of the "collision probability", i.e., of the probability that two independent [[random variables]], both described by $p$, will take the same value.

## Related concepts

[[!include Renyi entropies -- table ]]

## References

* G. M. Bosyk, M. Portesi, A. Plastino, *Collision entropy and optimal uncertainty*, Phys. Rev. A 85 (2012) 012108 ([arXiv:1112.5903](https://arxiv.org/abs/1112.5903))

See also:

* Wikipedia, *[Collision entropy](https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy#Collision_entropy)*

[[!redirects collision entropies]]


colocalization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[localization]] is a universal functor from a given category $C$ with respect to the inversion of some family $\Sigma$ of morphisms in $C$; sometimes one says also quotient category. In topos theory and some other subjects one often restricts to the situation when $\Sigma$ is a [[calculus of left fractions]], and the corresponding localization functor has a [[right adjoint]] (which is then necessarily fully faithful); even more, one often requires that the localization functor is also [[left exact]], hence [[exact functor|exact]].

One may then speak of _colocalization_ as the [dual](duality#abstractformalaxiomatic_duality) version of localization, a universal functor into a quotient category where now $\Sigma$ admits a [[calculus of right fractions]] and the corresponding quotient functor has a [[left adjoint]].  See also at _[[right Bousfield localization]]_.

The theory of colocalization in [[co-Grothendieck categories]] has some features of its own as compared to the localization in [[Grothendieck categories]]. Namely, while by Gabriel-Popescu's theorem, every Grothendieck category is a localization of a category of modules over a fixed unital ring, their dual categories may be presented
in terms of the theory of linear topological rings with some compactness properties, which is the content of [[Gabriel-Oberst duality]] theory.

## Related concepts

* [[right Bousfield delocalization]]

## References

* C. Năstăsescu, B. Torrecillas, _Colocalization on Grothendieck categories with applications to coalgebras_, J. Algebra __185__ (1996), 108–124 ([pdf](http://www.sciencedirect.com/science/article/pii/S0021869396903154))

A textbook exposition is in the chapter 6, _Duality_ of

* [[Nicolae Popescu]], _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

In derived and triangulated categories:

* Shoham Shamir, _Colocalization functors in derived categories and torsion theories_, [arxiv/0910.4724](http://arxiv.org/abs/0910.4724)
* Hvedri Inassaridze, Tamaz Kandelaki, Ralf Meyer, _Localisation and colocalisation of triangulated categories at thick subcategories_, [arxiv/0912.2088](http://arxiv.org/abs/0912.2088)

category: algebra
[[!redirects colocalisation]]


Colombeau algebra

#Content#
* table of contents
{:toc}

## Idea

The [[product of distributions]] is not canonically defined on all distributions; it is only partially defined on pairs of distributions whose sum of [[wave front sets]] does not intersect zero ("[[microlocal analysis]]").

In contrast, [Colombeau 92](#Colombeau92) defines algebra structures on distributions, now called _Colombeau algebras_ (see [Gratus 13](#Gratus13) for review) which are globally defined, but at the cost that

1. they are not uniquely defined,

1. they do not restrict to the usual pointwise product on all functions.

Briefly: Colombeau considers sequences of functions that converge to distributions (weakly) and defines the product of two distributions as the product of the sequences. This product is not independent of the chosen sequences, which means that the level of abstraction achieved by distribution theory is abandoned.

## References

* {#Colombeau92} Jean François Colombeau: _Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics._, Springer 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0815.35002&format=complete))

* {#Gratus13} Jonathan Gratus, _Colombeau Algebra: A pedagogical introduction_ ([arViv:1308.0257](https://arxiv.org/abs/1308.0257))

* Hans Vernaeve, _Algebras of generalized functions and Nonstandard analysis_, 2008 ([pdf](http://fibonacci.dm.unipi.it/cluster-pages/ultramath/slides/vernaeve-slides.pdf))

* Wikipedia, _[Colombeau algebra](https://en.wikipedia.org/wiki/Colombeau_algebra)_

[[!redirects Colombeau algebras]]


color branes and flavor branes

[[!redirects flavor brane]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[geometric engineering of quantum field theory]] in [[intersecting D-brane models]] the [[gauge theory]] which is thought to appear on [[coincident brane|coincident]] [[D-branes]] (see at _[[gauge enhancement]]_) may play two different roles:

1. *[[color charge|color]]* -- it may be a [[Yang-Mills theory]] of an "actual" [[gauge field]] (carried by [[gluons]]) coupled to [[color charges]] (carried by [[quarks]]) -- like [[quantum chromodynamics]];

1. *[[flavor (particle physics)|flavor]]* -- it may be the ("[[chiral perturbation theory|chiral]]") [[gauge theory]] of a [[hidden local symmetry|hidden local gauge field]] (carried by [[mesons]]) coupled to [[flavour (particle physics)|flavor charges]] (carried by [[baryons]]) -- like [[quantum hadrodynamics]].

In the first case one speaks of _color branes_, in the second of _flavor branes_. Typically one indicates the [[number]] of [[coincident brane|coincident]] such branes with

1.  $N_c \in \mathbb{N}$ for the number of color branes, leading (in the absence of [[orientifolds]]) to [[gauge group]] [[special unitary group|SU]]$(N_c)$;

1.  $N_f \in \mathbb{N}$ for the number of flavor branes, leading to [[flavor (particle physics)|flavor]]-[[symmetry group]] ("[[chiral symmetry]]") [[unitary group|SU]]$(N_f)$ (e.g. [[isospin]] for $N_f = 2$).

| | [[color charge]] | [[flavor physics|flavor charge]] |
|----|----|-----|
| **[[gauge bosons]]** | [[gluons]]
  ([[gauge group]]-[[local symmetry]])  | [[mesons]]
 ([[flavor physics|flavor]]-[[hidden local symmetry]]) |
| **[[fermions]]** | [[quarks]] | [[baryons]] |

In common constructions of [[holographic QCD]] in the [[large-N limit]] (large number of [[color charges]]) in which the [[AdS/QCD correspondence]] applies, [[color branes]] are modeled  as $N_c \ggt 1$ [[black branes]] while [[flavor branes]] are modeled as $N_f \sim 1$ [[probe branes]] ([Karch-Katz 02](#KarchKatz02)).

From [Ouyang 03, p. 2](#Ouyang03):

>  the important feature seems to be that the added branes must be  extended along the radial [[AdS spacetime|AdS]] direction; then, volume factors suppress the dynamics of the NN strings on these "flavor branes", which then contribute states to the gauge theory with [[global symmetries]] rather than [[gauge symmetries]].

## Examples

### Witten-Sakai-Sugimoto model for quantum chromodynamics

For example, in the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]] realized on [[D4-D8 brane intersections]], the [[D4-branes]] play the role of _color branes_ while the [[D8-branes]] play the role of _flavor branes_.

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Here we are showing

1. the [[color brane|color]] [[D4-branes]];

1. the [[flavor brane|flavor]] [[D8-branes]];

   with

   1. the [[5d Chern-Simons theory]] on their [[worldvolume]]

   1. the corresponding [[4d WZW model]] on the boundary

   exhibiting the [[vector meson fields]] in the [[Skyrmion model]];

1. the [[baryon]] [[D4-branes]]

   (see below at _[Baryons](#Baryons)_);

1. the [[Yang-Mills monopole]] [[D6-branes]]

   (see at _[[D6-D8-brane bound state]]_);

1. the [[NS5-branes]] (often not considered here).

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Phenomenology

The [[geometric engineering of QFT]] on flavor branes (as in the [[Witten-Sakai-Sugimoto model]]) realizes, at least qualitatively, the following [[experiment|experimentall]] phenomena:

* [[hadron supersymmetry]]

* [[hadron Kaluza-Klein theory]]

## Related concepts

* [[probe brane]]

## References

### General

The concept of [[flavor branes]] in the context of [[holographic QCD]] properly originates with:

* {#KarchKatz02} [[Andreas Karch]], [[Emanuel Katz]],  _Adding flavor to AdS/CFT_, JHEP 0206:043, 2002 ([arxiv:hep-th/0205236](https://arxiv.org/abs/hep-th/0205236))

based on the concept of [[probe branes]] due to

* [[Andreas Karch]], [[Lisa Randall]], Section 3 of: _Open and Closed String Interpretation of SUSY CFT's on Branes with Boundaries_, JHEP 0106:063, 2001 ([arXiv:hep-th/0105132](https://arxiv.org/abs/hep-th/0105132))

Other early discussion:

* {#Ouyang03} [[Peter Ouyang]], _Holomorphic D7-Branes and Flavored $\mathcal{N}=1$ Gauge Theories_, Nucl. Phys. B699:207-225, 2004 ([arXiv:hep-th/0311084](https://arxiv.org/abs/hep-th/0311084))

* Thomas S. Levi, [[Peter Ouyang]], _Mesons and Flavor on the Conifold_, Phys. Rev. D76:105022, 2007 ([arXiv:hep-th/0506021](https://arxiv.org/abs/hep-th/0506021))

* [[Carlos Nunez]], A. Paredes, A.V. Ramallo, _Flavoring the gravity dual of $\mathcal{N}=1$ Yang-Mills with probes_, JHEP 0312:024, 2003 ([arXiv:hep-th/0311201](https://arxiv.org/abs/hep-th/0311201))

See also:

* Daniel Arean, _Adding flavor on the Higgs branch_, Fortsch. Phys. 56:888-894, 2008 ([arXiv:0805.3447](https://arxiv.org/abs/0805.3447))

[[!include flavor on heterotic M5-branes -- references]]

[[!redirects flavor branes]]

[[!redirects flavour brane]]
[[!redirects flavour branes]]

[[!redirects color brane]]
[[!redirects color branes]]


color charge

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

#Contents#
* table of contents
{:toc}

## Idea

The [[charge]] carried by [[quarks]] and [[gluons]] in the description of the [[strong nuclear force]] via [[quantum chromodynamics]] is called _color charge_.

If the [[gauge group]] is [[SU(N)]] and [[fermions]] are in the [[fundamental representation]] $\mathbf{N}$ of [[dimension]] $N$, so that $\mathbf{N}$ has a [[linear basis]] of $N$ elements, one says that these are $N$ colors of the given [[fermion]] [[fundamental particle|particle]].

Hence, more generally one could reasonably say that the "number of colors" of a given [[fundamental particle]] in [[gauge theory]] is physics synonym for the choice of [[linear representation]] of the [[gauge group]] that defines this particle.

## Related concepts

* [[large N limit]], [[large 1/N limit]]

* [[color brane]]

* [[flavour (particle physics)]]

* [[electric charge]], [[magnetic charge]]

## References

* [[Howard Georgi]], §16  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

* Wikipedia, _[Color charge](https://en.wikipedia.org/wiki/Color_charge)_

[[!redirects color charges]]

[[!redirects colour charge]]
[[!redirects colour charges]]

[[!redirects number of colors]]
[[!redirects numbers of colors]]

[[!redirects number of colours]]
[[!redirects numbers of colours]]


colorable knot

# Colourability
* table of contents
{: toc}

## Idea

The colourability of a knot tells one information about its [[knot group]] yet has a simple, and visually attractive aspect that seems almost to avoid all mention of groups, presentations, etc., except at a fairly naive level.

The easiest form of colourability to examine is $3$-colourability.

## $3$-colourability

+--{: .un_defn}
###### Definition
A [[knot diagram]] is _$3$-colourable_ if we can assign colours to its arcs such that

1. each arc is assigned one  colour;

1.  _exactly_ three colours are used in the assignment;

1. at each crossing, either all the arcs have the same colour, or arcs of all three colours meet in the crossing.

=--

+--{: .un_example}
###### Examples and non-examples

* The usual diagram for the trefoil knot is $3$-colourable. (Just do it! Each arc is given a separate colour and it works.)

* The [[figure-8 knot]] diagram

  +-- {: style="text align:center"}
  [[!include figure 8 knot - SVG]]
  =--

  is not $3$-colourable. (Try it!)

=--

+--{: .un_theorem}
######Theorem
$3$-colourability is a [[knot  invariant]].

=--

The proof is amusing to work out oneself. You have to show that if a knot diagram $D$ is $3$-colourable and you perform a [[Reidemeister move]] on it then the result is also $3$-colourable. The thing to note is that any arcs that leave the locality of the move must be coloured the same before and after the move is done.

+--{: .un_note}
###### Notes
*  We can now use phrases such as 'the trefoil knot is $3$-colourable' as its validity does not depend on what diagram is used to represent it, (by the above and by [[Reidemeister moves|Reidemeister's theorem]].)

* As the trefoil knot is $3$-colourable and the unknot is not, *non-trivial knots exist*.  Moreover, the trefoil is $3$-colourable and the figure $8$ is not, so these are different. We also get that the [[bridge number]] of the trefoil is $2$, as this provides the missing piece of the argument found in that entry.

=--

 There are two comments to make here.  First what does this all mean at a deeper topological level?  The other is : why stop at $3$? What about $n$-colourability? We will handle the second one first.

##$n$-colourability##
 Let $n$ be an integer - in practice $n = 1$ or 2 are not that interesting, so usually $n \geq 3$. Let $\mathbb{Z}_n$ be the additive group of integers modulo $n$.

+--{: .un_defn}
###### Definition
An _$n$-colouring_ of a knot diagram, $D$, is an assignment to each arc of an element of $\mathbb{Z}_n$ in such a way that, at each crossing, the sum of the values assigned on the underpass arcs is twice that on the overpass, and such that in the assignment at least two elements of $\mathbb{Z}_n$ are used.

=--

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with $a+c \equiv_n 2b$.

What, of course, needs to be checked (left to 'the reader') is

* this notion is a generalisation of 3-colourability (i.e. coincides in the case of $n = 3$);

* $n$-colourability is preserved under [[Reidemeister moves]] so can be applied to a knot, not just to a knot diagram;

and then to find some examples of, say, 5-colourability. What knots are 5-colourable? Which $(2,k)$-[[torus knots]] are 5-colorable, and so on.  This is not important mathematically, but is quite fun and, in fact, does give insight and experience in working with the Reidemeister moves.

## Coloring by a quandle

There is an even more general notion of coloring a knot $K$ by the elements of a [[quandle]] $(Q,\rhd : Q \times Q \to Q)$.  Formally, a coloring of $K$ by $Q$ corresponds to a quandle homomorphism from the fundamental quandle $Q(K)$ of $K$ to $Q$.  Concretely, this says that at each crossing with arcs labelled $a$, $b$, and $c$ (as in the above diagram), the identity $c = a \rhd b$ must be respected.  In particular, $n$-coloring corresponds to coloring by the set $\mathbb{Z}_n$ equipped with the quandle operation $a \rhd b = 2b - a\,(mod\,n)$, known as the dihedral quandle.

## References

* [[Ralph Fox]], _A quick trip through Knot theory_ [pdf file]( (http://homepages.math.uic.edu/~kauffman/QuickTrip.pdf)

* Wikipedia articles on [tricolorability](http://en.wikipedia.org/wiki/Tricolorability) and [Fox n-coloring](http://en.wikipedia.org/wiki/Fox_n-coloring)

[[!redirects colorable knot]]
[[!redirects colorable knots]]
[[!redirects colourable knot]]
[[!redirects colourable knots]]

[[!redirects colorability]]
[[!redirects colourability]]

[[!redirects coloring knots]]
[[!redirects knot coloring]]
category: knot theory


colored Jones polynomial

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

#Contents#
* table of contents
{:toc}

## Idea

The **colored Jones polynomial** of a [[knot]] or [[link]] is a generalization of the [[Jones polynomial]]. The latter may be seen as 'colored' by the defining 2-dimensional [[complex number|complex]] [[linear representation]] of [[SU(2)]], where the $N$-colored Jones polynomial $J_N(K, q)$ involves instead the $N$-dimensional [[irreducible representation]].

##Related concepts

* [[volume conjecture]]

## References

* [[Stavros Garoufalidis]], Martin Loebl, _A non-commutative formula for the colored Jones function_ ([arXiv:math/0411505](https://arxiv.org/abs/math/0411505))

Relation of the [[colored Jones polynomial]] to [[Lie algebra weight systems]] on [[chord diagrams]]:

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], _On the Melvin–Morton–Rozansky conjecture_, Invent math (1996) 125: 103 ([doi:10.1007/s002220050070](https://doi.org/10.1007/s002220050070), [[BarNatanGaroufaldis96.pdf:file]])

In relation to the [[volume conjecture]]:

* {#MurakamiMurakami01} [[Hitoshi Murakami]] and [[Jun Murakami]], _The Colored Jones Polynomial And The Simplicial Volume Of A Knot_, Acta Math. 186 (2001) 85-104 ([euclid.acta/1485891370](https://projecteuclid.org/euclid.acta/1485891370))

[[!redirects colored Jones polynomials]]


colored symmetric sequence

[[!redirects symmetric colored sequence]]
[[!redirects symmetric color sequence]]
[[!redirects symmetric color sequence]]

#Contents#
* table of contents
{:toc}

## Idea

A [[symmetric sequence]] in a [[symmetric monoidal category]] $C$ can be thought of as a functor $\Sigma\to C$, where $\Sigma$ is the category of finite ordinals with bijections between them[^1]. One use of this idea is to define [[operads]] as monoids in symmetric sequences (with a certain composition product[^2]). However, operads defined in this way are necessarily operads with _one color_. To define [[colored operads]] as a type of "symmetric sequence", one has to generalize the category $\Sigma$. The category of colored symmetric sequences in a symmetric monoidal category $C$, as defined below, admits a [[composition product]], with respect to which the algebras are precisely the [[colored operads]] in $C$.

[^1]: Cf. [[species]], another name for symmetric sequence.

[^2]: Essentially synonymous with [substitution product](https://ncatlab.org/nlab/show/club#substitution_product_as_monoidal_product) as explained at [[club]], a concept due to [[Max Kelly]].

Note that the terminology "colored symmetric sequence" doesn't really make any sense, since we know longer have a _sequence_ at all, much less a "colored" one. [[Mike Shulman]] has suggested that what we call a colored symmetric sequence in $C$ might be better referred to as a _$C$-enriched multigraph_. The only drawback of such terminology is that it obscures the relevance of this construction to symmetric sequences (which it generalizes) and its relationship to colored operads.

## Definition

Let $\mathfrak{C}$ be a set that we will refer to as the set of _colors_. Define the category of $\mathfrak{C}$-profiles, denoted $P(\mathfrak{C})$, to be the groupoid whose objects are finite lists of elements of $\mathfrak{C}$ and in which there is an isomorphism $\vec{c}=\{c_1,\ldots,c_n\}\to \vec{d}=\{d_1,\ldots,d_n\}$ if $\vec{d}$ is a permutation of $\vec{c}$. Note that if $\mathfrak{C}=1$ then we recover the category $\Sigma$ of finite [[ordinals]] and [[bijections]].

A **symmetric $\mathfrak{C}$-colored sequence** in a [[category]] $C$ is then a functor $P(\mathfrak{C})\times\mathfrak{C}\to C$.

The idea of course is that the target of the pair $(\vec{c},d)=(\{c_1,\ldots,c_n\},d)$ (where some of the $c_i$ might be the same element of $\mathfrak{C}$) is the object in the colored operad governing operations that take as input an object of color $c_i$ for each $c_i$ composing $\vec{c}$ and have output of color $d$.

## Conceptual significance

The concept of colored symmetric sequence, seemingly rediscovered several times, involves concepts originally developed by Kelly (reference to be added).

The underlying conceptual point is that for a *category* $C$ (more generally than for a mere set $C$), the construct $Set^{\Sigma(C)^{op}}$ is the free symmetric [[monoidally cocomplete category]] generated by $C$, in the sense that given any [[cocomplete category|cocomplete]] $D$ with a symmetric monoidal structure whose tensor product is [[cocontinuous functor|cocontinuous]] in each variable, any functor $C \to D$ extends (uniquely up to coherent [[monoidal transformation|symmetric monoidal isomorphism]]) to a cocontinuous symmetric monoidal functor $Set^{\Sigma(C)^{op}}$, where the symmetric monoidal structure there is given by the [[Day convolution product]] induced by the symmetric monoidal structure on the free symmetric monoidal category $\Sigma(C)$.

It follows that the hom-category of functors and natural transformations $Hom(C, D)$ is equivalent to $SymMonCocont(Set^{\Sigma(C)^{op}}, D)$ (the category of symmetric monoidal cocontinuous functors and symmetric monoidal natural transformations between them). Hence, taking $D = Set^{\Sigma(C)^{op}}$, the evident endofunctor composition on $SymMonCocont(Set^{\Sigma(C)^{op}}, Set^{\Sigma(C)^{op}})$ gives a monoidal structure which may be transferred across the equivalence to a monoidal structure on $Hom(C, Set^{\Sigma(C)^{op}}) \cong Set^{\Sigma(C)^{op} \times C}$, and monoids in the latter monoidal structure are of course colored [[operads]].

Baez and Dolan made use of the same concepts in [Higher Dimensional Algebra III](http://arxiv.org/pdf/q-alg/9702014v1.pdf) (where their theory of [[opetopic sets]] was developed), calling the category $\Sigma(C) \times C$ for a set of colors $C$ being the category of *$C$-profiles*, and [[presheaves]] on that category the category of *$C$-signatures*.

## Colored Bisymmetric Sequences

The above definition can be generalized to support colored [[properads]] instead of just [[operads]]. The way to do this is to include the possibility of multiple outputs as well as multiple inputs. Thus a **colored bisymmetric sequence** in a category $C$, called a $\Sigma_{S(\mathfrak{C})}$-bimodule in [HRY](#HRY), is a functor $P(\mathfrak{C})\times P(\mathfrak{C})^{op}\to C$.

## References

* {#HRY} [[Philip Hackney]], [[Marcy Robertson]] and [[Donald Yau]]. _Infinity Properads and Infinity Wheeled Properads_,  Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. [(arxiv version)](http://arxiv.org/pdf/1410.6716v2.pdf)


colour superconductivity

+-- {: .rightHandSide}
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum chromodynamics]], _colour superconductivity_ is a [[phase of matter|phase of]] [[quark|quark matter]] of high [[baryon]] density and relatively small [[temperature]].

## Related concepts

* **[[QCD]]**

  * [[quark]], [[gluon]]

  * [[bound states]]:

    [[glueballs]], [[hadrons]] ([[mesons]], [[baryons]])

  * [[colour charge]], [[flavour (particle physics)]]

  * [[instanton in QCD]], [[caloron]], [[skyrmion]]

  * [[confinement]]

  * [[mass gap]]

  * [[quark-gluon plasma]]

## References

Review:

* Mark Alford, _Color superconducting quark matter_, Ann. Rev. Nucl. Part. Sci. 51:131-160, 2001 ([arXiv:hep-ph/0102047](https://arxiv.org/abs/hep-ph/0102047))

* Mark G. Alford, Krishna Rajagopal, [[Thomas Schaefer]], Andreas Schmitt, _Color superconductivity in dense quark matter_, Rev. Mod. Phys. 80:1455-1515, 2008 ([arXiv:0709.4635](https://arxiv.org/abs/0709.4635))

See also

* Wikipedia, _[Colour superconductivity](https://en.wikipedia.org/wiki/Color_superconductivity)_

Discussion via [[AdS/CFT]]:

* Kazem Bitaghsir Fadafan, Jesus Cruz Rojas, [[Nick Evans]], _A Holographic Description of Colour Superconductivity_, Phys. Rev. D **98** (2018) 066010  [[arXiv:1803.03107](https://arxiv.org/abs/1803.03107), [doi:10.1103/PhysRevD.98.066010](https://doi.org/10.1103/PhysRevD.98.066010)]


combinatorial category

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

# Combinatorial category
* table of contents
{: toc}

## Definition

A **combinatorial category** [[Pultr (1973, Def. 1.7)](#Pultr73)] $C$ is a [[locally finite category]] satisfying the _unnatural isomorphism property_:

For all objects $A, B \in C$, if there is an [[unnatural isomorphism]] $C(-, A) \cong C(-, B)$, then $A \cong B$. Equivalently if $\left|C(X, A)\right| = \left|C(X, B)\right|$ for all objects $X \in C$.

## Properties

\begin{proposition}
Let $C$ be a combinatorial category and let $A, B, X \in \C$. If $A \times X \cong B \times X$ and $X$ is [[weakly terminal]], then $A \cong B$.
\end{proposition}

\begin{proof}
For all $Y \in C$, we have
\begin{equation}
\left|C(Y, A)\right| \times \left|\C(Y, X)\right| = \left|C(Y, A \times X)\right| = \left|C(Y, B \times X)\right| = \left|C(Y, B)\right| \times \left|C(Y, X)\right|
\end{equation}
Hence we may divide both sides by $\left|C(Y, X)\right|$ (which is nonzero since $X$ is weakly terminal) and so $\left|C(Y, A)\right| = \left|C(Y, B)\right|$ for all $Y \in \C$, from which we have $A \cong B$.
\end{proof}

## Related concepts

* [[finite category]]

* *not* related is the notion of *[[combinatorial model category]]*

## References

* {#Pultr73} A. Pultr, *Isomorphism types of objects in categories determined by numbers of morphisms*, Acta Scientiarum Mathematicarum **35** (1973) 155-160 [[pdf](http://acta.bibl.u-szeged.hu/14440/1/math_035_155-160.pdf)]

[[!redirects combinatorial categories]]


combinatorial design

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

__Combinatorial design__ is a generic term for combinatorial structures described by families of finite sets satisfying some symmetries or other combinatorial properties of mutual arrangement. For example, the __block designs__ generically describe a set with family of subsets satisfying some combinatorial properties.

A concrete structures of this kind are block $t$-designs: if $t$ is an integer
a __$t$-design__ is a set $X$ with a family of $k$-element subsets of $X$ (called blocks) such that every $x\in X$ appears in exactly $r$ blocks, and every $t$-element subset $T$ appears in exactly $\lambda$ blocks. One also says $t-(v,k,\lambda)$-design if $v$ is the cardinality of $X$. The number of blocks $b$ and $r$ are determined by the other data. The applications include algebraic codes, finite geometries, algorithm design etc.

(Non)existence of combinatorial designs with specific properties often has profound consequences on classification of various other mathematical structures (not necessarily finite ones); in particular lattices, finite geometries, finite groups etc.

## Literature

Related $n$Lab items: [[binary linear code]], [[synthetic projective geometry]], [[species|Joyal species]], [[matroid]], [[building]], [[incidence geometry]]

* wikipedia: [combinatorial design](https://en.wikipedia.org/wiki/Combinatorial_design)
* Handbook of Combinatorial Designs, CRC Press 2006
* P. Dembowski, Finite geometries, Springer-Verlag 1968
* Jens Zumbrägel, _Designs and codes in affine geometry_, [arXiv:1605.03789](http://arxiv.org/abs/1605.03789)
* Peter Keevash, _The existence of designs_, [pdf](http://people.maths.ox.ac.uk/keevash/papers/designs.pdf)

> We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.

category: combinatorics

[[!redirects block design]]
[[!redirects combinatorial design]]


combinatorial functor

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[functor]]

* [[combinatorics]]

* [[Schanuel topos]]

## References

* J. N. Crossley, Anil Nerode, _Combinatorial Functors_, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, Spinger 1974 ([ISBN:9783642859359](https://www.springer.com/gp/book/9783642859359))

[[!redirects combinatorial functors]]