compactum

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### Context
#### Topology
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[[!include topology - contents]]
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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

A _compact Hausdorff space_ or _compactum_, for short, is a [[topological space]] which is both a [[Hausdorff space]] as well as a [[compact space]].
This is precisely the kind of topological space in which every [[limit of a sequence|limit]] of a [[sequence]] or more generally of a [[net]] that should exist does exist ([this prop.](net#CompactSpacesEquivalentlyHaveConvergetSubnets)) and does so uniquely ([this prop](#NetsDetectHausdorff)).

One may consider the analogous condition for [[convergence spaces]], or for [[locales]] (see also at _[Hausdorff locale](Hausdorff+space#HausdorffLocale)_ and _[[compact locale]]_). Even though these are all different contexts, the resulting notion of compactum is (at least assuming the [[axiom of choice]]) always the same.  Interestingly, there is even an algebraic definition, not one that uses only finitary operations, but one which uses a [[monad]].

## Definitions

If you know what a [[compact space]] is and what a [[Hausdorff space]] is, then you know what a compact Hausdorff space is, so let\'s be fancy. (Full justifications will be provided in section on [compacta as algebras](compactum#algebras).)

Given a [[set]] $S$, let $\beta S$ be the set of [[ultrafilters]] on $S$.  Note that $\beta$ is an [[endofunctor]] on [[Set]]; every [[function]] $f: S \to T$ induces a function $\beta f: \beta S \to \beta T$ using the usual application of functions to [[filters]].  In fact, $\beta$ is a [[monad]]; it comes with a [[natural transformation|natural]] (in $S$) unit $\eta_S: S \to \beta S$, which maps a point $x$ to the [[principal ultrafilter]] that $x$ generates, and multiplication $\mu_S: \beta \beta S \to \beta S$, which maps an ultrafilter $U$ on ultrafilters to the ultrafilter of sets whose principal ultrafilters of ultrafilters belong to $U$.  That is,

*  $ A \in \eta x \;\Leftrightarrow\; x \in A $, so $ \eta x = \{ A \subseteq S \;|\; x \in A \} $;
*  $ A \in \mu U \;\Leftrightarrow\; \{ F \in \beta S \;|\; A \in F \} \in U $.

Then a __compactum__ is simply an [[algebra for a monad|algebra]] for this monad; that is, a set $X$ together with a function $\lim: \beta X \to X$, such that

*  each point $x$ is the limit ($\lim$) of the principal ultrafilter $\eta x$, and
*  given an ultrafilter $U$ on ultrafilters, the limit of $\mu U$ is the limit of $(\beta \lim) U$.

It is then a theorem that this $\lim$ generates a [[convergence space|convergence]] on $S$ that is [[compact space|compact]], [[Hausdorff space|Hausdorff]], and [[topological space|topological]].  The converse, that every compact Hausdorff topological convergence is of this form, is equivalent to the [[ultrafilter principle]].

Every compact Hausdorff space is [[regular space|regular]] and [[sober space|sober]] and so defines a compact regular locale.  Again, the [[axiom of choice]] gives us a converse: every compact regular locale is spatial and so comes from a compactum.
>Probably this is also equivalent to the ultrafilter principle, but I need to check.

Note that every compact Hausdorff space (topological or localic) is not only regular but also [[normal space|normal]]. See _[[compact Hausdorff spaces are normal]]_.

## Compacta as algebras {#algebras}

Throughout this section, $CH$ will be used to denote the category of compact Hausdorff spaces (compacta).

### The space of ultrafilters

Let $Bool$ be the category of [[Boolean algebras]]. The functor $\hom(-, \mathbf{2}): Bool^{op} \to Set$ has a left [[adjoint]] $P: Set \to Bool^{op}$ given by power sets, and we define the **ultrafilter monad** to be the composite $\beta \coloneqq \hom(P -, \mathbf{2})$.

For a set $S$, [[topology|topologize]] $\beta S$ by declaring a basic [[open set]] to be one of the form

$$\hat{A} \coloneqq \{F \in \beta S: A \in F\}$$

for $A$ a subset of $S$. Notice $\hat{\emptyset}$ is [[empty set|empty]]. Indeed, $\widehat{(-)}$ defines a Boolean algebra map

$$P(S) \to P(\beta S)$$

so that in particular $\widehat{A \cap B} = \hat{A} \cap \hat{B}$, which immediately implies that the $\hat{A}$ form a [[basis]] of a topology.

+-- {: .num_remark #rem}
###### Remark
In fact, $\widehat{(-)}$ is the component at $P(S)$ of the counit of the adjunction $P \dashv \hom(-, \mathbf{2})$, as a morphism in $Bool^{op}$. If $f: S \to T$ is a function, the commutativity of the naturality square

$$\array{
P(T) & \stackrel{\widehat{(-)}}{\to} & P(\beta T) \\
\mathllap{P(f)} \downarrow & & \downarrow \mathrlap{P(\beta f)} \\
P(S) & \overset{\widehat{(-)}}{\to} & P(\beta S)
}$$

implies that if $U = \hat{B} \in P(\beta T)$ is a basic (cl)open, then so is
$(\beta f)^{-1}(U) = P(\beta f)(\hat{B}) = \widehat{f^{-1}(B)}$. It then follows that $\beta f$ is continuous.
=--

These results show that the monad $\beta: Set \to Set$ lifts through the forgetful functor $U: Top \to Set$.

The _unit_ of the monad $\beta$ is given componentwise by functions

$$prin_X: X \to \beta X$$

where $prin_X$ takes $x \in X$ to the principal ultrafilter

$$prin_X(x) = \{A \in P(X): x \in A\}.$$

It is evident that $prin_X$ is injective.

+-- {: .num_prop #dense}
###### Proposition
The injection $prin_X: X \to \beta X$ exhibits $X$ as a [[dense subset]] of $\beta X$.
=--

+-- {: .proof}
###### Proof
If $\hat{A}$ is a basic open neighborhood containing an ultrafilter $F$, then $A$ is nonempty and hence contains some $x \in X$, which is to say $A \in prin_X(x)$ or that $prin_X(x) \in \hat{A}$.
=--

### Ultrafilters form a compactum

+-- {: .num_prop}
###### Proposition
$\beta S$ is [[Hausdorff space|Hausdorff]].
=--

+-- {: .proof}
###### Proof
Let $F, G$ be distinct ultrafilters, so there is $A \subseteq S$ with $A \in F$ and $\neg A \in G$. Then $\hat{A}$ and $\widehat{\neg A}$ are disjoint neighborhoods which contain $F$ and $G$ respectively.
=--

+-- {: .num_prop}
###### Proposition
$\beta S$ is [[compact space|compact]].
=--

+-- {: .proof}
###### Proof
It is enough to show that if $\mathcal{O}$ is a collection of opens such that the union of any finite subcollection is a proper subset, then the union of $\mathcal{O}$ is also proper.

If $\mathcal{O}$ covers $U$, it admits a refinement by basic clopens also covering $U$, and thus we may assume WLOG that $\mathcal{O}$ consists of basic clopens $\hat{A}$. If every finite union of elements of $\mathcal{O}$ is a proper subset of $\beta S$, then every finite intersection $\widehat{\neg A_1} \cap \ldots \cap \widehat{\neg A_n}$ is nonempty, so that the $\neg A$ generate a filter, which is contained in some ultrafilter $F$. This $F$ lies outside the union of all the $\hat{A}$'s.
=--

### Convergence

+-- {: .num_defn}
###### Definition
Let $X$ be a topological space, and $F$ an ultrafilter on the underlying set $U X$. We say $F$ _converges_ to a point $x$ (in symbols, $F \rightsquigarrow x$) if the [[neighborhood|neighborhood filter]] $N_x$ of $x$ is contained in $F$.
=--

Convergence defines a [[relation]] $\xi$ from $\beta(U X)$ to $U X$.

+-- {: .num_prop}
###### Proposition
If $X$ is Hausdorff, then the relation $\xi$ is well-defined, or [[functional relation|functional]] (i.e., there is at most one point to which a given ultrafilter $F$ converges).
=--

+-- {: .proof}
###### Proof
If $x \neq y$, then there are disjoint neighborhoods $U$, $V$ of $x$ and $y$. We cannot have both $U \in F$ and $V \in F$ (otherwise $\emptyset = U \cap V$ would be an element of $F$), so at most one of the neighborhood filters $N_x, N_y$ can be contained in $F$.
=--

+-- {: .num_prop}
###### Proposition
If $X$ is compact, then the relation $\xi$ is [[total relation|total]] (i.e., there exists a point to which a given ultrafilter $F$ converges).
=--

+-- {: .proof}
###### Proof
If not, then for each $x \in X$ there is an open neighborhood $U_x$ that does not belong to $F$. Then $\neg U_x \in F$. Some finite number of neighborhoods $U_{x_1}, \ldots, U_{x_n}$ covers $X$. Then $\neg U_{x_1} \cap \ldots \cap \neg U_{x_n} = \emptyset \in F$, which is a contradiction.
=--

+-- {: .num_prop #coun}
###### Proposition
If $X$ is compact Hausdorff, then the function $\xi: \beta(U X) \to X$ is continuous.
=--

+-- {: .proof}
###### Proof
Let $U$ be an open neighborhood of $x \in X$; we must show that $\xi^{-1}(U)$ contains an open neighborhood of any of its points (i.e., ultrafilters $F$ such that $F \rightsquigarrow x$). Since $X$ is $T_3$ ([[separation axioms|Hausdorff regular]]), we may choose a neighborhood $V \in N_x$ whose closure $\bar{V}$ is contained in $U$. Then $\hat{V}$ is an open neighborhood of $F$ in $\beta(U X)$, and we claim $\hat{V} \subseteq \xi^{-1}(U)$.

For this, we must check that if $G \in \hat{V}$ and $G \rightsquigarrow y$, then $y \in U$. But if $y \in \neg U \subseteq \neg \bar{V}$, then $\neg \bar{V} \in N_y$, whence $G \rightsquigarrow y$ implies $\neg \bar{V} \in G$. This contradicts $G \in \hat{V}$, i.e., contradicts $V \in G$, since $V \cap \neg \bar{V} = \emptyset$.
=--

### Spaces of ultrafilters are universal

+-- {: .num_prop #exist}
###### Proposition
If $S$ is a set and $X$ is a compact Hausdorff space, then any function $f: S \to X$ can be extended (along $prin_S: S \to \beta S$) to a continuous function $\hat{f}: \beta S \to X$.
=--

+-- {: .proof}
###### Proof
We define $\hat{f}$ to be the composite

$$\beta S \stackrel{\beta(f)}{\to} \beta (U X) \stackrel{\xi}{\to} X$$

where $\beta(f)$ is continuous by Remark \ref{rem} and $\xi$ is continuous by Proposition \ref{coun}. It remains to check that the following diagram is commutative:

$$\array{
S & \stackrel{f}{\to} & U X & & \\
\mathllap{prin_S} \downarrow & & \mathllap{prin_{U X}} \downarrow & \searrow \mathrlap{1_{U X}} & \\
\beta S & \underset{\beta (f)}{\to} & \beta (U X) & \underset{\xi}{\to} & U X.
}$$

The square commutes by naturality of $prin$, and commutativity of the triangle simply says that the ultrafilter $prin_{U X}(x)$ converges to $x$, or that $N_x \subseteq prin(x)$, which reduces to the tautology that $x \in V$ for every neighborhood $V \in N_x$.
=--

+-- {: .num_theorem #free}
###### Theorem
For any set $S$, the function $prin_S: S \to \beta S$ is [[universal property|universal]] among functions from $S$ to compact Hausdorff spaces. Hence the functor $F: Set \to CH$ that takes $S$ to the compact Hausdorff space $\beta S$ is [[left adjoint]] to the forgetful functor $CH \to Set$.
=--

+-- {: .proof}
###### Proof
Proposition \ref{exist} shows that for any function $f: S \to U X$ to a compact Hausdorff space, there exists continuous $g: \beta S \to X$ such that $g \circ prin_S = f$. All that remains is to establish uniqueness of such $g$. But if two maps $g, g': \beta S \to X$ to a Hausdorff space $X$ agree on a dense subspace, in this case the subspace $prin_S : S \hookrightarrow \beta S$ by Proposition \ref{dense}, then they must be equal. Indeed, the pullback of the closed diagonal defines a closed subspace $D$ of $\beta S$,

$$\array{
D & \to & X \\
\downarrow & & \downarrow \mathrlap{\delta_X} \\
\beta S & \underset{\langle g, g' \rangle}{\to} & X \times X,
}$$

and $D$ contains a dense subspace $S$, therefore $D = \beta S$; i.e., the equalizer of $g$ and $g'$ is all of $\beta S$, hence these two maps are equal.
=--

### Compact Hausdorff spaces are monadic over sets

We recall hypotheses of Beck's precise [[monadicity theorem]]: a functor $U: C \to D$ is monadic if and only if

1. $U$ has a left adjoint,

1. $U$ reflects isomorphisms: a morphism $f: X \to Y$ of $C$ is an isomorphism if $U f: U X \to U Y$ is an isomorphism in $D$,

1. $D$ has, and $U$ preserves, coequalizers of parallel pairs that are $U$-split. (We say
$$X \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y$$
is $U$-**split** if there is a coequalizer
$$U X \stackrel{\overset{U f}{\to}}{\underset{U g}{\to}} U Y \stackrel{h}{\to} Z$$
that is split in $D$: there exists $i: Z \to U Y$ and $j: U Y \to U X$ such that $U h \circ i = 1_Z$, $U g \circ j = i \circ h$, and $U f \circ j = 1_{U Y}$.)

In the case where $D = Set$, we have the following useful lemma:

+-- {: .num_lemma #split}
###### Lemma
Suppose given a coequalizer in $Set$

$$X \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y \stackrel{h}{\to} Z$$

split by $i: Z \to Y$, $j: Y \to X$ (so that $f j = 1_Y$, $h i = 1_Z$, $g j = i h$). Let $R \hookrightarrow Y \times Y$ be the image of $\langle f, g \rangle : X \to Y \times Y$, and let $\langle p_1, p_2 \rangle : E \to Y \times Y$ be the equivalence relation given by the kernel pair $(p_1, p_2)$ of $h$. Then $E = R \cdot R^{op}$, the relational composite given by taking the image of the [[span]] composite $R \times_Y R^{op} \to Y \times Y$.
=--

+-- {: .proof}
###### Proof
Clearly $R \subseteq E$ since $h f = h g$, and we have $R^{op} \subseteq E^{op} = E$ and $R \cdot R^{op} \subseteq E \cdot E \subseteq E$ by symmetry and transitivity of $E$. In the other direction, suppose $(y_1, y_3) \in E$, so that $h(y_1) = h(y_3)$. Put $x = j(y_1)$ and $x' = j(y_3)$ (so that $f(x) = y_1$ and $f(x') = y_3$), and put $y_2 = g(x)$. Clearly then $(y_1, y_2) \in R$. Moreover,

$$y_2 = g(x) = g j(y_1) = i h(y_1) = i h(y_3) =  g j(y_3) = g(x')$$

so that $(y_3, y_2) \in R$, or $(y_2, y_3) \in R^{op}$. Hence $(y_1, y_3) \in R \cdot R^{op}$, and we have shown $E \subseteq R \cdot R^{op}$.
=--

+-- {: .num_theorem}
###### Theorem
The forgetful functor $U: CH \to Set$ is monadic.
=--

+-- {: .proof}
###### Proof
By theorem \ref{free}, $U$ has a [[left adjoint]]. Since [[bijection|bijective]] continuous maps between compact Hausdorff spaces are homeomorphisms, we have that $U$ reflects isomorphisms. Finally, suppose $(f, g)$ is a $U$-split pair of morphisms $X \to Y$ in $CH$; let $h: Y \to Z$ be their coequalizer in $Top$, given by a suitable quotient space. Being a quotient of a compact space, $Z$ is compact. Since $CH$ is a full subcategory of $Top$, the map $h$ is a coequalizer in $CH$ once we prove the following claim:

* **Claim:** $Z$ is Hausdorff.

Furthermore, since the forgetful functor $Top \to Set$ has a right adjoint (given by taking indiscrete topologies on sets), the underlying function of $h$ (again denoted $h$) is the coequalizer of $(U f, U g)$ in $Set$, so that $U$ would preserve the claimed coequalizer.

In other words, to complete the proof, it suffices to verify the claim. Letting $p_0, p_1: E \stackrel{\to}{\to} Y$ be the kernel pair of $h$ in $Top$, to show $Z = Y/E$ is Hausdorff, it suffices to prove that the equivalence relation $\langle p_0, p_1 \rangle : E \to Y \times Y$ in $Top$ is closed. Let $R \hookrightarrow Y \times Y$ be the image of $\langle f, g \rangle: X \to Y \times Y$. By lemma \ref{split}, the subset $E$ of $U Y \times U Y$ coincides with the subset $R \cdot R^{op} \subseteq U Y \times U Y$. Now $R$ is the image of the compact space $X$ under the continuous map $\langle f, g \rangle$, so $R$ is a closed subset of $Y \times Y$. Similarly $R^{op}$ is a closed subset of $Y \times Y$. Under their subspace topologies, their fiber product $R \times_Y R^{op}$ is compact, and so its image $R \cdot R^{op}$ under the (continuous) span composite $R \times_Y R^{op} \to Y \times Y$ is also closed. This completes the proof.
=--

## Weak versions

Every [[Hausdorff space]], hence every compactum, satisfies the [[separation axiom]] $T_0$.  As is usual with separation axioms, we can also look for a non-$T_0$ version.  A priori, this is a compact [[preregular space]]; however, since every such space is [[regular space|regular]], we can speak instead of a __compact regular space__.

In the absence of the axiom of choice, and especially in [[constructive mathematics]], the best definition of compactum seems to be a compact regular locale.  That is, it is the category of compact regular locales that has all of the nice properties, forming a [[nice category of spaces]], and that has the desired examples, such as the [[unit interval]].  (See the discussion at [[Tychonoff theorem]] for an example of how the category of compact Hausdorff topological spaces might fail to be nice; see [Frank Waaldijk's PhD thesis](http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf) (pdf) for a thorough discussion of what is needed to make the unit interval a compact Hausdorff topological space.)

The monadic definition, in particular, falls quite flat without some form of the axiom of choice; even [[excluded middle]] and [[COSHEP]] are powerless here.  In fact, it is quite consistent to assume that every ultrafilter is principal (a strong denial of the ultrafilter principle), in which case $\beta$ is the [[identity monad]].  Then a compactum would be just a set if that were the definition used.

On the other hand, it is the monadic definition that gives an [[algebraic category]] with a nice relationship to [[Set]].  Without the ultrafilter principle, there is no reason to think that the set-of-points functor from compact regular locales to sets is even [[continuous functor|continuous]].

## Properties

### General

* [[compact Hausdorff spaces are normal]]

* [[open subspaces of compact Hausdorff spaces are locally compact]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

### Stone--Čech compactification
 {#StoneCechCompactification}

By general nonsense, every $\beta S$, regarded as a free $\beta$-algebra, is a compactum, and the functor

$$\beta: Set \to Comp$$

is [[left adjoint]] to the [[forgetful functor]] $Comp \to Set$.  Assuming the ultrafilter principle, this functor extends to a functor $\beta: Top \to Comp$ (identifying a set with its [[discrete space]]) that is left adjoint to the forgetful functor $Comp \to Top$.  This is the __[[Stone–Čech compactification]]__ functor (N.B.: for many authors, Stone--Čech compactification refers to the restriction of this functor to [[Tychonoff spaces]] $X$, which are precisely those spaces where the unit $X \to \beta X$ is an [[embedding]] so that we have a [[compactification]] in the technical sense).

A classical construction of the Stone--Čech compactification starts with the unit [[interval]] $I =[0, 1]$ and proceeds to the [[codensity monad]] induced from the functor

$$\hom(-, I) \colon Top^{op} \to Set.$$

The monad is given on objects by $X \mapsto I^{\hom(X, I)}$; this lands in compact Hausdorff spaces under the ultrafilter principle. Let $\bar{X}$ be the closure of the [[image]] of the unit $u_X: X \to I^{\hom(X, I)}$; this $\bar{X}$ is compact Hausdorff.

+-- {: .num_prop}
###### Proposition
If $X$ is a [[Tychonoff space]], then the unit $u_X: X \to I^{\hom(X, I)}$ is a subspace embedding, so that $I$ is a [[cogenerator]] in the category of Tychonoff spaces. (In particular, $u_C$ is an embedding if $C$ is compact Hausdorff, so $I$ is also a [[cogenerator]] in the category of compact Hausdorff spaces.)
=--

The proof is essentially [[Urysohn's lemma]]; see also related discussion at [[Tychonoff space]] and at [[uniform space]] (noting that compact Hausdorff spaces are uniform spaces for a unique uniformity).

+-- {: .num_theorem}
###### Theorem
The natural map $i_X: X \to \bar{X}$ is universal among maps from $X$ to compact Hausdorff spaces, thus giving a [[left adjoint]] $Top \to Comp$ to the (fully faithful) forgetful functor $U: Comp \to Top$.
=--

+-- {: .proof}
###### Proof
Let $f: X \to C$ be a map, where $C$ is a compact Hausdorff space. Since $I$ is a cogenerator in the category of compact Hausdorff spaces, the unit for the codensity monad $M_I$,

$$u_C: C \to I^{\hom(C, I)},$$

is a continuous injection (and hence a closed subspace embedding, since $C$ is compact Hausdorff). Let $\hat{f} = M_I(f)$, and consider the pullback square

$$\array{
\hat{f}^{-1}(C) & \to & I^{\hom(X, I)} \\
\mathllap{\pi} \downarrow & & \downarrow \mathrlap{\hat{f}} \\
C & \underset{u_C}{\to} & I^{\hom(C, I)}.
}$$

From an evident naturality square for the unit $u$, we have a map $h: X \to \hat{f}^{-1}(C)$, i.e., the map $u_X: X \to I^{\hom(X, I)}$ factors through the closed subspace $\hat{f}^{-1}(C) \hookrightarrow I^{\hom(X, I)}$. Therefore $h$ factors as

$$X \stackrel{i_X}{\to} \bar{X} \subseteq \hat{f}^{-1}(C)$$

and since $\pi \circ h = f$, we conclude that $f$ factors through $i_X$. And moreover, there is at most one $k: \bar{X} \to C$ such that $k \circ i_X = f$, because $i_X$ maps $X$ onto a dense subspace of $\bar{X}$, and dense subspaces are epic in the category of Hausdorff spaces. This completes the proof.
=--

We have a similar Stone--Čech compactification functor $Loc \to Comp$; we do not need the ultrafilter principle here if $Comp$ is defined in terms of locales.

### Category of compacta

The category $Comp$ of compact Hausdorff spaces and continuous maps is

* [[exact category|Barr exact]], since $U: Comp \to Set$ is monadic,

* an [[extensive category]], and

* a [[total category]] (by monadicity over $Set$), and

* a [[total category|cototal category]] (because it is [[complete category|complete]], [[well-powered category|well-powered]], and has a [[cogenerator]] given by the unit interval $I = [0, 1]$).

From the first two properties, it follows that $Comp$ is a [[pretopos]], meaning that $Comp$ enjoys the same finitary exactness properties that hold in a [[topos]]; in particular, first-order intuitionistic logic may be enacted within $Comp$.

In ([Marra-Reggio 18](#MarraReggio)), the authors give a characterization of $Comp$ up to equivalence as the unique non-trivial [[pretopos]] which is [[well-pointed category|well-pointed]], [[filtrality|filtral]] and admits all set-indexed [[copowers]] of its [[terminal object]]. They contrast this result with Lawvere’s characterisation of the category of sets in [[ETCS]], noting that the main divergence concerns

> the existence of infinite “discrete” objects. While the third axiom of ETCS postulates the existence of a natural numbers object, we prescribe filtrality which forbids the existence of infinite discrete objects. ([Marra-Reggio 18, p. 2](#MarraReggio))

In the context of [[ultracategories]], $Comp$ is equivalent to $Fun_{RUlt}(\ast, Set)$, the category of right ultrafunctors between the terminal ultracategory and $Set$ ([Lurie, p. 4](#Lurie)).

The [[infinitary pretopos]] of [[condensed sets]]
is the completion of the [[pretopos]] of [[compact Hausdorff spaces]].

### Relation to compactly generated topological spaces

\begin{proposition}\label{kSpacesAreTheColimitsInTopOfCompactHausdorffSpaces}
**([[k-spaces]] are the [[Top#UniversalConstructions|colimits in Top]] of [[compact Hausdorff spaces]])**
\linebreak
  A [[topological space]] is a [[k-space]] (Def. \ref{kSpace})
iff it is a [[colimit]] as formed in [[Top]] (according to [this Prop.](Top#DescriptionOfLimitsAndColimitsInTop)) of a [[diagram]] of [[compact Hausdorff spaces]].
\end{proposition}
([Escardo, Lawson & Simpson 2004, Lem. 3.2 (v)](#EscardoLawsonSimpson04))

Such colimits of compact Hausdorff spaces are also equivalently [[quotient topological spaces]] of [[locally compact Hausdorff spaces]], see [there](locally+compact+topological+space#kSpacesAreTheQuotientSpacesOfLocallyCompactHausdorffSpaces).

## Related concepts

* [[compact space]], [[locally compact space]], [[compactly generated topological space]]

* [[paracompact Hausdorff space]]

* [[pyknotic set]]

## References

* [[Saunders MacLane]], §VI.9 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#MarraReggio} Vincenzo Marra, Luca Reggio, _A characterisation of the category of compact Hausdorff spaces_, ([arXiv:1808.09738](https://arxiv.org/abs/1808.09738))

* {#Lurie} [[Jacob Lurie]], _Ultracategories_, ([pdf](http://www.math.harvard.edu/~lurie/papers/Conceptual.pdf))

Relation to [[compactly generated topological spaces]]:

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

[[!redirects compactum]]
[[!redirects compacta]]
[[!redirects compact Hausdorff space]]
[[!redirects compact Hausdorff spaces]]
[[!redirects compact Hausdorff topological space]]
[[!redirects compact Hausdorff topological spaces]]
[[!redirects compact regular locale]]
[[!redirects compact regular locales]]

[[!redirects compact regular space]]
[[!redirects compact regular spaces]]
[[!redirects compact regular topological space]]
[[!redirects compact regular topological spaces]]


companion pair

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

# Companion pairs
* table of contents
{: toc}

## Idea

A **companion pair** in a [[double category]] is a way of saying that a [[horizontal  morphism]] and a [[vertical morphism]] are "[[isomorphism|isomorphic]]", even though they do not live in the same [[1-category]]/[[2-category]].

A **connection pair** in a [[double category]] is a [[strict 2-functor|strictly 2-functorial]] choice of companion pairs for every vertical morphism.

## Definition

Let $f\colon A\to B$ be a [[vertical morphism]] and $f'\colon A\to B$ a [[horizontal morphism]] in a [[double category]].  These are said to be a **companion pair** if they come equipped with [[2-morphisms]] of the form:
$$
  \array{
    A & \overset{f'}{\longrightarrow} & B
    \\
    \mathllap{^f}\big\downarrow & \mathllap{^\phi}\swArrow & \big\downarrow\mathrlap{^id}
    \\
    B & \underset{id}{\longrightarrow} & B
  }
  \qquad
  \qquad
  \text{and}
  \qquad
  \qquad
  \array{
    A & \overset{id}{\longrightarrow} & A
   \\
   \mathllap{^id}\big\downarrow & \mathllap{^\psi}\swArrow & \big\downarrow\mathrlap{^f}
   \\
   A & \underset{f'}{\longrightarrow} & B
  }
$$
such that $\phi \circ_h \psi = id_{f'}$ and $\phi \circ_v \psi = id_{f}$, where $\circ_h$ and $\circ_v$ denote horizontal and vertical composition of 2-cells.

Given such a companion pair, we say that $f$ and $f'$ are **companions** of each other.

## Examples

\begin{example}
In the double category $\mathbf{Sq}(K)$ of squares ([[quintets]]) in any [[2-category]] $K$, a companion pair is simply an invertible 2-cell between two parallel 1-morphisms of $K$.
\end{example}

\begin{example}
In the [[double category of algebras | double category $T$-$\mathbf{Alg}$ of algebras, lax morphisms, and colax morphisms]] for a [[2-monad]] $T$, an arrow (of either sort) has a companion precisely when it is a *strong* (= pseudo) $T$-morphism.  This is important in the theory of [[doctrinal adjunction]].
\end{example}

## Properties

* The horizontal (or vertical) dual of a companion pair is a [[conjunction]].

* Companion pairs (and conjunctions) have a [[mate]] correspondence generalizing the calculus of mates in 2-categories.

* If every vertical arrow in some double category $D$ has a companion, then the functor $f\mapsto f'$ is a [[pseudofunctor]] $V D\to H D$ from the vertical 2-category to the horizontal one, which is the identity on objects, and [[locally fully faithful 2-functor|locally fully faithful]] by the mate correspondence.  A choice of companions that make this a strict 2-functor is called a [[connection on a double category|connection]] on $D$ (an arbitrary choice of companions may be called a "pseudo-connection").  A double category with a connection is thereby equivalent to an [[F-category]].  If every vertical arrow also has a [[conjoint]], then this makes $D$ into a [[proarrow equipment]], or equivalently a [[framed bicategory]].

* Companion pairs and mate-pairs of 2-cells between them in any double category $D$ form a 2-category $Comp(D)$.  The functor $Comp\colon DblCat \to 2Cat$ is right adjoint to the functor $Sq\colon 2Cat \to DblCat$ sending a 2-category to its double category of [[quintet construction|squares]].

## Related concepts

* [[retrocell]]

## References

* [[Marco Grandis]] and [[Robert Pare]], *Adjoints for double categories*, [NUMDAM](http://www.numdam.org/item/CTGDC_2004__45_3_193_0.pdf)

* [[Robert Dawson]] and [[Robert Pare]] and [[Dorette Pronk]], *The Span construction*, [TAC](http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html).

* [[Michael Shulman]], *Framed bicategories and monoidal fibrations*, [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html)

This latter reference explains the relationship between companions to connection pairs and **foldings**:

* [[Ronnie Brown]] and C.B. Spencer, [Double groupoids and crossed modules](http://www.numdam.org/item/CTGDC_1976__17_4_343_0), _Cahiers de Topologie et Géométrie Différentielle Catégoriques_ **17** (1976), 343--362.

* [[Ronald Brown]] and Ghafar H. Mosa, _Double categories, 2-categories, thin structures and connections_, Theory and Application of Categories 5.7 (1999): 163-1757.

* [[Thomas M. Fiore]], _Pseudo Algebras and Pseudo Double Categories_, _Journal of Homotopy and Related Structures_, Volume 2, Number 2, pages 119-170, 2007. 51 pages.

[[!redirects companion]]
[[!redirects companions]]
[[!redirects companion pair]]
[[!redirects companion pairs]]
[[!redirects companion pair in a double category]]
[[!redirects companion pairs in a double category]]
[[!redirects companion pairs in double categories]]
[[!redirects connection pair]]
[[!redirects connection pairs]]


comparison

+-- {: .rightHandSide}
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### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--

\tableofcontents

## Definition

A **comparison** or **cotransitive relation** or **weakly linear relation** on a set $A$ is a (binary) [[relation]] $\sim$ on $A$ such that in every pair of related elements, any other element is related to one of the original elements in the same order as the original pair:
$$\forall (x, y, z: A),\; x \sim z \;\Rightarrow\; x \sim y \;\vee\; y \sim z$$
which generalises from $3$ to any (finite, positive) number of elements.  To include the case where $n = 1$, we must explicitly state that the relation is [[irreflexive relation|irreflexive]].

Alternatively, this is the same condition as
$$\forall (x, z: A),\; x \sim z \;\Rightarrow\; \forall (y: A),\; x \sim y \;\vee\; y \sim z$$

Comparisons are most often studied in [[constructive mathematics]].  In particular, the relation $\lt$ on the (located Dedekind) [[real numbers]] is an [[irreflexive comparison]], even though its [[negation]] $\geq$ is not constructively [[total relation|total]].  (Indeed, $\lt$ is a [[pseudo-order]], even though $\geq$ is not constructively a [[total order]].)

A comparison is a [[cartesian monoidal category|cartesian monoidal]] [[semicategory]] [[enriched category|enriched]] on the [[co-Heyting algebra]] $TV^\op$, where $TV$ is the [[Heyting algebra]] of [[truth values]].

## Related concepts

* [[pseudo-order]]

* [[total relation]]

## References

* Wikipedia, [Apartness relation](https://en.wikipedia.org/wiki/Apartness_relation)

* Wikipedia, [Pseudo-order](https://en.wikipedia.org/wiki/Pseudo-order)

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Auke Booij]], *Analysis in univalent type theory* (2020) $[$[etheses:10411]( 	http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$

[[!redirects comparisons]]
[[!redirects comparison relation]]
[[!redirects comparison relations]]

[[!redirects cotransitive]]
[[!redirects cotransitive relation]]
[[!redirects cotransitive relations]]
[[!redirects cotransitivity]]

[[!redirects weakly linear]]
[[!redirects weakly linear relation]]
[[!redirects weakly linear relations]]
[[!redirects weak linearity]]


comparison functor

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

#Contents#
* table of contents
{:toc}

## Idea

While every [[functor]] might be understood as constituting a "comparison" between its [[domain]] and [[codomain]]-[[categories]], the term "comparison functor" is often understood by default as referring, in [[categorical algebra]], to the unique functor that relates any [[adjoint functors|adjunction]] to its [[monadic adjunction]] -- this is the case we discuss here.

But other instances of functorial "comparison" are bound to be relevant. For instance, for the "[[comparison lemma]]" in [[topos theory]] see there.

## Statement

\begin{proposition}
**(the [[comparison functor]] from any [[adjoint functors|adjunction]] to its [[monadic adjunction]])**
\linebreak
Every [[pair]] of [[adjoint functors]] $(F \dashv U) \;\colon\; \mathbf{C}' \leftrightarrow \mathbf{C}$ (with [[unit of an adjunction|unit]] $\eta^{U F}$ and [[counit of an adjunction|counit]] $\epsilon^{F U}$) between [[categories]] $\mathbf{C}$, $\mathbf{C}'$ fits into a [[commuting diagram]] in [[Cat]] of the following form:

\begin{imagefromfile}
    "file_name": "AdjunctionsAndMonadicity-221108.jpg",
    "width": "470",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Here:

* $\mathcal{E} \;\coloneqq\; U F \;\coloneqq\; U \circ F$ denotes the induced [[endofunctor]] on $\mathbf{C}$

  which carries the [[structure]] of a [[monad]] with

  * [[unit of a monad|unit]] given by

    $$
      \eta^{\mathcal{E}}
      \;\;
      \colon
      \;\;
      id_{\mathbf{C}}
      \xrightarrow{\;\;
        \eta^{U F}
      \;\;}
      U F
      \;=\;
      \mathcal{E}
    $$

  * product given by

    $$
      \mu^{\mathcal{E}}
      \;\;
      \colon
      \;\;
      \mathcal{E} \mathcal{E}
      \;=\;
      U F U F
      \xrightarrow{\;\;
        U ( \epsilon^{F U} ) F
      \;\;}
      U F
      \;=\;
      \mathcal{E}
    $$

* $\mathbf{C}^{\mathcal{E}}$ denotes the ("[[Eilenberg-Moore category|Eilenberg-Moore]]"-)[[category]] of $\mathcal{E}$-[[algebras over a monad|algebras]] in $\mathbf{C}$

  $$
    A
    \;=\;
    \Big(
      \mathcal{E}
      U^{\mathcal{E}}(A)
      \xrightarrow{\;\; \rho_A  \;\;}
      U^{\mathcal{E}}(A)
    \Big)
  $$

   in $\mathbf{C}$ with [[homomorphisms]] between them,

* $K^{U F}$ denotes the "[[comparison functor]]" given by

  $$
    \array{
      \mathbf{C}'
      &
      \xrightarrow{\;\;\;\;\;\;\;\;\;\;
         K^{U F}
      \;\;\;\;\;\;\;\;\;\;}
      &
      \!\!\!\!\!\!\!\!\!\!\!\!\!\!
      \mathbf{C}^{\mathcal{E}}
      \\
      C'
      &\mapsto&
      {
        \Big(
          \mathcal{E}
          U(C')
          \,=\,
          U F U (C')
          \xrightarrow{\;
            U \epsilon^{F U}_{C'}
          \;}
          U (C')
        \Big)
      }
    }
  $$

* $\mathbf{C}_{\mathcal{E}}$ denotes the ("[[Kleisli category|Kleisli]]"-)[[category]] of *[free](algebra+over+a+monad#FreeAlgebras)* $\mathcal{E}$-[[algebras over a monad|algebras]]

  $$
    \begin{array}{rl}
    F^{\mathcal{E}}(C)
    &
    \;=\;
    \Big(
      \mathcal{E}
      \mathcal{E}
      (C)
      \xrightarrow{\;\;
        \mu^{\mathcal{E}}_C
      \;\;}
      \mathcal{E}(C)
    \Big)
    \\
    \;=\;
    K^{U F} F(C)
    &
    \;=\;
    \Big(
      \mathcal{E}
      U
      F
      (C)
      \xrightarrow{\;\;
        U \epsilon^{ F U }_{F(C)}
      \;\;}
      U
      F
      (C)
    \Big)
    \,.
    \end{array}
  $$

* Here the last line makes explicit that

  $$
    U^{\mathcal{E}} F^{\mathcal{E}}
    \;=\;
    U F
    \;=\;
    \mathcal{E}
    \,.
  $$

  In fact, $F^{\mathcal{E}} \dashv U^{\mathcal{E}}$ and this realizes the [[monadic adjunction]] whose induced monad is $\mathcal{E}$.

  Hence the original $F \dashv U$ is [[monadic adjunction|monadic]] iff the [[comparison functor]] $K^{U F}$ is an [[equivalence of categories]].

\end{proposition}

(See for instance [MacLane (1971), §VI.3](#MacLane71))

## Properties

* $K^{UF}$ is [[fully faithful]] if and only if $U$ is of [[descent type]]. In this case, $K^{UF}$ has a [[right adjoint]] if and only if it is [[essentially surjective]] if and only if it is an [[equivalence of categories]], since [[coreflections]] create colimits, and every algebra is a colimit of a free algebra (see [here](https://ncatlab.org/nlab/show/Eilenberg-Moore+category#AsColimitCompletionOfKleisliCategory)).

* When $F \dashv U$ is the adjunction associated to the [[Kleisli category]] of a monad, $K^{UF}$ is [[dense functor|dense]] (a corollary of [the pullback theorem](https://ncatlab.org/nlab/show/Eilenberg-Moore+category#AsColimitCompletionOfKleisliCategory)) and [[fully faithful]].

## Related concepts

* [[monadic functor]], [[monadicity theorem]]

## References

The [[initial object|initiality]] of the [[Kleisli category]] under [[monad morphisms]] was first observed in

* {#Maranda66} [[Jean-Marie Maranda]], *On Fundamental Constructions and Adjoint Functors*, Canadian Mathematical Bulletin **9** 5  (1966) 581-591 [[doi:10.4153/CMB-1966-072-9](https://doi.org/10.4153/CMB-1966-072-9)]

Textbook account:

* {#MacLane71} [[Saunders MacLane]], §VI.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects comparison functors]]


comparison lemma

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#### Topos Theory
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[[!include topos theory - contents]]
=--
=--
=--

> This page is about a general theorem in [[topos theory]]. For other meanings see e.g. [[comparison theorem (étale cohomology)]].

# The comparison lemma
* table of contents
{: toc}

## Idea

A [[functor]] from a [[category]] to a [[site]] induces a [[Grothendieck topology|topology]] on the source category.  The **comparison lemma** says that, under certain conditions, such a functor induces an [[equivalence of categories|equivalence]] between the [[category of sheaves|categories of sheaves]] on the sites.

## Statement

\begin{definition}
\label{InducedTopology}
Let $u \colon B \to C$ be a functor with $C$ a [[site]]. The **induced topology** on $B$ by $u$ is the finest one such that $u$ is a [[continuous functor]], i.e. such that the map $G \mapsto G \circ u$ takes [[sheaves]] on $C$ to sheaves on $B$.
\end{definition}

The classical comparison lemma ([Verdier 1972](#Verdier72)):

+-- {: .num_theorem #SGA4}
###### Theorem

Let $B$ be a [[small category]], $C$ a [[site]], and $u : B \to C$ a [[fully faithful]] functor.  Consider $B$ as a [[site]] with the topology induced by $u$ (Def. \ref{InducedTopology}).
  If every object $x \in C$ has a covering $(u(a_\alpha) \to x)$ by objects of $B$, then $u \colon B \to C$ induces an [[equivalence]] of [[category of sheaves|categories of sheaves]] (of sets) $B^\sim \to C^\sim$.

=--

[Beilinson 2012](#Beilinson2011) proves the following generalisation of the classical comparison lemma:

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

Let $B$ be an [[essentially small category]] and $C$ be an essentially small [[site]].  Suppose that $u \colon B \to C$ is a faithful functor which exhibits $B$ (with the induced topology) as a [[dense subsite]] of $C$, i.e. which satisfies the following condition:

* For every object $x \in C$ and finite [[family]] $(x \to u(a_\alpha))_\alpha$, with $a_\alpha \in B$, there exists a [[covering family]] $(u(b_\beta) \to x)_\beta$ of $x$ such that every composite $u(b_\beta) \to x \to u(a_\alpha)$ lies in the image of $\Hom(b_\beta, a_\alpha) \hookrightarrow \Hom\big(u(b_\beta), u(a_\alpha)\big)$.

Then:

1. The topology on $B$ induced by $u$ has the following simple description: a [[sieve]] $(x_\gamma \to x)_\gamma$ is [[covering sieve|covering]] iff the sieve generated by the family $(u(x_\gamma) \to u(x))_\gamma$ is covering in $C$.

1. The functor $u \colon B \to C$ induces an [[equivalence]] of [[categories of sheaves]] (of sets) $Sh(B) \to Sh(C)$.

=--

## Related concepts

* [[(∞,1)-comparison lemma]]

* [[dense subsite]]

* "[[comparison functor]]" is mostly understood to refer to a different concept in [[categorical algebra]]

## References

* {#Verdier72} [[Jean-Louis Verdier]], _Fonctorialité de catégories de faisceaux_, in: _Théorie des topos et cohomologie étale de schémas_ ([[SGA4]]), Tome 1, Lect. Notes in Math. **269**, Springer (1972) 265-298.

* [[Anders Kock]], [[Ieke Moerdijk]], Section 2 of: _Presentations of étendues_. [[Cahiers|Cahiers Topologie Géom. Différentielle Catég.]] **32** 2 (1991) 145–164 [[EuDML](https://eudml.org/doc/91476)]

* [[Peter Johnstone]], Section C2.2 of: _[[Sketches of an Elephant]]_, Oxford University Press (2002)

* {#Beilinson2011} [[Alexander Beilinson]], _$p$-adic periods and derived de Rham cohomology_, J. Amer. Math. Soc. **25** (2012) 715-738  [[arXiv:1102.1294](http://arxiv.org/abs/1102.1294), [doi:10.1090/S0894-0347-2012-00729-2](https://doi.org/10.1090/S0894-0347-2012-00729-2), [pdf](https://www.ams.org/journals/jams/2012-25-03/S0894-0347-2012-00729-2/S0894-0347-2012-00729-2.pdf)]

[[!redirects Comparison lemma]]
[[!redirects comparison lemma]]
[[!redirects the comparison lemma]]


comparison map

* [[comparison map between algebraic and topological K-theory]]

[[!redirects comparison map]]
[[!redirects comparison maps]]

category: disambiguation


comparison map between algebraic and topological K-theory

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[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For suitable setups, there is a canonical [[homomorphism]] from [[algebraic K-theory]] to [[topological K-theory]] with given [[coefficients]].

This usually goes just by "the comparison map". (e.g. [Rosenberg, theorem 2.1](#Rosenberg))

## Statement

### For complex K-theory

There is a canonical [[homomorphism]] of [[spectra]]

$$
  K \mathbb{C} \longrightarrow KU
$$

from the [[algebraic K-theory]] spectrum of the [[complex numbers]] to [[KU]] (e.g. [Paluch 01, lemma 2.6](#Paluch01)).

In fact in terms of [[cohesion]] of [[smooth spectra]], this is a component of a [[natural transformation]], this we discuss [below](#FormulationInCohesion).

### Formulation in cohesive homotopy-type theory
 {#FormulationInCohesion}

Given a [[cohesive (infinity,1)-topos]] and a [[symmetric monoidal (∞,1)-category]] $V\in CMon_\infty(Cat_\infty(\mathbf{H}))$, [[internal (∞,1)-category|internal]] to $\mathbf{H}$ write

$$
  \mathcal{K}(V)\in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}
$$

for its [[K-theory of a symmetric monoidal (∞,1)-category]] (formed locally and then [[∞-stackification|∞-stackified]]). This is a [[stable homotopy type]] in the [[tangent cohesive (∞,1)-topos]].

The [[flat modality]] part $\flat \mathcal{K}(V)$ is the [[algebraic K-theory]] of $\flat V \in CMon_\infty(Cat_\infty)$. The [[shape modality]] part on the other hand is a "topological" version of this

The [[points-to-pieces transform]] $\flat \to \Pi$ provides a [[natural transformation|natural]] comparison map

$$
  \flat (\mathcal{K}(V)) \longrightarrow \Pi (\mathcal{K}(V))
  \,.
$$

For instance for $\mathbf{H}= $ [[Smooth∞Grpd]] and $V = \mathbf{Vect}^{\oplus}$ the [[stack]] of smooth [[vector bundles]] with [[direct sum]] monoidal structure, then $\flat \mathcal{K}(V)\simeq K \mathbb{C}$ is the [[algebraic K-theory]] of the [[complex numbers]] and $\Pi \mathcal{K}(V)\simeq $ [[KU]] is complex [[topological K-theory]].

See at _[[differential cohomology diagram]]_ for more on this.

## References

* [[Henri Gillet]], _Comparing algebraic and topological K-theory_, in _Higher Algebraic K-Theory: an overview_ Lecture Notes in Mathematics Volume 1491, 1992, pp 55-99

Discussion relating [[algebraic K-theory]] of [[varieties]] to complex topological K-theory is in

* {#Paluch01} Michael Paluch, _Algebraic $K$-theory and topological spaces_, K-theory 471 (2001) ([web](http://www.math.uiuc.edu/K-theory/0471/))

and for [[sheaves of spectra]] of [[twisted K-theory]] in

* {#AntieuWilliams11} [[Benjamin Antieau]], [[Ben Williams]], section 2.3 of _The period-index problem for twisted topological K-theory_, Geometry & Topology ([arXiv:1104.4654](http://arxiv.org/abs/1104.4654))

Discussion in terms of [[Banach algebras]] is in

* {#Rosenberg} [[Jonathan Rosenberg]], _Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras_   ([pdf](http://www2.math.umd.edu/~jmr/algtopK.pdf))

[[!redirects comparison between algebraic and topological K-theory]]
[[!redirects comparison mapso between algebraic and topological K-theory]]

[[!redirects comparison map between algebraic K-theory and topological K-theory]]


comparison theorem

Many [[theorems]] can be "comparison theorems". In [[geometry]]/[[cohomology]] the term is traditionally used to refer to statements that identify one [[cohomology theory]] with another over a suitable kind of [[spaces]], such as

* [[comparison theorem (étale cohomology)]]

* [[comparison theorem (crystalline cohomology)]]

[[!redirects comparison theorems]]


comparison theorem (crystalline cohomology)

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A comparison between [[p-adic cohomology]] for [[varieties]] in [[characteristic]] $p \gt 0$ and [[crystalline cohomology]] was was discussed in

* [[Pierre Berthelot]], _Cohomologie cristalline des schémas de caractéristique $g \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer- Verlag, Berlin, 1974. MR 0384804

The _comparison theorem_ shown there (theorem V2.3.2) says that the crystalline cohomology over $\mathbb{Z}/p$ is canonically identified with the [[de Rham cohomology]] of a lift to [[p-adic geometry]] if one exists.

Discussion of this in terms of [[Cech cohomology]] is in

* [[Bhargav Bhatt]], [[Aise Johan de Jong]], _Crystalline cohomology and de Rham cohomology_ ([pdf](http://www.math.columbia.edu/~dejong/papers/crystalline-comparison.pdf))

## Related concepts

* [[comparison theorem]]

* [[comparison theorem (étale cohomology)]]


comparison theorem (étale cohomology)

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=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[étale cohomology]] of [[schemes]], the _Artin comparison theorem_ ([Artin 66](#Artin66)) says that under some conditions on a [[scheme]] and a [[sheaf]] of [[coefficients]],  the [[étale cohomology]] of the scheme coincides with the [[ordinary cohomology]] (e.g. [[singular cohomology]]) of its underlying [[complex analytic topology]].

Historically this kind of statement was a central motivation for the development of [[étale cohomology]] in the first place.

Specifically, let $A$ be either of

* a [[finite group|finite]] [[field]];

* the [[p-adic integers]] $\mathbb{Z}_p$ for some [[prime]] $p \geq 2$;

* the [[p-adic numbers]] $\mathbb{A}_{p}$

(but _not_ for instance the [[integers]] themselves).

Then for $X$ a [[variety]] over the [[complex numbers]] and $X^{an}$ its [[analytification]] to the [[topological space]] of complex points $X(\mathbb{C})$  with its [[complex analytic topology]], then there is an [[isomorphism]]

$$
  H^\bullet(X_{et}, A) \simeq H^\bullet(X^{an}, A)
$$

between the [[étale cohomology]] of $X$ and the [[ordinary cohomology]] of
$X^{an}$.

Notice that on the other hand for instance if instead $X = Spec(k)$ is the [[spectrum of a commutative ring|spectrum]] of a [[field]], then its [[étale cohomology]] coincides with the [[Galois cohomology]] of $k$. In this way [[étale cohomology]] interpolates between "[[topology|topological]]" and "[[number theory|number theoretic]]" notions of cohomology.

## Related concepts

* [[analytification]], [[GAGA]]

* [[comparison theorem]]

* [[comparison theorem (crystalline cohomology)]]

* [[Chow's theorem]]

## References

The original reference is

* [[Michael Artin]], _The étale topology of schemes_ , Proc. Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 44–56
 {#Artin66}

* [[Alexander Grothendieck]], [[SGA]] I, Exposé XII

A textbook account is in

* {#Danilov91} [[Vladimir Danilov]], chapter 3 of _Cohomology of algebraic varieties_, in I. Shafarevich (ed.), _Algebraic Geometry II_, volume 35 of _Encyclopedia of mathematical sciences_, Springer 1991 ([GoogleBooks](http://books.google.de/books?id=ZhzXJHUgcRUC&lpg=PA67&ots=aVQoeMkBwc&dq=analytification&pg=PA61&redir_esc=y#v=onepage&q=analytification&f=false)))

Reviews and lecture notes include

* [[eom]], _[Comparison theorem](http://www.encyclopediaofmath.org/index.php/Comparison_theorem_(algebraic_geometry)_

* [[James Milne]], section 21 of _[[Lectures on Étale Cohomology]]_

In the context of [[Berkovich analytic spaces]] see also

* [[Vladimir Berkovich]], _On the comparison theorem for étale cohomology of non-archimedean analytic spaces_ ([pdf](http://www.wisdom.weizmann.ac.il/~vova/IJM_1995_92_compar.pdf))

[[!redirects comparison theorem (etale cohomology)]]


compatible localization

There are several notions under a name of compatible localization; where localization pertains to the localization of rings or of localization of categories.

## (Pairwise) compatibility of a family of localizations

Consider a family of localizations sharing the same source ring (or source category). Two Gabriel localizations $i_{R k}: R\to R_k$, $k = 1,2$ of a ring $R$, with
corresponding localization endofunctors for modules $Q_{k}:M\to M_k$ are mutually compatible if the consecutive localization of $Q_2 Q_1 R$ is naturally a ring (or equivalently $Q_1 Q_2 R$ is a ring), such that the corresponding localization arrow is a morphism of rings. This is equivalent to the following more general criterium.

Two [[affine localization]]s with the inverse image functors
$Q_k^*$ and direct image functors $Q_{k*}: A\to A_k$, $k = 1,2$ are mutually compatible if the endofunctors $Q_k:=Q_{k*}Q_k^*$ mutually commute as endofunctors of $A$.

According to Thomason-Trobaugh, an [[algebraic scheme]] is **semiseparated** if it has an affine cover such that the intersection of any two affines in the cover is affine (semiseparated cover). For example, every [[separated scheme]] is semiseparated. [[Alexander Rosenberg]] extends this terminology to [[noncommutative scheme]]s by observing that an affine cover is semiseparated iff the corresponding localization functors restricting the category of quasicoherent sheaves on the whole scheme to an element of a cover, form a compatible family of affine localizations.

## Coaction compatible localizations

Given a right $B$-comodule algebra $(E, \rho)$ for a bialgebra $H$, with coaction $\rho : E\to E\otimes H$, an Ore localization
of rings (or more general affine localization) $j : E\to E'$ is $\rho$-compatible if there exists a morphism $\rho': E'\to E\otimes H$
such that the following diagram commutes:

$$\array{
E & \stackrel{j}\to & E'\\
\downarrow\rho && \downarrow \rho'\\
E\otimes H &\stackrel{j\otimes H}\to & E'\otimes H
}$$

## Compatibility of localizations and endofunctors

A localization functor $Q^*=Q^*_\Sigma : A\to B$
(not necessarily having a right adjoint)
universally inverting a family $\Sigma$ of morphisms in $A$
is compatible with an endofunctor $G:A\to A$ if $G(\Sigma)\subset \Sigma$. By the universality property of the localization functor, this is equivalent to the existence of a functor $G_\Sigma\in B$
such that $Q^* G = G_\Sigma Q^*$.

If $Q^*$ is a localization having a fully faithful right adjoint $Q_*$, with counit $\epsilon$ (which is hence iso) and unit $\eta$; then the compatibility of $Q^*$ with an endofunctor $G$ is equivalent to any of the following:

(i) there exists a distributive law $l : Q_* Q^* G\to G Q_* Q^*$ where $Q_* Q^*$ is understood as (the underlying functor of) the idempotent monad induced by the adjunction $Q^*\dashv Q_*$

(ii) the natural transformation $Q^* G\eta : Q^* G\to Q^* G Q_* Q^*$ is invertible

(iii) there is a functor $G'$ and a natural isomorphism $Q^* G\cong G' Q^*$.

If (i) holds then $l$ is invertible and uniquely determined
by $G$ and the monad $Q^* Q_*$. Notice that $Q_* G_\Sigma \neq G Q_*$ in general. See [[zoranskoda:distributive law for idempotent monad]] for more.

One can also consider the distributive laws $\tilde{l} : G Q_* Q^*\to Q_* Q^* G$. The inverse $l^{-1}$ of the distributive law $l: G Q_* Q^*\to Q_* Q^* G$ as above is the unique invertible example of such. It is not clear (to me at least -- [[Zoran Škoda|Zoran]]) if there are also noninvertible examples for $\tilde{l}$.

## Literature

For the pairwise compatibility of localizations see

* [[Fred Van Oystaeyen]], _Compatibility of kernel functors and localization functors_, Bull. Soc. Math. Belg. __28__ (1976), no. 2, 131--137.

* [[Alain Verschoren]], _Compatibility and stability_,
Notas de Matemática [Mathematical Notes], 3. Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. xii+81 pp. ISBN: 84-7684-934-6

* J. Mulet, A. Verschoren, _On compatibility. II._, Comm. Algebra 20 (1992), no. 7, 1897--1905. [doi](http://dx.doi.org/10.1080/00927879208824438)

* M. I. Segura, D. Tarazona, A. Verschoren, _On compatibility_,  Comm. Algebra __17__ (1989),  no. 3, 677--690.

The compatibility of coactions with Ore localizations is introduced in

* [[Zoran Škoda]], _Localizations for construction of quantum coset spaces_, in "Noncommutative geometry and Quantum groups", W.Pusz,  P.M. Hajac, eds. Banach Center Publications __61__, pp. 265--298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090).

The compatibilities between localization functors
with endofunctors is formulated in

* [[V. A. Lunts]], [[A. L. Rosenberg]], _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3894)

A version with distributive laws is introduced in

* [[Zoran Škoda]], Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770).

See also

* [[Zoran Škoda]], _Compatibility of (co)actions and localization_, [arXiv:0902.1398](http://arxiv.org/abs/0902.1398) (preliminary version).

[[!redirects compatible localizations]]


CompBoolAlg

__$Comp Bool Alg$__ is the [[category]] whose [[objects]] are [[complete boolean algebras]] and whose [[morphisms]] are complete lattice [[homomorphisms]], that is [[functions]] which preserve all [[joins]] and [[meets]] (including the [[bottom]] and [[top]] elements); it follows that they preserve the boolean [[negation]] operation.  $Comp Bool Alg$ is a [[subcategory]] of [[Pos]].

$Comp Bool Alg$ is given by a [[variety of algebras]], or equivalently by an [[algebraic theory]], so it is an [[equationally presented category]]; however, it requires operations of arbitrarily large arity.  In fact, it is *not* a [[monadic category]] (over [[Set]]), because it lacks some [[free objects]].  Specifically, the __free complete boolean algebra__ on a [[set]] $X$, while it exists (by general abstract nonsense) as a [[class]], is small only if $X$ is [[finite set|finite]] (in which case it is finite and equals the [[free boolean algebra]] $2^{2^X}$ on $X$).

## In weak foundations

For all practical purposes, $Comp Bool Alg$ is not available in [[predicative mathematics]].  The definitions go through, but we cannot prove that it has any [[infinite set|infinite]] objects.  (More precisely, the [[power class]] of a nontrivial small complete boolean algebra must also be small.)  Generally speaking, predicative mathematics treats infinite complete boolean algebras only as [[proper class|large]] objects.

In impredicative [[constructive mathematics]], it no longer holds that the free complete boolean algebra on a finite set equals the free boolean algebra on that set.  In particular, the free complete boolean algebra on the [[empty set]] is the set of [[truth values]], while the free boolean algebra on the empty set is the set $\{\bot, \top\}$ of decidable truth values.  (I\'m not sure whether the free complete boolean algebra a positive finite number of elements is even small.)

In predicative constructive mathematics, even the free complete boolean algebra on the empty set is a proper class.

category: category

[[!redirects CompBoolAlg]]
[[!redirects Comp Bool Alg]]
[[!redirects CompBoolLat]]
[[!redirects Comp Bool Lat]]
[[!redirects CompBoolRing]]
[[!redirects Comp Bool Ring]]

[[!redirects free complete boolean algebra]]
[[!redirects free complete Boolean algebra]]
[[!redirects free complete boolean lattice]]
[[!redirects free complete Boolean lattice]]
[[!redirects free complete boolean ring]]
[[!redirects free complete Boolean ring]]


CompLat

__$Comp Lat$__ is the [[category]] whose [[objects]] are [[complete lattices]] and whose [[morphisms]] are complete lattice [[homomorphisms]], that is [[functions]] which preserve all [[joins]] and [[meets]] (including the [[bottom]] and [[top]] elements).  $Comp Lat$ is a [[subcategory]] of [[Pos]].

$Comp Lat$ is given by a [[variety of algebras]], or equivalently by an [[algebraic theory]], so it is an [[equationally presented category]]; however, it requires operations of arbitrarily large arity.  In fact, it is *not* a [[monadic category]] (over [[Set]]), because it lacks some [[free objects]].  Specifically, the __free complete lattice__ on a [[set]] $X$, while it exists (by general abstract nonsense) as a [[class]], is small only if $X$ has at most $2$ elements (in which case it is finite and equals the [[free lattice]] on $X$).

## In weak foundations

For all practical purposes, $Comp Lat$ is not available in [[predicative mathematics]].  The definition goes through, but we cannot prove that $Comp Lat$ has any [[infinite set|infinite]] objects.  (More precisely, the [[power class]] of a nontrivial small complete lattice must also be small.)  Generally speaking, predicative mathematics treats infinite complete lattices only as [[proper class|large]] objects.

In impredicative [[constructive mathematics]], it no longer holds that the free complete lattice on a set with at most $2$ elements equals the free lattice on that set.  In particular, the free complete lattice on the [[empty set]] is the set of [[truth values]], while the free lattice on the empty set is the set $\{\bot, \top\}$ of decidable truth values.  (I\'m not sure whether the free complete lattice on $1$ or $2$ elements is even small.)

In predicative constructive mathematics, even the free complete lattice on the empty set is a proper class.

category: category

[[!redirects CompLat]]
[[!redirects Comp Lat]]

[[!redirects free complete lattice]]
[[!redirects free complete lattices]]


complement

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## Idea

### In sets

The __complement__ $X \setminus S$ of a [[subset]] $S \subset X$ of a [[set]] $X$ is the set of all [[elements]] of $X$ not contained in $S$:

$$
  X \setminus S
  \;\coloneqq\;
  \big\{
    x \in X \;|\;
    x \notin S
  \big\}
  \,.
$$

(Besides $X \setminus S$, there are many other notations, such as $X - S$, $\bar{S}$, $S^c$, $\neg{S}$, and so forth.)

Notice that $S \cap (X \setminus S) = \empty$, while $S \cup (X \setminus S) = X$ by the principle of [[excluded middle]].

### In lattices

A __complement__ of an element $S$ of a [[lattice]] is an element $T$ such that $S \wedge T = \bot$ and $S \vee T = \top$. Note that in general, complements need not be unique; for example, in the lattice of vector subspaces of a 2-dimensional [[vector space]] over a [[field]] $k$, a 1-dimensional subspace will have as many complements as there are elements of $k$. However, in some cases complements will be unique, for example in a [[distributive lattice]], in which case it is denoted $\tilde{S}$ (or $\neg S$, etc.).

If every element has a complement, one speaks of a [[complemented lattice]]. Examples are [[Boolean algebras]], and in fact complemented distributive lattices are the same thing as Boolean algebras (in the sense that the [[category]] of Boolean algebras is [[equivalence|equivalent]] to the category of complemented distributive lattices).

More generally, the __pseudocomplement__ of an element $S$ of a [[Heyting algebra]] is given by $\tilde{S} = S \Rightarrow \bot$.  This satisfies $S \wedge \tilde{S} = \bot$ but not $S \vee \tilde{S} = \top$ in general.  This case includes the complement of a subset even in [[constructive mathematics]].

### In any category

In another direction, the __complement__ of a [[complemented subobject]] $S$ of an [[object]] $X$ in a [[coherent category]] is the unique subobject $\tilde{S}$ such that $S \cap \tilde{S}$ is the [[initial object]] and $S \cup \tilde{S} = X$.

### In truth values

The complement of a [[truth value]] (seen as a subset of the [[point]]) is called its _[[negation]]_.

### Strict complements

In [[constructive mathematics]], the denial inequality is not a [[tight apartness relation]], and [[sets]] in general do not have a [[tight apartness relation]]. This means that every [[set]] with a [[tight apartness relation]] has two notions of inequality, the normal [[denial inequality]] $\neq$ and the [[tight apartness relation]] $\#$. Thus, there are two notions of complement. There is the usual notion of complement, where given a set $X$, every element in the complement of a subset $S \subseteq X$ is not equal to any element in $S$. Then there are **strict complements**, where given a set $X$ with a tight inequality relation $\#$, every element in the strict complement of a subset $S \subseteq X$ is apart from every element in $S$.

## Related concepts

* [[orthocomplemented lattice]]

* [[open subset]], [[closed subset]]

* [[complemented subobject]]

* [[decidable subset]]

* [[decidable object]]

* [[pushout complement]]

[[!redirects complement]]
[[!redirects complements]]
[[!redirects complementation]]

[[!redirects pseudocomplement]]
[[!redirects pseudocomplements]]

[[!redirects strict complement]]
[[!redirects strict complements]]

[[!redirects complemented]]


complemented lattice

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## Idea

An [[lattice]] is _complemented_ if every element has a [[complement]]. It is _orthocomplemented_ if it is equipped with an [[involution]] that sends each element to a complement.

## Examples

These are both orthocomplemented:

* any [[Boolean algebra]], as in [[classical logic]]

* the [[subspaces]] of a [[Hilbert space]], as in Birkhoff--von Neumann [[quantum logic]]/[[Hilbert lattice]]

## Related concepts

* [[orthomodular lattice]]

## References

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

* P. Pták and S. Pulmannová, _Orthomodular structures as quantum logics_, ser. Fundamental theories of physics. Kluwer Academic Publishers,
1991.

[[!redirects orthocomplemented lattice]]
[[!redirects orthocomplemented lattices]]

[[!redirects complemented lattice]]
[[!redirects complemented lattices]]


complemented subobject

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#### Constructivism, Realizability, Computability
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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[subobject]] given by a [[monomorphism]] $i: S \to X$ in a [[coherent category]] is called _complemented_ (also: _decidable_) if it has a [[complement]]: a subobject $\tilde{i} \colon \tilde{S} \to X$ such that

1. their [[intersection]] is [[generalized the|the]] [[initial object]]

   $$
     S \cap \tilde{S}
     \;\simeq\;
     \emptyset
   $$

1. their [[union]] is all of $X$

   $$
     S \cup \tilde{S} \simeq X
   $$

## Examples

In [[constructive mathematics]], a complemented subobject in [[Set]] is called a _[[decidable subset]]_.

## Related entries

* [[constructive model structure on simplicial sets]]

## Properties

Every subobject is complemented in a [[Boolean category]].

In [[classical mathematics]], every subset is decidable.  Indeed, the  [[law of excluded middle]] may be taken to say precisely that every subset of the [[point]] is complemented.

More generally, if every subobject of the [[terminal object]] of a [[well-pointed topos|well-pointed]] [[coherent category]] $C$ is complemented, then every subobject in $C$ is complemented.

## References

* [[Nicola Gambino]], [[Christian Sattler]], [[Karol Szumiło]], Section 1.1 of _The constructive Kan-Quillen model structure: two new proofs_ ([arXiv:1907.05394](https://arxiv.org/abs/1907.05394))

[[!redirects decidable inclusion]]
[[!redirects decidable inclusions]]


complete algebraic variety

#Contents#
* table of contents
{:toc}

## Idea

An [[algebraic variety]] $X$ is called _complete_ if for every algebraic variety $Y$ the [[projection]]

$$
  X \times Y \longrightarrow Y
$$

out of their [[Cartesian product]] is a [[closed map]].

This is an algebraic analog of being a [[compact topological space]].

## References

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

[[!redirects complete algebraic varieties]]

[[!redirects complete variety]]
[[!redirects complete varieties]]


complete augmented algebra

## Definition

A __complete augmented algebra__ over a [[field]] $k$ is an [[augmented algebra]] $R$ over $k$ equipped with a decreasing [[filtration]] of two-sided [[ideals]]
$$R=F_0 R\supset F_1 R\supset F_2 R \supset \cdots,$$
where $1\in F_0 R$ and $F_p R \cdot F_q R \subset F_{p+q} R$,
and, furthermore,
the map $F_1 R\to F_0 R$ is the [[augmentation ideal]], i.e., the [[kernel]] of the [[augmentation]] map $R\to k$,
the [[associated graded algebra]] of $R$ is generated by its degree $1$ component,
and the [[filtration]] is complete, i.e., the canonical map
$$R\to lim_n F_n R$$
is an [[isomorphism]].

A __morphism of complete augmented algebras__ is a morphism of [[augmented algebras]] that is also a morphism of [[filtered algebras]].

## Examples

The [[forgetful functor]] from complete augmented algebras to [[augmented algebras]]
admits a [[left adjoint functor]], which sends an [[augmented algebra]] $B$
to the complete augmented algebra $\hat B=lim_n B/I^n$,
where $I$ is the [[augmentation ideal]] of $B$.

In particular, for any set $I$, the algebra $k\langle\langle x_i\rangle\rangle_{i\in I}$ of [[noncommutative formal power series]] in variables $x_i$ ($i\in I$)
is a complete augmented algebra, since it is the completion of the algebra
of noncommutative polynomials in the same variables.
Such algebras are precisely the [[projective objects]]
in the category of complete augmented algebras.
(Corollary A.1.9 in [Quillen](#Quillen).)

The [[quotient]] $R/J$ of a complete augmentation algebra $R$ by a closed ideal $J$
such that the [[augmentation map]] vanishes on $J$
is again a complete augmented algebra.

## Properties

Any complete augmented algebra is a [[quotient]] of the algebra of [[noncommutative formal power series]] by a closed [[ideal]].
The quotient map can be chosen to induce an isomorphism of [[associated graded algebras]] in degree 1.
(Corollary A.1.7 in [Quillen](#Quillen).)

A morphism of complete augmented algebras is an [[effective epimorphism]]
if and only if it is surjective,
equivalently, the degree 1 component of its associated graded is surjective.
(Proposition A.1.8 in [Quillen](#Quillen).)

The category of complete augmented algebras admits [[small limits]]
and has a [[projective generator]], namely, $k\langle\langle x\rangle\rangle$.
(Proposition A.1.10 in [Quillen](#Quillen).)

## Constructions

The [[forgetful functor]] from the category of complete augmented algebras
to the [[category of groups]] that sends a complete augmented algebra $R$ to the [[group]] $1+\bar R$, where $\bar R$ denotes the [[augmentation ideal]] of $R$,
admits a [[left adjoint functor]],
which sends a group $G$ to the completion of its [[group algebra]].

The [[forgetful functor]] from the category of complete augmented algebras
to the [[category of Lie algebras]] that sends a complete augmented algebra $R$ to the [[Lie algebra]] $\bar R$ with [[Lie bracket]] $[x,y]=xy-yx$, where $\bar R$ denotes the [[augmentation ideal]] of $R$,
admits a [[left adjoint functor]],
which sends a [[Lie algebra]] $\mathfrak{g}$ to the completion of its [[universal enveloping algebra]].

See (1.12) in [Quillen](#Quillen).

## Monoidal structure

The category of complete augmented algebras admits a [[symmetric monoidal structure]],
given by the completion of the tensor product of the underlying filtered vector spaces.

The [[associated graded]] functor is a [[strong monoidal functor]]
from the category of complete augmented algebras to the category
of graded algebras.

The monoidal product has a universal property:
morphisms $R\otimes R'\to S$ are in a natural bijection
with pairs of morphisms $R\to S$ and $R'\to S$ whose images in $S$ commute.

## Related concepts

* [[augmented algebra]]

* [[complete Hopf algebra]]

* [[Malcev group]]

* [[Malcev completion]]

## References

* {#Quillen} [[Daniel Quillen]], _Rational homotopy theory_, Annals of Mathematics 90:2 (1969), 205.  [doi](http://dx.doi.org/10.2307/1970725).

[[!redirects complete augmented algebras]]


complete Boolean algebra

# Complete boolean algebras
* table of contents
{: toc}

## Definition

A __complete Boolean algebra__ is a [[complete lattice]] that is also a [[Boolean algebra]].  Since lattice homomorphisms of Boolean algebras automatically preserves the Boolean structure, the complete Boolean algebras form a [[full subcategory]] [[CompBoolAlg]] of [[CompLat]].

## Category of complete Boolean algebras

A natural notion of [[morphism]] for complete Boolean algebras
is that of a continuous homomorphism of Boolean algebras,
also known as complete Boolean homomorphisms.
These can be defined as homomorphisms of Boolean algebras that preserve [[suprema]],
or, equivalently, [[infima]].
It suffices to require preservation of [[suprema]] of [[directed subsets]].

With this notion of morphisms, complete Boolean algebras form a category.

## Relation to Stonean locales and Boolean locales

The category of complete Boolean algebras is equivalent
to the category of [[Boolean locales]]
and [[Stonean locales]].
The latter fact is also known as the (localic) __Stonean duality__.

In the presence of the [[axiom of choice]],
the category of [[Stonean locales]] is equivalent
to the category of [[Stonean spaces]],
so the latter is contravariantly equivalent to the category
of complete Boolean algebras.
The latter fact is also known as the (traditional) __Stonean duality__.

## Stonean duality

The [[Stone duality]] establishes a [[contravariant equivalence]]
of [[categories]] between the category of [[Boolean algebras]]
and the category of [[Stone spaces]].
The latter is a [[full subcategory]] of the category [[Top]]
of [[topological spaces]] and [[continuous maps]]
on [[compact]] [[totally disconnected]] [[Hausdorff]] [[topological spaces]].

Recall that a [[Stonean space]] is a [[compact]] [[extremally disconnected]]
[[Hausdorff]] [[topological space]].
[[Morphisms]] of [[Stonean spaces]] are defined to be [[open]] [[continuous maps]].
Restricting the [[Stone duality]] produces a [[contravariant equivalence]]
between the category of complete Boolean algebras
and the category of [[Stonean spaces]].
See Corollary 6.10(2) in [Bezhanishvili](#SDGC).

## CABAs

Assuming [[excluded middle]], complete *[[atomic Boolean algebra|atomic]]* Boolean algebras are (up to [[isomorphism]]) precisely [[power sets]].  In fact, taking power sets defines a [[fully faithful functor]] from the [[opposite category]] of [[Set]] to [[Comp Bool Alg]] whose [[essential image]] consists of the complete atomic boolean algebras.  See at _[Set -- Properties -- Opposite category](Set#OppositeCategory)_. These abstract representations of power sets are important enough to have their own abbreviation: 'CABA'.

This property of CABAs is not applicable in [[constructive mathematics]], where power sets are rarely boolean algebras.  However, we can use [[discrete space|discrete]] [[locales]] instead (or rather, their corresponding [[frames]]).  That is, define a __CABA__ to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$) are [[open map|open]] (as locale maps).  (Of course, that $X \to 1$ is open is the condition that $X$ is [[overt locale|overt]].)  Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above).

Another approach is via [[overlap algebra]]s.  An overlap algebra is a [[frame]] with two extra conditions (one of which is [[overt locale|overtness]] of the corresponding [[locale]]).  Classically, overlap algebras are the same thing as complete Boolean algebras; constructively, atomic overlap algebras are the same thing as powersets.  See [Ciraulo 2010](#Ciraulo2010).

## Algebraicity

Complete Boolean algebras are the models of an [[algebraic theory]] (in which the operations, notably $j$-indexed [[suprema]] and infima, have arities $j$ unbounded by any cardinal). It follows from general principles that the underlying-set functor $U: CompBoolAlg \to Set$ preserves and reflects limits and isomorphisms.

However, this functor $U$ is _not_ [[monadic functor|monadic]]; in fact, it does not even possess a left adjoint. Indeed, while the free complete Boolean algebra on a _finite_ set $X$ exists and coincides with the free Boolean algebra on $X$ (it is finite, being isomorphic to the double power set $P(P X)$), we have

+-- {: .un_theorem}
###### Theorem (Gaifman-Hales; Solovay)
There is no free complete Boolean algebra on countably many generators.
=--

As a consequence, $CompBoolAlg$ is not cocomplete (otherwise there would exist a countable coproduct of copies of $P(P 1)$, which is ruled out by the previous theorem).

## Related concepts

* [[Boolean locale]]
* [[Stonean locale]]

## References

For instance around theorem 2.4 of

* [[Jaap van Oosten]], _Basic category theory_ ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))

and

* PlanetMat, _[representing a complete atomic Boolean algebra by power set](http://planetmath.org/RepresentingACompleteAtomicBooleanAlgebraByPowerSet.html)_

The [[Stone duality]] for complete Boolean algebras is explained in

*  [[Guram Bezhanishvili]],
_Stone duality and Gleason covers through de Vries duality_.
Topology and its Applications 157 (2010), 1064--1080.
{#SDGC}

For overlap algebras, see

*  [[Francesco Ciraulo]], 2010.  _The Role of the Overlap Relation in Constructive Mathematics_.  [Slides](https://www.math.unipd.it/~ciraulo/documents/Ciraulo%20Chiemsee%202010.pdf).
{#Ciraulo2010}

[[!redirects complete Boolean algebra]]
[[!redirects complete boolean algebra]]
[[!redirects complete Boolean algebras]]
[[!redirects complete boolean algebras]]
[[!redirects complete Boolean lattice]]
[[!redirects complete boolean lattice]]
[[!redirects complete Boolean lattices]]
[[!redirects complete boolean lattices]]
[[!redirects complete Boolean ring]]
[[!redirects complete boolean ring]]
[[!redirects complete Boolean rings]]
[[!redirects complete boolean rings]]

[[!redirects complete atomic Boolean algebra]]
[[!redirects complete atomic boolean algebra]]
[[!redirects complete atomic Boolean algebras]]
[[!redirects complete atomic boolean algebras]]
[[!redirects complete atomic Boolean lattice]]
[[!redirects complete atomic boolean lattice]]
[[!redirects complete atomic Boolean lattices]]
[[!redirects complete atomic boolean lattices]]
[[!redirects complete atomic Boolean ring]]
[[!redirects complete atomic boolean ring]]
[[!redirects complete atomic Boolean rings]]
[[!redirects complete atomic boolean rings]]
[[!redirects caba]]
[[!redirects CABA]]
[[!redirects cabas]]
[[!redirects CABAs]]
[[!redirects caba's]]
[[!redirects CABA's]]
[[!redirects caba’s]]
[[!redirects CABA’s]]
[[!redirects caba\'s]]
[[!redirects CABA\'s]]


complete category

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# Complete categories
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{: toc}

## Definition

A [[category]] $C$ is **complete** if it has all [[small limit|small]] [[limit|limits]]: that is, if every [[small diagram|small]] [[diagram]]

$$ F: D \to C$$

where $D$ is a [[small category]] has a [[limit]] in $C$.

Sometimes one says that $C$ is __small-complete__ to stress that $D$ must be small; compare [[finitely complete category]].  Also compare [[complete small category]], which is different; here we see that any [[small category]] that is also small-complete must be [[thin category|thin]] (at least classically).

## Examples

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

As hinted above, every [[complete lattice]] is complete as a category.

A common situation is that of a category of algebras for a [[monad]] in a complete category:  If there exists a [[monadic  functor]] $C\to D$ and $D$ is complete, then $C$ is complete (as [[monadic functor|monadic functors]] [[created limit|create limits]]).  This includes some obvious examples (such as [[Grp]], [[Ab]], and [[Vect]]), as well as some less-obvious examples, such as [[complete lattice|complete lattices]] and [[compact space|compact]] [[Hausdorff space|Hausdorff]] spaces.

The contravariant presheaf category $[C,Set]$ is complete, and the dual [[Yoneda embedding]] exhibits it as the [[free completion]] of $C$.

## Related concepts

* [[cocomplete category]]

* [[bicomplete category]]

* [[M-complete category]]

[[!redirects complete category]]
[[!redirects complete categories]]
[[!redirects small-complete category]]
[[!redirects small-complete categories]]
[[!redirects smally complete category]]
[[!redirects smally complete categories]]


complete field

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## Definition

A [[field]] (as an [[associative algebra|algebraic structure]]) equipped with a [[metric]] is **complete** if its operations are [[continuous function|continuous]] with respect to the induced [[topology]] and it is a [[complete metric space]].

The most important case is the case of [[valued fields]] (i.e. multiplicatively [[normed fields]]).

Given a multiplicative [[discrete valuation]] on a field $k$, there is a [[topology]] on $k$ induced by the metrics induced by the [[valuation]]. A complete valued field is a field complete with respect to the valuation metric.

## Properties

One of [[Ostrowski's theorems]] says that for $k$ a field [[complete field|complete]] with respect to an [[absolute value]] ${\vert - \vert}$ either the absolute value is archimedean, in which case $k$ is either the field of [[real numbers]] or of [[complex numbers]], or the absolute value is non-archimedean.

## Related concepts

* [[archimedean field]], [[archimedean valued field]]

[[!include analytic geometry ingredients -- table]]

## References

See also:

* Wikipedia, _[Complete field](https://en.wikipedia.org/wiki/Complete_field)_

[[!redirects complete fields]]


complete Heyting algebra

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## Idea

A *complete Heyting algebra* is a [[Heyting algebra]] which is also a [[complete lattice]]; that is, it is a [[poset]] with arbitrary [[limits]] and [[colimits]], that is also [[cartesian closed]].

## Properties

### Relation to frames

By the [[adjoint functor theorem]], one can demonstrate that every [[frame]] is a complete Heyting algebra, and vice versa, so far as the underlying [[poset]] goes.

However, [[morphisms]] of frames needn't preserve [[exponential objects]] or infinitary [[meets]], as would most naturally be required of morphisms of complete Heyting algebras.  Also, when considering *[[large category|large]]* lattices which are only *small*-[[complete category|complete]], then frames and complete Heyting algebras are different objects.

## References

See at *[[Heyting algebra]]*.


complete Hopf algebra

## Definition

A __complete Hopf algebra__ is a [[complete augmented algebra]] $A$
equipped with a diagonal map $\Delta\colon A\to A\otimes A$
that is a morphism of [[complete augmented algebras]],
is coassociative and cocommutative
and has the [[augmentation map]] as a counit map.
(See the article [[Hopf algebra]] for a definition of these terms.)

## Examples

The completion of a [[Hopf algebra]] as
an [[augmented algebra]] (see [[complete augmented algebra]] for a construction)
is a complete Hopf algebra.

In particular, the completion of the [[group algebra]] of a [[group]]
and the [[universal enveloping algebra]] of a [[Lie algebra]] is a complete Hopf algebra.

## Properties

The [[monoidal product]] of the underlying [[complete augmented algebras]]
of complete Hopf algebras can be equipped in the obvious way
with a structure of a complete Hopf algebra.
The resulting [[monoidal structure]] $\hat\otimes$ on complete Hopf algebras
is the [[cartesian monoidal structure]],
with the projection maps $A\otimes A'\to A$ and $A\otimes A'\to A'$
induced by the [[augmentation maps]] of $A'$ respectively $A$.

The [[associated graded]] functor sends complete Hopf algebras
to graded [[Hopf algebras]].

## Constructions

The [[forgetful functor]] from the category of complete Hopf algebras
to the [[category of groups]]
sends a complete Hopf algebra $A$ to the [[group]]
$$G(A)=\{x\in 1+\bar A\mid \Delta x=x\hat\otimes x\}$$
of _group-like elements_ in $A$.
Its [[left adjoint functor]] sends a [[group]] to the completion
of its [[group algebra]].
(See Proposition A.2.5 in [Quillen](#Quillen).)

In the case $k=\mathbf{Q}$,
the forgetful functor is [[fully faithful]] and its [[essential image]]
comprises precisely [[Malcev groups]].
Thus, the category of rational complete Hopf algebras
is equivalent to the category of [[Malcev groups]].
(Theorem A.3.3 in [Quillen](#Quillen).)

The [[monad]] induced by the [[adjunction]] between
groups and complete Hopf algebras
is known as the [[Malcev completion]] of groups.

The [[forgetful functor]] from the category of complete Hopf algebras
to the [[category of Lie algebras]]
sends a complete Hopf algebra $A$ to the [[Lie algebra]]
$$P(A)=\{x\in \bar A\mid \Delta x=x\hat\otimes 1+1\hat\otimes x\}$$
of _primitive elements_ in $A$.
Its [[left adjoint functor]] sends a [[Lie algebra]] to the completion
of its [[universal enveloping algebra]].
(See Proposition A.2.5 in [Quillen](#Quillen).)

In the case $k$ has characteristic 0,
the forgetful functor is [[fully faithful]] and its [[essential image]]
comprises precisely [[Malcev Lie algebras]].
Thus, the category of complete Hopf algebras over a [[field]] of characteristic 0
is equivalent to the category of [[Malcev Lie algebras]] over the same [[field]].
(Theorem A.3.3 in [Quillen](#Quillen).)

## Properties in the case of zero characteristic

Assume that the [[field]] $k$ has [[characteristic]] 0.

Then the [[exponential]] map
$$\exp\colon P(A)\to G(A)$$
can be defined using [[formal power series]]
for any complete Hopf algebra $A$.

The exponential map is an isomorphism of sets,
both of which have natural filtrations induced from $A$,
which turn them into a complete [[filtered Lie algebra]] and a complete [[filtered group]] respectively.

The exponential map induces an isomorphism of the [[associated graded]] [[Lie algebra]] over [[integers]].
For the filtered [[group]] $G(A)$, the [[Lie bracket]] on its [[associated graded]] [[abelian group]] is given by the [[commutator]].
The [[associated graded]] [[abelian group]] of $G(A)$ is also equipped
with a structure of a $k$-[[module]] transferred from the [[associated graded]]
$k$-[[module]] of $P(A)$ via the exponential isomorphism.

## Related concepts

* [[augmented algebra]]

* [[complete augmented algebra]]

* [[Malcev group]]

* [[Malcev Lie algebra]]

* [[Malcev completion]]

## References

* {#Quillen} [[Daniel Quillen]], _Rational homotopy theory_, Annals of Mathematics 90:2 (1969), 205.  [doi](http://dx.doi.org/10.2307/1970725).

[[!redirects complete Hopf algebras]]


complete lattice

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## Definition

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###### Definition

A **complete lattice** is a [[partial order|poset]] which has all small [[join|joins]] and [[meet|meets]] (as opposed to just finite joins and meets).

In particular, it is a [[lattice]].

Complete lattices and complete lattice homomorphisms form a [[concrete category]] [[CompLat]].

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###### Remark

By the [[adjoint functor theorem]] for posets, having either all joins or all meets is sufficient for the other.  However, a [[suplattice]] morphism may preserve only joins, while dually an [[inflattice]] morphism may preserve only meets.  Furthermore, a *large* poset with all *small* joins or meets need not have the other.

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###### Remark

Regarded as a [[small category]], a complete lattice is [[complete category|complete]].  Conversely, in [[classical logic]] and in any [[Grothendieck topos]], any [[complete small category]] is in fact a preorder, and hence a complete lattice.

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## Examples

* Any [[power set]];
* Any finite inhabited [[total order|toset]];
* The ordered set of [[real number]]s, if a [[top]] element $\infty$ and a [[bottom]] element $-\infty$ are added;
* The [[unit interval]] $[0,1]$.

Complete lattices are harder to come by in [[constructive mathematics]] and nearly impossible in [[predicative mathematics]] (at least if they are to be small).  In particular, one must use the Mac Neille reals (and be a bit careful about infinity) for the analytic examples to work constructively.

## Related concepts

* [[completely distributive lattice]]

[[!redirects complete lattices]]


complete loop graph

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## Definition

A **complete loop graph** is a [[loop graph]] $G$ where between any two vertices $a:G$ and $b:G$ there is exactly one edge $p:a \to_G b$.

In the context of [[homotopy type theory]], a complete loop graph is a type $G$ with a type family $\to_G$ such that the [[dependent type]] $a \to_G b$ is a [[contractible type]] for all elements $a:G$ and $b:G$.

A complete loop graph is **univalent**, a **[[subsingleton]]**, or a **[[proposition]]** if the canonical function
$$idtoedge:(a =_G b) \to (a \to_G b)$$
is an equivalence of types.

## See also

* [[loop graph]]
* [[directed loop graph]]
* [[equivalence relation]]
* [[pregroupoid]]
* [[precategory]]
* [[Segal space]]


complete normed group

## Idea

A [[normed group]] which is [[complete space|complete]]

## Related concepts

[[!include analytic geometry ingredients -- table]]

[[!redirects complete normed groups]]