locally compact topological space

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

## Idea

A [[topological space]] is called _locally compact_ if every point has a [[compact topological space|compact]] [[neighbourhood]].

Or rather, if one does not at the same time assume that the space is [[Hausdorff topological space]], then one needs to require that these compact neighbourhoods exist in a controlled way, e.g. such that one may find them inside every prescribed open neighbourhood (def. \ref{LocalCompactnessViaCompactNeighbourhoodBase} below) and possibily such that they are topological closures of smaller open neighbourhoods (def. \ref{LocallyCompactSpace} below).

There are various definitions in use, which all coincide if the space is also [[Hausdorff topological space|Hausdorff]] (prop. \ref{InHausdorffSpacesDefinitionsOfLocalCompactnessAgree} below).

A locally compact [[Hausdorff space|Hausdorff]] space may also be called a _local compactum_; compare at _[[compactum]]_.

Local compactness is one of the conditions that are often required by default for working with topological spaces: locally compact spaces are a class of  "[[nice topological spaces]]".

## Definition

+-- {: .num_defn #LocalCompactnessViaCompactNeighbourhoodBase}
###### Definition
**(local compactness via compact neighbourhood base)**

A [[topological space]] is _locally compact_ if every point has a [[neighborhood base]] consisting of [[compact subspaces]]. This means that for every point $x \in X$ every [[open neighbourhood]] $U_x \supset \{x\}$ contains a [[compact topological space|compact]] [[neighbourhood]] $K_x \subset U_x$.

=--

Alternatively:

+-- {: .num_defn #LocallyCompactSpace}
###### Definition
**(local compactness via compact closures inside neighbourhoods)**

A [[topological space]] $X$ is called _locally compact_ if for every point $x \in X$ and every [[open neighbourhood]] $U_x \supset \{x\}$
there exists a smaller open neighbourhood $V_x \subset U_x$ whose [[topological closure]]
is [[compact topological space|compact]] and still contained in $U$:

$$
  \{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x
  \,.
$$

=--

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

If $X$ is a [[Hausdorff topological space]] then definition \ref{LocalCompactnessViaCompactNeighbourhoodBase} is equivalent to definition \ref{LocallyCompactSpace}.

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

Generally definition \ref{LocallyCompactSpace} implies definition \ref{LocalCompactnessViaCompactNeighbourhoodBase}. We need to show that Hausdorffness implies the converse.

Hence assume that for every point $x \in X$ then every open neighbourhood $U_x \supset \{x\}$ contains a compact neighbourhood. We need to show that it then also contains the closure $Cl(V_x)$ of a smaller open neighbourhood and such that this closure is compact.

So let $K_x \subset U_x$ be a compact neighbourhood. Being a neighbourhood, it has a non-trivial [[interior]] which is an open neighbourhood

$$
  \{x\} \subset Int(K_x) \subset K_x \subset U_x \subset X
  \,.
$$

Since [[compact subspaces of Hausdorff spaces are closed]], it follows that $K_x \subset X$ is a closed subset. This implies that the [[topological closure]] of its interior as a subset of $X$ is still contained in $K_x$ (since the topological closure is the smallest closed subset containing the given subset): $Cl(Int(K_x)) \subset K_x$. Since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]], $Cl(Int(K_x))$ is also closed as a subset of the compact subspace $K_x$.
Now since [[closed subsets of compact spaces are compact]], it follows that this closure is also compact as a subspace of $K_x$, and since [[continuous images of compact spaces are compact]], it finally follows that it is also compact as a subspace of $X$:

$$
  \{x\} \subset Int(K_x) \subset \underset{\text{compact}}{Cl(Int(K_x))} \subset \underset{}{K_x} \subset U_x \subset X
  \,.
$$

=--

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

As for [[compact spaces]] ([this remark](compact+space#DifferingTerminology)), some authors choose to include the [[Hausdorff space|Hausdorff]] condition as a matter of course, calling locally compact not-necessarily-Hausdorff spaces 'locally quasi-compact'. We will not follow that convention here, but the reader should be warned that without the Hausdorff hypothesis, there are several inequivalent notions of local compactness in the literature; see [the English Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Locally_compact_space) for a survey and counterexamples.

Note, however, that a topological space $X$ satisfying Definition \ref{LocallyCompactSpace} is regular, because, as is immediate from the definition, closed neighbourhoods then form a neighbourhood basis of $X$, which is equivalent to regularity. Thus we only need that $X$ be $T_{0}$ for it to in fact be Hausdorff.

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## Examples

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

Every [[discrete space]] is locally compact.

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+-- {: .num_example}
###### Example
**([[open subspaces of compact Hausdorff spaces are locally compact]])**

Every [[open subset|open]] [[topological subspace]] $X \underset{\text{open}}{\subset} K$ of a [[compact topological space|compact]]
[[Hausdorff space]] $K$ is a [[locally compact topological space]].

In particular every [[compact Hausdorff space]] itself is [[locally compact topological space|locally compact]].

Conversely, every locally compact Hausdorff space $X$ arises in this way, since it can be considered an open subspace in its [[one-point compactification]] $X \sqcup \{\infty\}$. See there _[this example](one-point+compactification#LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace)_.

=--

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

The [[real numbers]], [[complex numbers]], and $\mathfrak{p}$-[[adic completions]] of [[algebraic number fields]] (with respect to a [[prime ideal]] $\mathfrak{p}$ in the [[ring of integers]]) are locally compact. In [[positive characteristic]] $p$, the field of [[Laurent series]] $\mathbb{F}_q((t))$ over a [[finite field]] with $q$ elements, topologized with respect to a discrete valuation, is locally compact. In fact, any non-discrete locally compact [[field]] must be of one of these types; they are called [[local fields]].

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

Finite [[product topological spaces]] of locally compact spaces are locally compact.

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

[[closed subspace|Closed subspaces]] of locally compact spaces are locally compact. (Hence locally compact spaces form a [[finitely complete category]].)

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+-- {: .num_example #TopologicalManifoldsAreLocallyCompact}
###### Example
**([[topological manifolds]] are [[locally compact topological spaces]])**

[[topological manifolds|Topological manifolds]], being locally [[homeomorphism|homeomorphic]] to the [[Euclidean space|Euclidean]] [[metric spaces]] $\mathbb{R}^n$, are locally compact, via examples \ref{real} and \ref{product}.

This applies also to [[locally Euclidean spaces]] which are not necessarily [[paracompact topological space|paracompact]] [[Hausdorff topological spaces]], such as the [[long line]].

=--

\begin{example}\label{CountablyInfiniteProductsOfNonCompactSpacesAreNotLocallyCompact}
**(countably infinite products of non-compact spaces are NOT locally compact)**
\linebreak
Let $X$ be a [[topological space]] which is not [[compact topological space|compact]]. Then the [[product topological space]] of a [[countable set|countably infinite set]] of copies of $X$

$$
  \underset{n \in \mathbb{N}}{\prod} X
$$

is not locally compact.
\end{example}

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

Since the [[continuous image of a compact space is compact]], and since the [[projection]] maps $p_i \;\colon\; \underset{\mathbb{N}}{\prod} X \longrightarrow X$ are continuous, it follows that every compact subspace of the product space is contained in one of the form

$$
  \underset{i \in \mathbb{N}}{\prod} K_i
$$

for $K_i \subset X$ compact.

But by the nature of the [[Tychonoff topology]], a [[base for a topology|base for the topology]] on $\underset{\mathbb{N}}{\prod} X$ is given by subsets of the form

$$
  \left(
    \underset{i \in \{1,\cdots,n\}}{\prod}
    U_{i}
  \right)
  \times
  \left(
    \underset{j \in \mathbb{N}_{\gt n}}{\prod}
    X
  \right)
$$

with $U_i \subset X$ open. Hence every compact neighbourhood in $\underset{\mathbb{N}}{\prod} X$ contains a subset of this kind, but if $X$ itself is non-compact, then none of these is contained in a product of compact subsets.

=--

\begin{example}\label{RationalsAreNotLocallyCompact}
**(non-example)**
The space of [[rational numbers]] as a subspace of the [[real numbers]] with the Euclidean topology is not locally compact since its compact subsets all have empty interior.
\end{example}

## Properties

### General

+-- {: .num_prop}
###### Proposition
**([[proper maps to locally compact spaces are closed]])**

Let

1. $(X,\tau_X)$ be a [[topological space]],

1. $(Y,\tau_Y)$ a [[locally compact topological space]] according to def. \ref{LocallyCompactSpace},

1. $f \colon X \to Y$ a [[continuous function]].

Then:

If $f$ is a [[proper map]], then it is a [[closed map]].

=--

* [[locally compact and sigma-compact spaces are paracompact]]

* [[locally compact and second-countable spaces are sigma-compact]]

### Category-theoretic properties

Perhaps the most important consequence of local compactness (as defined above) for categorical topology is that locally compact spaces are [[exponential object|exponentiable]], i.e., if $Y$ is locally compact, then $Y \times -: Top \to Top$ has a [[adjunction|right adjoint]] $(-)^Y: Top \to Top$. In fact, this is almost an abstract definition of local compactness: for [[sober spaces]], local compactness is equivalent to being exponentiable. Cf. the situation for [[locales]]: a result of Hyland is that locale is [[locally compact locale|locally compact]] if and only if it is exponentiable.  (See [[exponential law for spaces]] and [[compact-open topology]] for more details.)

As noted above, locally compact spaces form a finitely complete [[full subcategory]] of $Top$. It is not true that arbitrary products of locally compact spaces are locally compact. However, some important examples of locally compact spaces are constructed as restricted direct products, as follows.

Let $(X_p, K_p)_{p \in P}$ be a collection of pairs of spaces where each $X_p$ is locally compact and $K_p \subseteq X_p$ is a compact _open_ subspace. The **restricted direct product** of the collection is the colimit of the [[filtered category|filtered diagram]] consisting of spaces
$$D_F = \prod_{p \in F} X_p \times \prod_{p \notin F} K_p$$
where $F$ ranges over all finite subsets of $P$, together with inclusions $D_F \subseteq D_{F'}$ where $F \subseteq F'$. We observe that each of the $D_F$ is locally compact, and that a filtered colimit or union of a system of _open_ inclusions of locally compact spaces is again locally compact. Therefore, restricted direct products are locally compact, under the hypotheses stated above.

These hypotheses are of course pretty severe; important examples of such restricted direct products include topologized [[adele ring]]s and [[idele group]]s. In the case of adele rings, the collection of pairs is $(K_{\mathfrak{p}}, O_{\mathfrak{p}})$ where $K_{\mathfrak{p}}$ is the $\mathfrak{p}$-adic completion of a [[number field]] $K$ and $O_{\mathfrak{p}}$ is the $\mathfrak{p}$-adic completion of the ring of integers $O \subseteq K$.

In any event, the category of locally compact spaces does not admit general infinite products. If it did, then so would the category of locally compact Hausdorff spaces, and so would the category of locally compact Hausdorff abelian groups. However, there is no product of countably many copies of the real numbers in $LCHAb$, for if there were, then by utilizing the universal property of the product, it would become a Hausdorff TVS over the real numbers, in contradiction to the fact that the only locally compact Hausdorff TVS are finite-dimensional.

Locally compact spaces _are_ closed under [[coproduct]]s in $Top$. They do not admit many types of [[colimit]]s generally; in some sense this is a _raison d\'être_ for [[compactly generated topological spaces]]: they are precisely the colimits in $Top$ of diagrams of locally compact spaces (see also Prop. \ref{HausdorffkSpacesAreQuotientsOfLCHausSpaces} below).

### Relation to compactly generated spaces
 {#RelationToCompactlyGeneratedSpaces}

\begin{prop}\label{LocallyCompactHausdorffSpacesAreCompactlyGeneratedWeaklyHausdorff}
  Every [[locally compact Hausdorff space]] is
  [[compactly generated topological space|compactly generated]]
  and [[weakly Hausdorff topological space|weakly Hausdorff]].
\end{prop}
(e.g. [Dugundji 1966, XI Thm. 9.3](#Dugundji66); [Strickland 09, Prop. 1.7](compactly+generated+topological+space#Strickland09))

\begin{prop}
  The [[product topological space]] of a [[locally compact Hausdorff space]] with a [[k-space]] is already a k-space (i.e. without need of [[k-ification]]).
\end{prop}
(e.g. [Piccinini 1992, Thm. B.6](#Piccinini92), [Strickland 2009, Prop. 2.6](compactly+generated+topological+space#Strickland09))

\begin{proposition}\label{kSpacesAreTheQuotientSpacesOfLocallyCompactHausdorffSpaces}
**([[k-spaces]] are the [[quotient topological space|quotient spaces]] of [[locally compact Hausdorff spaces]])**
\linebreak
  A [[topological space]] is a [[k-space]]
iff it is a [[quotient topological space]] of a [[locally compact Hausdorff space]].
\end{proposition}
This is proven in [Dugundji 1966, XI Thm. 9.4](#Dugundji66) (also [Piccinini 92, Thm. B.4](#Piccinini92)) assuming Hausdorffness, and without that assumption in [Escardo, Lawson & Simpson 2004, Cor. 3.4 (iii)](#EscardoLawsonSimpson04). Moreover:
\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]]
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))

### Gelfand duality

Under [[Gelfand duality]] the category of compact Hausdorff topological spaces is equivalent to the [[opposite category]] of commutative [[C-star algebras]]. With some care there are generalizations of this also to locally compact topological spaces. See at _[[Gelfand duality]]_ for more.

### Further properties

\begin{proposition}
Locally compact Hausdorff spaces are [[paracompact space|paracompact]] whenever they are also [[second-countable space|second-countable]].
\end{proposition}

\begin{example}\label{LocallyCompactHausdorffSpacesAreCompletelyRegular}
  Locally compact Hausdorff spaces are [[completely regular topological spaces]].
\end{example}
(e.g. [Dugundji 1966, XI Thm. 6.4](#Dugundji66); [Engelking 1989, Thm. 3.3.1](#Engelking89))
\begin{remark}
  Example \ref{LocallyCompactHausdorffSpacesAreCompletelyRegular} plays a key role in discussion of [[slice theorem|slice theorems]], see there for more.
\end{remark}

## Related concepts

* [[locally proper map]]

* [[locally compact locale]]

* [[compact topological space]], [[countably compact topological space]], [[paracompact topological space]], [[locally compact topological space]], [[strongly compact topological space]]

* [[Gelfand duality]]

* [[vanishing at infinity]]

## References

Textbook accounts:

* {#Dugundji66} [[James Dugundji]], Section XI.6 of: *Topology*, Allyn and Bacon 1966 ([pdf](https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf))

* {#Engelking89} [[Ryszard Engelking]], *General Topology*, Sigma series in pure mathematics **6**, Heldermann 1989 ([ ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm))

* {#Piccinini92} [[Renzo A. Piccinini]], *Lectures on Homotopy Theory*, Mathematics Studies **171**, North Holland 1992 ([ISBN:978-0-444-89238-6](https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C))

Further discussion in 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 locally compact topological space]]
[[!redirects locally compact topological spaces]]

[[!redirects locally compact Hausdorff topological space]]
[[!redirects locally compact Hausdorff topological spaces]]
[[!redirects locally compact Hausdorff]]

[[!redirects locally compact space]]
[[!redirects locally compact spaces]]
[[!redirects locally compact]]

[[!redirects locally compact Hausdorff space]]
[[!redirects locally compact Hausdorff spaces]]
[[!redirects local compactness]]
[[!redirects local compactum]]
[[!redirects local compactums]]
[[!redirects local compacta]]
[[!redirects locally compactum]]
[[!redirects locally compactums]]
[[!redirects locally compacta]]


locally confluent

Let $(X,\rightarrow)$ be an abstract rewriting system ie. a set equipped with a binary relation $\rightarrow$. Write $\rightarrow^{*}$ for the [[reflexive-transitive closure]] of $\rightarrow$. We say that $\rightarrow$ is locally confluent iff for every $a,b,c \in X$ such that $a \rightarrow b$ and $a \rightarrow c$, there exists $d \in X$ such that $b \rightarrow^{*} d$ and $c \rightarrow^{*} d$.

[[!redirects local confluence]]


locally connected geometric morphism

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
+ automatic table of contents goes here
{:toc}

## Idea

A [[geometric morphism]] is *locally connected* if it behaves as though its [[fiber]]s are [[locally connected space]]s.  In particular, a [[Grothendieck topos]] $E$ is [[locally connected topos|locally connected]] iff the unique [[geometric morphism]] to [[Set]] (the terminal Grothendieck topos, i.e. the [[point]] in the [[category]] [[Topos]] of toposes) is locally connected.

## Definition
 {#Definition}

A [[geometric morphism]] $ (f^* \dashv f_*) : F \underoverset{f_*}{f^*}{\leftrightarrows} E$ is **locally connected** if it satisfies the following equivalent conditions:

1. It is [[essential geometric morphism|essential]], i.e. $f^*$ has a [[left adjoint]] $f_!$, and moreover $f_!$ can be made into an $E$-[[indexed functor]].

1. For every $A\in E$, the functor $(f/A)^* \colon E/A \to F/f^*A$ is [[cartesian closed functor|cartesian closed]].

1. $f^*$ commutes with [[dependent products]] -- For any morphism $h\colon A\to B$ in $E$, the canonically defined [[natural transformation]] $(f/B)^* \circ \Pi_h \to \Pi_{f^*h} \circ (f/A)^*$ is an [[isomorphism]].

## Properties

### Relation to connectedness

If $f$ is locally connected, then it makes sense to think of the left adjoint $f_!$ as assigning to an object of $F$ its "set of connected components" in $E$.  In particular, if $f$ is locally connected, then it is moreover [[connected geometric morphism|connected]] if and only if $f_!$ preserves the [[terminal object]].  However, not every connected geometric morphism is locally connected. {#connect}

### Over $Set$

Over the [[base topos]] $E = $ [[Set]] every [[connected topos]] which is [[essential geometric morphism|essential]] is automatically locally connected.

This is because the required [[Frobenius reciprocity]] condition

$$
  f_!(A \times f^* (B)) \simeq f_!(A) \times B
$$

is automatically satisfied, using that [[cartesian product]] with a [[set]] is equivalently a [[coproduct]]

$$
  A \times B = \coprod_{a \in A} B
  \,,
$$

that the [[left adjoint]] $f_!$ preserves coproducts, and that for $f^*$ [[full and faithful]] we have $f_! f^* \simeq Id$.

### Strong adjunctions
 {#StrongAdjunctions}

The pair of [[adjoint functors]] $(f_! \dashv f^*)$ in a locally connected geometric morphisms forms a "strong adjunction" in that it holds also for the [[internal homs]] in the sense that there is a [[natural isomorphism]]

$$
  [f_!(X), A] \simeq f_* [X, f^* A]
$$

for all $X, A$. This follows by duality from the [[Frobenius reciprocity]] that characterizes $f^*$ as being a [[cartesian closed functor]]:

by the [[Yoneda lemma]], the morphism in question is an [[isomorphism]]
if for all objects $A,B, X$ the morphism

$$
  Hom(X, [f_!(A), B])
   \stackrel{}{\to}
  Hom(X,f_*[A,f^*(B)])
$$

is a bijection. By [[adjunction]] this is the same as

$$
  Hom(X \times f_!(A), B) \stackrel{\simeq}{\to}
  Hom(f_!(f^*(X) \times A), B)
  \,.
$$

Again by Yoneda, this is a bijection precisely if

$$
  f_!(f^*(X) \times A) \to X \times f_!(A)
$$

is an [[isomorphism]]. But this is the [[Frobenius reciprocity]] condition on $f^*$.

### Coreflectivity

Locally connected toposes are [[coreflective subcategory|coreflective]] in [[Topos]]. See ([Funk (1999)](#Funk)).

### Other characterizations

* Let $(\mathcal{C}, J)$ be a [[site]] and $S$ be a [[sieve]] on the object $U$. $S$ is called _connected_ when $S$ viewed as a full subcategory of $\mathcal{C}/U$ is connected. The **site** is called _locally connected_ if every sieve is connected. For a [[bounded geometric morphism]] $p:\mathcal{E}\to\mathcal{S}$ the following holds:
_$p$ is locally connected iff there exists a locally connected internal site in $\mathcal{S}$ such that $\mathcal{E}\simeq Sh(\mathcal{C},J)$._ (cf. [Johnstone (2002)](#elephant), pp.656-658)

* [Caramello (2012)](#Cara12) gives syntactic characterizations of [[geometric theories]] whose [[classifying topos]] is locally connected.

The same paper also contains the following characterization:

* A Grothendieck topos is locally connected iff it has a [[separator|separating set]] of (coproduct) indecomposable objects.

### Variations in the context of the Nullstellensatz

[Johnstone (2011)](#JS11) studies several subclasses of locally connected geometric morphisms in the context of [[William Lawvere|Lawvere]]'s theory of [[cohesion]] and the [[Nullstellensatz]]. He calls a locally connected morphism $p$ _stably locally connected_ if $p_!$ preserves finite products. According to the [above remark](#connect) this implies that $p$ is connected. Slightly stronger is the preservation of all finite limits by $p_!$: these $p$ are called [[totally connected geometric morphisms]].

## Examples

* If the terminal [[global section]] geometric morphism $E \to Set$ is locally connected, one calls $E$ a [[locally connected topos]].  More generally, if $E\to S$ is locally connected, we may call $E$ a *locally connected $S$-topos*.

* Let $X$ be a [[topological space]] (or a [[locale]]) and $U\subseteq X$ an [[open subset]], with corresponding [[geometric embedding]] $j\colon Sh(U)\to Sh(X)$.  Then any $A\in Sh(X)$ can be identified with a space (or locale) $A$ equipped with a [[local homeomorphism]] $A\to X$, in such a way that $Sh(X)/A \simeq Sh(A)$.  Moreover, $j^*A \in Sh(U)$ can be identified with the pullback of $A\to X$ along $U$, and so $Sh(U)/j^*A \simeq Sh(j^*A)$ similarly.  Noting that $j^*A \to A$ is again the inclusion of an open subset, and using the fact that the inverse image part of any open [[geometric embedding]] is cartesian closed, we see that $(j/A)^*\colon Sh(X)/A \to Sh(U)/j^*A$ is cartesian closed for any $A$.  Hence $j$ is locally connected.

## Related entries

* [[connected geometric morphism]]

* [[local geometric morphism]]

* [[totally connected geometric morphism]]

## References

The case of $Sh(X)$ for a topological space $X$ was an exercise (p.417) in [[SGA4]]:

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, Springer LNM vol.269 (1972).

The concept relative to other bases was introduced in the following paper:

* [[Michael Barr]], [[Robert Paré]], _Molecular Toposes_ , JPAA **17** (1980) pp.127-152.

The standard reference is section C3.3 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. {#elephant}

Further references include

* [[Olivia Caramello]], _Syntactic Characterizations of Properties of Classifying Toposes_ , TAC **26** no.6 (2012) pp.176-193. ([pdf](http://www.tac.mta.ca/tac/volumes/26/6/26-06.pdf))
  {#Cara12}

* [[Jonathon Funk]], _The locally connected coclosure of a Grothendieck topos_, JPAA **137** (1999) pp.17-27.
  {#Funk}

* {#JS11} [[Peter Johnstone]], _Remarks on Punctual Local Connectedness_ , TAC **25** no.3 (2011) pp.51-63. ([pdf](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

* [[Ieke Moerdijk]], _Continuous fibrations and inverse limits of toposes_ , Comp. Math. **58** (1986) pp.45-72. ([pdf](http://archive.numdam.org/article/CM_1986__58_1_45_0.pdf))

* [[Ieke Moerdijk]], [[Gavin Wraith]], _Connected and locally connected toposes are path connected_ , Trans. AMS **295** (1986) pp.849-859. ([pdf](http://www.ams.org/journals/tran/1986-295-02/S0002-9947-1986-0833712-3/S0002-9947-1986-0833712-3.pdf))

[[!redirects locally connected geometric morphisms]]


locally connected locale

[[!redirects locally connected locales]]
[[!redirects locally connected locales]]
## Definition

A [[locale]] $L$ is __connected__ if $1\ne0$ and $1=A\cup B$
for disjoint opens $A$ and $B$ implies $A=0$ or $B=0$.

A [[locale]] $L$ is __locally connected__ if its connected opens
form a [[base]] (any open is a [[supremum]] of connected opens).

## Properties

For a locally connected locale one can define its set of connected components.

See [[cosheaf of connected components]] for a parametrized version of this construction.

## Related notions

* [[locally connected topos]]
* [[locally connected (∞,1)-topos]]

[[!redirects locally connected locales]]


locally connected site

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#Contents#
* table of contents
{:toc}

## Idea

A *locally connected site* is a [[site]] satisfying sufficient conditions to make its [[topos of sheaves]] into a [[locally connected topos]].

## Definition

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

Let $C$ be a small [[site]]; we say it is a **locally connected site** if all [[covering]] [[sieves]] of any object $U\in C$ are [[connected category|connected]], as [[full subcategories]] of the [[slice category]] $C_{/U}$.

(In particular, this means that all [[covering families]] are [[inhabited set|inhabited]].)

=--

## Locally connected toposes from sites
 {#LocallyConnectedToposesFromLocallyConnectedSites}

We discuss that the [[sheaf toposes]] over locally connected sites are [[locally connected toposes]].

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

If $C$ is locally connected, then every [[constant presheaf]] on $C$ is a [[sheaf]].

=--

The fact that all [[covering families]] are [[inhabited set|inhabited]] makes the constant presheaves be [[separated presheaves]] (see [this example](separated+presheaf#ConstantPresheafOnSiteWithAllInhabitedCoverings)) and then the connectedness condition further makes them be sheaves.

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

If $C$ is a locally connected site, then the [[sheaf topos]] $Sh(C)$ is a [[locally connected topos]].

=--

This means that the [[inverse image functor]] $L Const\colon Set \to Sh(C)$ has a [[left adjoint]] $\Pi_0$.

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

By remark \ref{ConstantPresheavesAreSheaves} it follows that the constant *[[presheaf]]* functor $Const \colon Set \to Psh(C)$ has a [[left adjoint]] given by taking [[colimits]] along $C^{op}$ (this is one of the equivalent _definitions_ of the colimit operation.) Since constant presheaves on $C$ are sheaves, $L Const$ is just a factorization of $Const$ through $Sh(C)$, and thus it also has a left adjoint given by the colimit operation.

=--

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

The colimit over a [[representable functor]] is always the [[singleton set]].

So for $X \in Sh(C)$ any sheaf, we may write it, using the [[co-Yoneda lemma]] as a [[coend]] over representables

$$
  X = \int^{U \in C} X(U) \cdot U
  \,.
$$

The left adjoint functor $\Pi_0$ commutes with the coend and the [[tensoring]] in the integrand to produce

$$
  \Pi_0(X) = \int^{U \in C} X(U) \cdot {*} = colim_{U \to X} {*}
  \,.
$$

We may think of this as computing the set of _plot-connected components_ of $X$.

=--

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

If $C$ furthermore has a [[terminal object]], then colimits along $C^{op}$ preserve the [[terminal object]], so that $Sh(C)$ is moreover a [[connected topos]].

=--

Note that a non-locally-connected site can still give rise to a locally connected topos of sheaves, but every locally connected topos can be defined by *some* locally connected site.

## Examples

* any small subcategory of [[Top]] on [[connected]] topological spaces (with the standard [[open cover]] [[coverage]]).

* [[CartSp]]

* Any site whose topology is generated by a [[singleton pretopology]], i.e. a [[Grothendieck pretopology]] in which all covering families consist of single arrows.  For if a covering sieve on $U$ is generated by a single arrow $p:V\to U$, then $p$ is a [[weakly terminal object]] of the sieve (*qua* full subcategory of $C/U$), so the sieve is connected.

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* **locally connected site** / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]] / [[∞-cohesive site]]

[[!redirects locally connected sites]]


locally connected topological space

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

#Contents#
* table of contents
{:toc}

## Idea

It is not generally true that a [[topological space]] is the [[disjoint union space]] ([[coproduct]] in [[Top]]) of its [[connected components]]. The spaces such that this is true for all open subspaces are the _locally connected topological spaces_.

## Definition
  {#Definition}

+-- {: .num_defn #LocallyConnected}
###### Definition
**(locally connected topological space)**

A [[topological space]] $X$ is **locally connected** if  every point has a [[neighborhood basis]] of [[connected topological space|connected]] [[open subsets]].

=--

+-- {: .num_prop #AlternativeCharacterizationsOfLocalConnectivity}
###### Proposition
**(alternative characterizations of local connectivity)**

For $X$ a [[topological space]], then the following are equivalent:

1. $X$ is locally connected (def. \ref{LocallyConnected});

1. every [[connected component]] of every open [[subspace]] of $X$ is open;

1. every [[open subset]], as a [[topological subspace]], is the [[disjoint union space]] ([[coproduct]] in [[Top]]) of its [[connected components]].

In particular, in a locally connected space, every connected component $S$ is a [[clopen subset]]; hence connected components and quasi-components coincide.

=--

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

$\,$

1) $\Rightarrow$ 2)

Assume $X$ is locally connected, and let $U \subset X$ be an open subset with $U_0 \subset U$ a connected component. We need to show that $U_0$ is open.

Consider any point $x \in U_0$. Since then also $x \in U$, the defintion of local connectedness, def. \ref{LocallyConnected}, implies that there is a connected open neighbourhood $U_{x,0}$ of $X$. Observe that this must be contained in $U_0$, for if it were not then $U_0 \cup U_{x,0}$ were a larger open connected open neighbourhood, contradicting the maximality of the connected component $U_0$.

Hence $U_0 = \underset{x \in U_0}{\cup} U_{x,0}$ is a union of open  subsets, and hence itself open.

2) $\Rightarrow$ 3)

Now assume that every connected component of every open subset $U$ is open. Since the connected components generally consitute a [[cover]] of $X$ by [[disjoint subsets]] this means that now they for an [[open cover]] by disjoint subsets. But by forming intersections with the cover this implies that every open subset of $U$ is the disjoint union of open subsets of the connected components (and of course every union of open subsets of the connected components is still open in $U$), which is the definition of the topology on the [[disjoint union space]] of the connected components.

3) $\Rightarrow$ 1)

Finally assume that every open subspace is the disjoint union of its connected components. Let $x$ be a point and $U_x \supset \{x\}$ a neighbourhood. We need to show that $U_x$ contains a connected neighbourhood of $x$.

But, by definition, $U_x$ contains an open neighbourhood of $x$ and by assumption this decomposes as the disjoint union of its connected components. One of these contains $x$. Since in a [[disjoint union space]] all summands are open, this is the required connected open neighbourhod.

=--

## Examples

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

Every [[discrete topological space]] is locally connected.

=--

+-- {: .num_example #LocallyConnectedEuclideanSpace}
###### Example
**([[Euclidean space]] is locally connected)

For $n \in \mathbb{N}$ the [[Euclidean space]] $\mathbb{R}^n$ (with its [[metric topology]]) is locally connected.

=--

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

By nature of the Euclidean metric topology, every neighbourhood $U_x$ of a point $x$ contains an [[open ball]] containing $x$. Moreover, every open ball clearly contains an open cube, hence a [[product space]] $\underset{i \in \{1, \cdots, n\}}{\prod} (x_i-\epsilon, x_i + \epsilon)$ of [[open intervals]] which is still a neighbourhood of $x$.

Now intervals are connected (by [this example](connected+space#ConnectedSubspacesOfRealLineAreTheIntervals)) and products of connected spaces are connected (by [this example](connected+space#ProductSpaceOfConnectedSpacesIsConnected)). This shows that ever open neighbourhood contains a connected neighbourhood.

=--

+-- {: .num_prop #LocallyConnectedOpenSubspaceOfLocallyConnectedSpace}
###### Proposition
**([[open subset|open]] [[subspace]] of locally connected space is locally connected)**

Every [[open subset|open]] [[subspace]] of a locally connected space is itself locally connected

=--

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

This is immediate from def. \ref{LocallyConnected}.

=--

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

A [[connected topological space]] need not be locally connected.

=--

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

The [[topologist's sine curve]] is connected but not locally connected.

=--

Examples of locally connected spaces include [[manifold|topological manifolds]].

Finally,

+-- {: .num_defn}
###### Definition
A space $X$ is **[[totally disconnected topological space]]** if its [[connected components]] are precisely the [[singletons]] of $X$.

=--

In other words, a space is totally disconnected if its coreflection into $LocConn$ is discrete. Such spaces recur in the study of [[Stone spaces]].

The category of totally disconnected spaces is a reflective subcategory of $Top$. The reflector sends a space $X$ to the space $X/\sim$ whose points are the connected components of $X$, endowed with the quotient topology induced by the projection $q: X \to X/\sim$. Details may be found at [[totally disconnected space]].

## Properties

### The category of locally connected spaces

Let $i \colon LocConn \hookrightarrow Top$ be the [[full subcategory]] inclusion of locally connected spaces into all of [[Top]]. The following result is straightforward but useful.

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

$LocConn$ is a [[coreflective subcategory]] of $Top$, i.e., the inclusion $i$ has a [[right adjoint]] $R$. For $X$ a given space, $R(X)$ has the same underlying set as $X$ and the coarsest locally connected topology that is finer than the original topology on $X$.

=--

Being a coreflective category of a complete and cocomplete category, the category $LocConn$ is also complete and cocomplete. Of course, limits and particularly _infinite_ products in $LocConn$ are not calculated as they are in $Top$; rather one takes the limit in $Top$ and _then_ retopologizes it according to Theorem \ref{coref}. (For _finite_ products of locally connected spaces, we can just take the product in $Top$ -- the result will be again locally connected.)

### Cohesion over sets
 {#CohesionOverSets}

Let $\Gamma \colon LocConn \to Set$ be the underlying set functor, and let $\nabla, \Delta \colon Set \to LocConn$ be the [[functors]] which assign to a set the same set equipped with the [[codiscrete topology|codiscrete]] and [[discrete topologies]], respectively. Let $\Pi_0 \colon LocConn \to Set$ be the functor which assigns to a locally connected space the set of its [[connected components]].

+-- {: .num_theorem}
###### Theorem
There is an [[adjoint quadruple]] of [[adjoint functors]]

$$
  \Pi_0 \dashv \Delta \dashv \Gamma \dashv \nabla \colon Set \to LocConn
$$

and moreover, the functor $\Pi_0$ preserves [[finite products]].

=--

While $LocConn$ is not a [[topos]], this is the adjunction structure as on a [[cohesive topos]].

\begin{proof}
The proof is largely straightforward; we point out that the continuity of the unit $X \to \Delta \Pi_0 X$ is immediate from a locally connected space's being the coproduct of its connected components. As for $\Pi_0$ preserving finite products, write locally connected spaces $X$, $Y$ as coproducts of connected spaces

$$X = \sum_i C_i; \qquad Y = \sum_j D_j;$$

then their product in $LocConn$ coincides with their product in $Top$, and is

$$X \times Y \cong \sum_{i, j} C_i \times D_j$$

where each summand $C_i \times D_j$ is connected by Result \ref{3}. From this it is immediate that $\Pi_0$ preserves finite products.
\end{proof}

Accordingly the [[category of sheaves]] on a locally connected space is a [[locally connected topos]]. For related discussions, see also [[cohesive topos]].

### Quotients of locally connected spaces

+-- {: .num_lemma #quot}
###### Lemma

A [[quotient space]] of a locally connected space $X$ is also locally connected.

=--

+-- {: .proof}
###### Proof
Suppose $q: X \to Y$ is a quotient map, and let $V \subseteq Y$ be an open neighborhood of $y \in Y$. Let $C(y)$ be the connected component of $y$ in $V$; we must show $C(y)$ is open in $Y$. For that it suffices that $C = q^{-1}(C(y))$ be open in $X$, or that each $x \in C$ is an interior point. Since $X$ is locally connected, the connected component $U_x$ of $x$ in $q^{-1}(V)$ is open, and the subset $q(U_x) \subseteq V$ is connected, and therefore $q(U_x) \subseteq C(y)$ (as $C(y)$ is the maximal connected subset of $V$ containing $q(x)$). Hence $U_x \subseteq q^{-1}(C(y)) = C$, proving that $x$ is interior to $C$, as desired.
=--

The conclusion does not follow if $q: X \to Y$ is merely surjective; e.g., there is a surjective (continuous) map from $\mathbb{R}$ to (a version of) the [[Warsaw circle]], but the latter is not locally connected.

## Related concepts

* [[totally disconnected topological space]]

* [[locally connected topos]]

[[!redirects locally connected space]]
[[!redirects locally connected spaces]]
[[!redirects locally connected topological space]]
[[!redirects locally connected topological spaces]]


locally connected topos

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

#Contents#
* table of contents
{:toc}

## Idea

A [[topos]] may be thought of as a generalized [[topological space]]. Accordingly, the notions of

* [[locally connected space]]

* locally 2-[[connected]] space

* etc. ...

* [[locally contractible space]]

have analogs for [[topos]]es and [[(∞,1)-topos]]es

* **locally connected topos**

* [[locally n-connected (n,1)-topos]]

* etc. ...

## Definition

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

An [[object]] $A$ in a [[topos]] $\mathcal{E}$ is called a **[[connected object]]** if the [[hom-functor]] $\mathcal{E}\big(A, -\big)$ [[preserved limit|preserves]] [[finite coproduct|finite]] [[coproducts]].

Equivalently, an object $A$ is connected if it is non-empty (in that it is not [[generalized the|the]] [[initial object]]) and cannot be expressed as a [[coproduct]] of two nonempty [[subobjects]].

=--

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

A [[Grothendieck topos]] $\mathcal{E}$ is called a **locally connected topos** if every object  is a [[coproduct]] of [[connected objects]] (Def. \ref{ConnectedObject}),
hence if for $A \in \mathcal{E}$ there exists $\big\{A_i \in \mathcal{E}\big\}_{i \in I}$ such that
$$
  A
  \;\simeq\;
  \coprod_{i \in I} A_i
  \,.
$$

=--

If this is the case, it  follows that the [[indexed set|index]] set $I$ is unique up to [[isomorphism]], and we denote it by

$$
  \pi_0(A) \;\coloneqq\; I
  \,.
$$

This construction defines a [[functor]]

$$
  \array{
    \Pi_0 \colon & \mathcal{E} &\longrightarrow& Set
    \\
    & A &\mapsto& \pi_0(A)
  }
$$

which is [[left adjoint]] to the [[locally constant sheaf]] functor, the [[left adjoint]] part of the [[global section]] [[geometric morphism]].

This is the **connected component functor**. It generalizes the functor, also denoted $\pi_0$ or $\Pi_0$, which to a [[topological space]] assigns the set of [[connected components]] of that space. See the [examples](#Examples) below.

In summary, for a locally connected topos the [[terminal geometric morphism]] extends to an [[adjoint triple]] of this form:

\begin{tikzcd}[column sep=24pt]
  \mathcal{E}
  \ar[
    rr,
    shift left=16pt,
    "{ \Pi_0 }"
  ]
  \ar[
    from=rr,
    "{ \mathrm{LConst} }"{description}
  ]
  \ar[
    rr,
    shift right=16pt,
    "{ \Gamma }"{swap}
  ]
  \ar[rr, phantom, shift left=9pt, "{\scalebox{.6}{$\bot$}}"]
  \ar[rr, phantom, shift right=9pt, "{\scalebox{.6}{$\bot$}}"]
  &&
  \mathrm{Set}
\end{tikzcd}

The following proposition asserts that the existence of $\Pi_0$ already characterizes locally connected toposes.

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

A [[Grothendieck topos]] $\mathcal{E}$ is locally connected precisely if the [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E} \to Set$ is an [[essential geometric morphism]] $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$.

=--

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

The "only if"-case was just claimed/argued above, we need to show the "if"-case.

Hence suppose that $\Pi_0$ with $(\Pi_0 \dashv LConst)$ exists. We will show that then every object is a coproduct of connected objects. (A proof also appears as ([Johnstone, Lemma C.3.3.6](#Johnstone)).)

First we claim that an object $A$ is connected in the
[above sense](#ConnectedObject) precisely if $\Pi_0(A) = \ast$.

To see this, observe that
\[
  \label{EmptyShapeImpliesInitialObject}
  \Pi_0(A) \;\simeq\; \varnothing
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  LConst \Pi_0(A) \;\simeq\; \varnothing
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  A \xrightarrow{\eta_{A}} \varnothing
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  A \;\simeq\; \varnothing
  \,,
\]
where we used that $LConst$ is a [[left adjoint]] and that [[left adjoints preserve colimits]] ([hence](initial+object#InitialObjectIsEmptyColimit) [[preserved colimit|preserve]] [[initial objects]]), we consideted the [[adjunction unit]] $\eta_A$, and
where the last implication follows since the [[initial object]] in any topos is [[strict initial object|strict]].

But this gives
$$
  \Pi_0(A) \simeq \ast
  \;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;
  A \;\text{is connected}
  \,,
$$
because if $A$ as on the left were the coproduct of non-initial $A_1$, $A_2$, then also $\Pi_0(A)$ would be the coproduct of non-initial $\Pi_0(A_1)$, $\Pi_0(A_2)$, by (eq:EmptyShapeImpliesInitialObject), which would contradict the assumption that $\Pi_0(A) \simeq \ast$.

Conversely, to see
$$
  A \;\text{is connected}
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  \Pi_0(A) \simeq \ast
$$
observe that the connectivity assumption implies in particular that
\[
  \label{ASequenceOfIsos}
  Set
  \big(
    \Pi_0(A)
    ,\,
    S
  \big)
  \;\simeq\;
  \mathcal{E}
  \left(
     A,\,
     LConst
     \coprod_S
     \ast
  \right)
  \;\simeq\;
  \mathcal{E}
  \left(
    A
    ,\,
   \coprod_S \,  LConst \ast
  \right)
  \;\simeq\;
  {\coprod}_S
  \,
  \mathcal{E}
  \big(
    A
    ,\,
    \ast
  \big)
  \;\simeq\;
  \coprod_S \ast
  \;\simeq\;
  S
  \,,
\]
for all $S \in Set$,
where in the first step we used the $\big(\Pi_0 \dashv LConst\big)$-[hom isomorphism](adjoint+functor#InTermsOfHomIsomorphism). But this says that $\Pi_0(A)$ is connected as an object of [[Sets]], hence that it is [[generalized the|the]] [[singleton set]].

With this equivalence
$$
  A \; \text{is connected}
  \;\;\;\;
  \Leftrightarrow
  \;\;\;\;
  \Pi_0(A) \simeq \ast
$$
in hand (given the extra left adjoint $\Pi_0$),
we are now reduced to showing that every object of $\mathcal{E}$ is a coproduct of objects for which $\Pi_0(-)$ is the [[terminal object|point]].

For that purpose consider for $A \in \mathcal{E}$ the following [[pullback]] diagram

$$
  \array{
    i_A^* {\underset{\underset{\Pi_0(A)}{\longrightarrow}}{\lim}} \ast
    &\xrightarrow{\phantom{----}}&
    {\underset{\underset{\Pi_0(A)}{\longrightarrow}}{lim}} \ast
    \\
    {}^{\mathllap{\simeq}}\big\downarrow
    &&
    \big\downarrow{}^{\mathrlap{\simeq}}
    \\
    A
    &\xrightarrow{\phantom{--}i_A\phantom{--}}&
    LConst \Pi_0 (A)
    \,,
  }
$$

where the bottom morphism is the $(\Pi \dashv L Const)$-[[unit of an adjunction|unit]] and the right [[isomorphism]] is the identification of any set as the [[colimit]] (here: [[coproduct]]) of the functor over the set itself that is constant on the point. Since isomorphisms are preserved under pullback ([here](pullback#PullbackPreservesMonomorphisms)), it follows that also the left morphism is an isomorphism, as shown.

Now, by the fact that a [[topos]] has "[[universal colimits]]", this left morphism is equivalently

$$
   \underset{
     \underset{s \in \Pi_0(A)}{\longrightarrow}
   }{\lim}
   (i_A^* \ast_s)
   \xrightarrow{\phantom{--}\sim\phantom{--}}
   A
$$

and hence expresses $A$ as a coproduct of objects $i_A^* *_s$, each of which is a [[pullback]]

$$
  \array{
    i_A^* \ast_s &\xrightarrow{\phantom{----}}& LConst *
    \\
    \big\downarrow && \big\downarrow{}^{\mathrlap{s}}
    \\
    A &\xrightarrow{\phantom{--}i_A\phantom{--}}& LConst \Pi_0 A
    \,,
  }
$$

where the right morphism includes the element $s$ into the set $\Pi_0 A$. By applying $\Pi_0$ to this diagram and [[pasting]] on the $(\Pi_0 \dashv L Const)$-[[counit of an adjunction|counit]] we get

$$
  \array{
    \Pi_0(i_A^* *_s)
    &\xrightarrow{\phantom{-----}}&
    \Pi_0 LConst \ast &\xrightarrow{\phantom{---}}& *
    \\
    \big\downarrow && \big\downarrow^{} && \big\downarrow
    \\
    \Pi_0(A)
    &\xrightarrow{\phantom{--}\Pi_0(i_A)\phantom{--}}&
    \Pi_0 LConst \Pi_0 A
    &\xrightarrow{\phantom{----}}&
    \Pi_0 A
  }
$$

and by the [[zig-zag identity]] the bottom morphism is the [[identity morphism|identity]]. This says that in

$$
  \Pi_0
  \left(
     \underset{\underset{\Pi_0 A}{\longrightarrow}}{\lim}
     \;
     i_A^* \ast_s
     \xrightarrow{\sim}
     A
  \right)
  \simeq
  \left(
     \underset{\underset{\Pi_0 A}{\longrightarrow}}{\lim}
     \;
     \Pi_0\big(i_A^* \ast_s\big)
     \xrightarrow{\sim}
     \Pi_0(A)
  \right)
$$

all the component maps out of the coproduct factor through the point. This means that this can only be an isomorphism if all these component maps are point inclusions, hence if $\Pi_0(i_A^* *_s) \simeq *$ for all $s \in \Pi_0 A$.

=--

\begin{remark}
Beware that Prop. \ref{LocalConnectednessByEssentialGeometricMorphism} only applies to [[terminal geometric morphisms]] (of locally connected toposes). The analogous "[[relative point of view|relative]]" characterization of more general [[locally connected geometric morphisms]] involves more than just the existence of the extra left adjoint functor, see [there](locally+connected+geometric+morphism#Definition).

\end{remark}

## Properties

### Characterization over locally connected sites

See at _[[locally connected site]]_.

### Equivalent conditions

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

For $C$ and $C$ [[cartesian closed categories]], a [[functor]] $F : C \to D$ that preserves [[product]]s is called a **[[cartesian closed functor]]** if the canonical [[natural transformation]]

$$
  F(B^A) \to (F(B))^{F(A)}
$$

(which is the [[adjunct]] of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an [[isomorphism]].

=--

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

The [[constant sheaf]]-functor $\Delta : \mathcal{S} \to \mathcal{E}$ is a [[cartesian closed functor]] precisely if $\mathcal{E}$ is a locally connected topos.

=--

### Locally connected and connected {#Connected}

A [[topos]] $E$ is called a [[connected topos]] if the [[left adjoint]] $L Const : Set \to E$ is a [[full and faithful functor]].

+-- {: .num_prop}
###### Proposition
If $\Gamma \colon E\to Set$ is a locally connected topos, then it is also a [[connected topos]] --- in that $L Const$ is full and faithful --- if and only if the [[left adjoint]] $\Pi_0$ of $L Const$ preserves the
[[terminal object]].
=--

This is ([Johnstone, C3.3.3](#Johnstone)).

Notice that for a connected and locally connected topos, the adjunction

$$
  Set \stackrel{\overset{\Pi_0}{\leftarrow}}{\hookrightarrow} E
$$

exhibits [[Set]] as a [[reflective subcategory]] of $E$. We may think then of [[Set]] as being the [[localization]] of $E$ at those morphisms that induce isomorphisms of connected components.

## Examples {#Examples}

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

For $X$ a [[topological space]], the [[category of sheaves]] $Sh(X) \coloneqq Sh(Op(X))$ is a locally connected topos precisely if $X$ is a [[locally connected space]]. The functor $\Pi_0$ sends a sheaf $F \in Sh(X)$ to the set of connected components of the corresponding [[etale space]].

=--

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

For $C = $ [[CartSp]] the [[site]] of [[Cartesian spaces]] with its [[good open cover]] [[coverage]], the topos $Sh(CartSp)$ of [[smooth spaces]] is locally connected.  An arbitrary $X \in Sh(CartSp)$ is sent to the [[colimit]] $\lim_\to X \in Set$. If $X$ is a [[diffeological space]] or even a [[smooth manifold]], then this is the set of connected components of the underlying topological space.

=--

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

Every locally connected geometric morphism is a [[locally cartesian closed functor]].

=--

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

Suppose that $C$ is a [[site]] such that constant [[presheaves]] on $C$ are [[sheaves]].  Then the left adjoint $\Pi_0$ exists and is given by the [[colimit]] functor: if we write $L : PSh(C) \to Sh(C)$ for [[sheafification]], then for any sheaf $X$, we have

  $$
    Hom_{Sh(C)}(X, L Const S)
    \simeq
    Hom_{PSh(C)}(X, L Const S)
    \simeq
    Hom_{PSh(C)}(X, Const S)
    \simeq
    Hom_{Set}(\lim_\to X, S)
    \,.
  $$

  In particular, this is the case if every covering [[sieve]] in $C$ is connected, i.e. $C$ is a [[locally connected site]].

  If $C$ furthermore has a terminal object $1$, then the global sections functor $\Gamma\colon Sh(C)\to Set$ (the right adjoint of $L Const$) is simply given by evaluation at $1$, and so the unit $S \to \Gamma L Const S \cong L Const S(1)$ is an isomorphism.  Thus in this case $Sh(C)$ is additionally [[connected topos|connected]].  This situation also applies to $C=CartSp$ in example \ref{SmoothSpaces} above.

=--

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

If $C$ is a category with all [[finite limits]] and if the unique functor $\pi \colon C \to \ast$ to the [[terminal category]] is a [[cover-preserving functor]] (for $\ast$ equipped with the trivial topology/[[coverage]]) then $Sh(C)$ is locally connected.
(In particular, this holds for presheaf toposes).
This is because the inclusion of the terminal object $i \colon \ast \to C$ provides a [[right adjoint]] to $\pi$, so that there is an
[[adjoint quadruple]] of functors on [[presheaf categories]]

$$
  (\pi_! \simeq Lan_\pi)
    \dashv
  (\pi^\ast \simeq i_! \simeq Lan_i)
   \dashv
  (\pi_\ast \simeq i^\ast )
    \dashv
  (\pi^! \simeq i_* \simeq Ran_i)
  \;\colon\;
  PSh(C) \leftrightarrow PSh(\ast) \simeq Sh(C) \simeq Set
  \,,
$$

where $Lan_{(-)}$ and $Ran_{(-)}$ denote let and right [[Kan extension]], respectively. Now if $C \to \ast$ indeed preserves covers and using that $C \to \ast$ trivially preserves finite limits and hence is a [[flat functor]], then by the discussion at _[[morphism of sites]]_ the first three functors here descend to [[sheaves]] and hence exhibit $Sh(C)$ as being locally connected.

But **beware** that the assumptions here are stronger than they may seem: that $C \to \ast$ preserves covers is not automatic, but is a strong condition. It is violated as soon as $C$ contains an empty object with [[empty set|empty]] cover, such as is the case in most categories of [[spaces]], notably in [[categories of open subsets]] $Op(X)$ of a [[topological space]] $X$, as in example \ref{LocallyConnectedTopologicalSpace}.

=--

## Related entries

* [[essential geometric morphism]]

* **locally connected topos** / [[locally ∞-connected (∞,1)-topos]]

* [[connected topos]] / [[∞-connected (∞,1)-topos]]

* [[totally connected topos]] / [[totally connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

## References
 {#References}

Section C1.5 and C3.3 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ .
{#Johnstone}

A variant is in

* [[Marta Bunge]], Funk, _Quasi locally connected toposes_ ([pdf](http://www.tac.mta.ca/tac/volumes/18/8/18-08.pdf))

Discussion of characterizations of [[sites]] of definition of locally connected toposes is in

* [[Olivia Caramello]], _Site characterizations for geometric invariants of toposes_, Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728. ([TAC](http://www.tac.mta.ca/tac/volumes/26/25/26-25abs.html))

[[!redirects locally connected toposes]]
[[!redirects locally connected topoi]]


locally constant function

# Locally constant functions
* table of contents
{: toc}

## Idea

A locally constant function is a [[function]] whose value never changes.  This is a weaker concept than that of a [[constant function]], which has only one value.  They correspond on a [[connected space]].  However, in general, a function may be locally constant but not constant, since it can take values on two distant components without the values\' ever changing between them (since there is no path between them).

## Definitions

If $X$ is a [[topological space]] and $Y$ is any [[set]], then a [[function]] $f$ from (the [[underlying set]] of) $X$ to $Y$ is __locally constant__ if, for every element $a$ of $X$, $f$ is [[constant function|constant]] when restricted to some [[neighbourhood]] of $a$.

We have $Y$ here as a set; but in fact, $Y$ may be given any [[topological structure]]; then every locally constant function $f$ will become a locally constant [[continuous map]].

## Examples

+-- {: .num_example #LocallyConstantFunctionIntoDiscreteSpace}
###### Example
**(continuous function into [[discrete space]] is locally constant)**

A function into a [[discrete topological space]] is [[continuous function|continuous]] precisely if it is locally constant.

=--

## Pattern

* A **locally constant function** is a section of a [[constant sheaf]];

* a [[locally constant sheaf]] is a section of a [[constant stack]];

* a [[locally constant stack]] is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

[[!redirects locally constant functions]]


locally constant infinity-stack

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that a [[locally constant sheaf]] (of sets) is a section of the [[constant stack]] with fiber the [[groupoid]] $Core(FinSet)$, the [[core]] of the [[category]] [[FinSet]].

This extends to a general pattern:

a **locally constant $\infty$-stack** is a section of the [[constant ∞-stack]] that is constant on the [[∞-groupoid]] $Core(\infty FinGrpd)$.

## Definition

For $\mathbf{H}$ an [[(∞,1)-sheaf (∞,1)-topos]] there is the [[terminal object|terminal]] [[(∞,1)-geometric morphism]]

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

consisting of the [[global section]] and the [[constant ∞-stack]] [[(∞,1)-functor]].

Write $\mathcal{S} := core(Fin \infty Grpd) \in \infty Grpd$ for the [[core]] [[∞-groupoid]] of the [[(∞,1)-category]] of finite $\infty$-groupoids. (We can drop the finiteness condition by making use of a larger [[universe]].) This is canonically a [[pointed object]] $* \to \mathcal{S}$.

Notice the for $X \in \mathbf{H}$ any [[object]], the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is the [[little topos|little]] $(\infty,1)$-topos of $X$. Objects in here we may regard as $\infty$-stacks on $X$.

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

For $X \in \mathbf{H}$ an [[object]] a **locally constant $\infty$-stack** on $X$ is an morphism $X \to LConst \mathcal{S}$.

The **[[∞-groupoid]] of locally constant $\infty$-stacks** on $X$ is

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

=--

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

An an object of the [[little topos|little]] [[(∞,1)-topos]] of $X$, the
[[over-(∞,1)-topos]] $\mathbf{H}/X$ the locally constant $\infty$-stack given by $\tilde \nabla$ is its [[(∞,1)-Grothendieck construction]] in $\mathbf{H}$

$$
  \array{
    P &\to& *
    \\
    \downarrow &\swArrow& \downarrow
    \\
    X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S}
  }
$$

the pullback of the [[universal fibration of (∞,1)-categories|universal fibration of finite ∞-groupoids]]

$$
  \array{
    P &\to& LConst \mathcal{Z}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S}
  }
  \,.
$$

=--

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

A locally constant $\infty$-stack is also called a [[local system]]. See there for more details.

=--

## Examples {#References}

Here are commented references that establish aspects of the above general abstract situation.

### Locally constant 1-stacks and 2-stacks on topological spaces

A discussion of locally constant 2-stacks over topological spaces is in

* [[Pietro Polesello]] and [[Ingo Waschkies]], _Higher monodromy_ , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 ([pdf](http://www.intlpress.com/HHA/v7/n1/a7/pdf)).

We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above:

The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of [[topological space]]s.

Prop. 1.1.9 gives the [[adjunction]]

$$
  (LConst \dashv \Gamma) : Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Grpd
$$

between forming constant stacks and taking global sections.

Then prop 1.2.5, 1.2.6, culminating in
[theorem 1.2.9, p. 121](http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=13) gives (somewhat implicitly) the other adjunction

$$
  (\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_1}{\to}} Grpd
$$

with the [[right adjoint]] to $LConst$ being the [[fundamental groupoid]] functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write $\Gamma(X,C_X)$ we think of this here equivalently as $Sh_{(2,1)}(X)(X,LConst(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq Grpd(\Pi_1(X),C)$).

Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in [theorem 2.2.5, p. 132](http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=24) gives (somewhat implicitly) the adjunction

$$
  (\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_2}{\to}} Grpd
$$

now with the [[path n-groupoid|path 2-groupoid]] operation (locally) left adjoint to forming constant 2-stacks. (Subject verbatim to a remark as above.)

### Locally constant $\infty$-stacks on topological spaces {#ooStackOnTopSpace}

A discussion of locally constant $\infty$-stacks over [[topological space]]s is in

* [[Bertrand Toen]], _Toward a Galoisian interpretation of homotopy theory_ ([arXiv:0007157](http://arxiv.org/abs/math/0007157))

In [theorem 2.13, p. 25](http://arxiv.org/PS_cache/math/pdf/0007/0007157v4.pdf#page=25) the author proves an equivalence of [[(∞,1)-categories]] (modeled there as [[Segal category|Segal categories]])

$$
  LConst(X) \simeq Fib(\Pi(X))
$$

of [[locally constant ∞-stack]]s on $X$ and [[Kan fibration]]s over the [[fundamental ∞-groupoid]] $\Pi(X) = Sing(X)$.

But by the right Quillen functor $Id : sSet_{Quillen} \to sSet_{Joyal}$ from the Quillen [[model structure on simplicial sets]] to the [[Joyal model structure on simplicial sets]] every Kan fibration is a categorical fibration and every categorical fibration over a [[Kan complex]] is a [[Cartesian fibration]] (as discussed there) and a coCartesian fibration. Finally, by the [[Grothendieck construction|(∞,1)-Grothendieck construction]], these are equivalent to [[(∞,1)-functor]]s $\Pi(X) \to \infty Grpd$.

In total this means that via the [[Grothendieck construction]] Toën's result does actually produce an equivalence

$$
  LConst(X) \simeq Func(\Pi(X), \infty Grpd)
  \,.
$$

## Pattern

* A [[locally constant function]] is a section of a [[constant sheaf]];

* a [[locally constant sheaf]] is a section of a [[constant stack]];

* a [[locally constant stack]] is a section of (... and so on...)

* a **locally constant $\infty$-stack** is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

## References

Section A.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_ .

See also the references at [[geometric homotopy groups in an (∞,1)-topos]].

[[!redirects locally constant infinity-stacks]]

[[!redirects locally constant ∞-stack]]
[[!redirects locally constant (∞,1)-sheaf]]

[[!redirects locally constant ∞-stacks]]
[[!redirects locally constant (∞,1)-sheaves]]


locally constant sheaf

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

#Contents#
* table of contents
{:toc}

## Idea

A locally constant [[sheaf]] $A$ over a [[topological space]] is a sheaf of [[section]]s of a [[covering space]] of $X$: there is a [[cover]] of $X$ by open subsets $\{U_i\}$ such that the [[inverse image|restrictions]] $A|_{U_i}$ are [[constant sheaf|constant sheaves]].

More elegantly said: locally constant sheaves are the sections of [[constant stack]]s:

Let $C = Core(FinSet)\in $ [[Grpd]] be the [[core]] of the [[category]] [[FinSet]] of finite set, let $const_C : Op(X)^{op} \to Grpd$ the [[constant presheaf|presheaf constant]] on $C$, i.e. the [[functor]] on the [[opposite category]] of the [[category of open subsets]] of $X$ that sends everything to (the identity on) $C$. Then the [[constant stack]] on $C$ is the [[stackification]] $\bar const_C : Op(X)^{op} \to Grpd$.

Write then $X$ for the space $X$ regarded as a sheaf or trivial [[covering space]] over itself, i.e. the [[terminal object]] $X$ in sheaves and hence in stacks over $X$. Then by definition of stackification morphisms

$$
  X \to \bar const_C
$$

are represented by

* an open cover $\{U_i\}$ of $X$;

* over each $U_i$ a choice $F_i \in C$ of object in $C$, hence a finite set in $C$;

* over each double overlap $U_{i j} = U_i \cap U_j$ an morphism $g_{i j} : F_i|_{U_{i j}} \stackrel{\simeq}{\to} F_j|_{U_{i j}}$, hence a bijection of finite sets;

* such that on triple overlaps we have $g_{i k}|_{U_{i j k}}= g_{j k}|_{U_{i j k}}\circ g_{i j}|_{U_{i j k}}$.

Such data clearly is the local data for a [[covering space]] over $X$ with typical fiber any of the $F_i$.

## Definition

Let $(\Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \mathcal{S}$ be the [[global section]] [[geometric morphism]] of a [[topos]] $\mathcal{E}$ over base $\mathcal{S}$.

Without further assumption on $\mathcal{E}$ we have the following definition.

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

For $U \to *$ an [[epimorphism]] in $\mathcal{E}$, an [[object]] $E \in \mathcal{E}$ is called **locally constant and split by $U$** if in the [[over category]] $\mathcal{E}/U$ we have an [[isomorphism]]

$$
  E \times U \simeq (\Delta S) \times U
  \,,
$$

for some $S \in $ [[Set]].

An object which is locally constant and $U$-split for _some_ $U$ is called **locally constant**.

A locally constant object $E$ which is in addition an $\Delta Aut(X)$-[[principal bundle]] is called a **[[Galois object]]** .

=--

If $\mathcal{E}$ is a [[locally connected topos]] there is another characterization of locally constant sheaves.

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

For $C$ and $C$ [[cartesian closed categories]], a [[functor]] $F \,\colon\, C \to D$ that preserves [[products]] is called a **[[cartesian closed functor]]** if the canonical [[natural transformation]]

$$
  F(B^A) \to (F(B))^{F(A)}
$$

(which is the [[adjunct]] of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an [[isomorphism]].

=--

From the discussion at [[locally connected topos]] we have that

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

The [[constant sheaf]]-functor $\Delta \,\colon\, \mathcal{S} \to \mathcal{E}$ is a [[cartesian closed functor]] precisely if $\mathcal{E}$ is a [[locally connected topos]].

=--

In this case the above definition is equivalent to the following one.

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

Let $\mathcal{E} = Sh(C)$ be a [[locally connected topos]]. Let $p \,\colon\, core(Set^\kappa_*) \to core(Set^\kappa)$ be the [[core]] of the [[generalized universal bundle]] for sets of [[cardinality]] less than some $\kappa$.

A **locally constant $\kappa$-bounded object** in $\mathcal{E}$ is the [[pullback]] of $\Delta(p)$ along a morphism $* \to core(Set^\kappa)$ in the [[2-topos|(2,1)-topos]] $Sh_{(2,1)}(C)$.

=--

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

This says that locally constant sheaves are the sections of the [[constant stack]] on the [[groupoid]] $core(Set^\kappa)$.

Notice that

$$
  core(Set \kappa) \simeq \coprod_i \mathbf{B}Aut(F_i)
  \,,
$$

where the [[coproduct]] is over all [[cardinal]]s smaller than $\kappa$ and where $\mathbf{B}Aut(F_i)$ denotes the [[delooping]] [[groupoid]] of the [[automorphism group]] of the set $F_i$: the [[symmetric group]] on $F_i$.

This means that on each connected component of $\mathcal{E}$ a locally constant sheaf is the $\Delta \rho$-[[associated bundle]] to an $\Delta Aut(F)$-[[principal bundle]] induced by the canonical [[permutation representation]] $\rho : \mathbf{B} Aut(F) \to Set$ of the [[automorphism group]] $Aut(F)$ on $F$.

Specifically for $g : * \to \Delta \mathbf{B} Aut(F) \simeq \mathbf{B} \Delta Aut(F) \hookrightarrow \Delta core(set)$ the classifying morphism of a locally constant sheaf and for $U \to *$ an [[epimorphism]] on which it trivializes, we have a pasting diagram of [[pullback]]s

$$
  \array{
    U \times \Delta F
    &\to&
    P \times_{\Delta Aut(F) (\Delta (F // Aut(F)))}
    &\to&
    \Delta(F // Aut(F))
    &\to&
    \Delta Set^\kappa
    \\
    \downarrow &&
    \downarrow &&
    \downarrow &&
    \downarrow
    \\
    U &\to& * &\stackrel{g}{\to}& \mathbf{B} \Delta Aut(F)
    &\hookrightarrow&
    \Delta core(Set^\kappa)
  }
  \,,
$$

where $F//Aut(F)$ is the [[action groupoid]], the [[2-colimit]] of $\rho \mathbf{B}Aut(F) \to Grpd$.

=--

## Applications

* Locally constant sheaves are sheaves of sections of [[covering space]]s.

* When used as coefficient objects in [[cohomology]] they are also called [[local system]]s.

* The [[action]] of the [[fundamental groupoid]] $\Pi_1(X)$ on the fibers of a local system give rise to the notion of [[monodromy]].

* This may be used to define [[homotopy group (of an infinity-stack)|homotopy groups of]] general objects in a [[topos]], and the [[fundamental group of a topos]].

* This is the content of [[Galois theory]].

## Pattern

In sufficiently highly locally connected cases, we have:

* A [[locally constant function]] is a section of a [[constant sheaf]];

* a **locally constant sheaf** is a section of a [[constant stack]];

* a [[locally constant stack]] is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

## Related concepts

* [[constructible sheaf]]

## References

The definition of _locally constant sheaf_ originates in the notion of _covering projection_

* [[SGA]] 4, Exposé IX, 2.0 .

Lecture notes are in

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_

The topos-theoretic definition is reproduced in the context of a discussion of the notion of [[Galois topos]] as definition 5.1.1 in

* [[Eduardo Dubuc]], _On the representation theory of Galois and Atomic Topoi_ ([arXiv:0208222](http://arxiv.org/abs/math/0208222))

and definition 2.2 in

* [[Eduardo Dubuc]], _The fundamental progroupoid of a general topos_ ([arXiv:0706.1771](http://arxiv.org/abs/0706.1771))

or as definition 1 in

* [[Michael Barr]], [[Radu Diaconescu]], _On locally simply connected toposes and their fundamental groups_ ([NUMDAM](http://www.numdam.org/item?id=CTGDC_1981__22_3_301_0))
{#BarrDiaconescu}

Discussion of the notions of locally constant sheaves is at

* [[Michael Shulman]], _Locally constant sheaves_ (blog discussion)

[[!redirects locally constant sheaves]]

[[!redirects locally constant object]]
[[!redirects locally constant objects]]


locally constant stack

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **constant stack** is a section of a constant 2-stack.

## Pattern

* A [[locally constant function]] is a section of a [[constant sheaf]];

* a [[locally constant sheaf]] is a section of a [[constant stack]];

* a **locally constant stack** is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

## References

In view of [[infinity-local systems|$\infty$-local systems]]:

* {#PoleselloWaschkies05} [[Pietro Polesello]], [[Ingo Waschkies]], _Higher monodromy_, Homology, Homotopy and Applications **7** 1 (2005) 109-150 [[arXiv:0407507](http://arxiv.org/abs/math/0407507), [eudml:51918](https://eudml.org/doc/51918)]

[[!redirects locally constant stacks]]


locally constant ∞-stack > history

< [[locally constant ∞-stack]]


locally contractible space

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

# Contents
* table of contents
{:toc}

## Definition

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

A [[topological space]] $X$ is said to be __locally contractible__ if it has a [[basis for a topology|basis]] of [[open subset]]s that consists of [[contractible]] topological spaces $U \hookrightarrow X$.
=--

Sometimes one requires just that the inclusions $U \to X$ are [[null-homotopic map]]s. This might be called **semi-locally contractible**.

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

One could also consider a basis of open sets such that the opens $U$ have (just) trivial [[homotopy group]]s, but this does not seem to crop up in practice.
=--

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

A [[locale]] $X$ is __locally contractible__ if, viewing a locale as a $(0,1)$-[[(0,1)-topos|topos]] and hence a (very special kind of) $(\infty,1)$-[[(infinity,1)-topos|topos]], it is [[locally infinity-connected (infinity,1)-topos|locally ∞-connected]].
=--

## Examples

* Any [[CW-complex]] is locally contractible (see [there](CW+complex#CWComplexesAreLocallyContractible)).

* Any [[paracompact space|paracompact]] [[manifold]] is locally contractible.

* Any [[contractible space]] is semi-locally contractible

* The [[cone]] on the [[Hawaiian earring space]] is contractible, hence semi-locally contractible, but is not locally contractible, as any neighbourhood of the 'bad point' is not simply connected.

## Properties

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

For $X$ a locally contractible topological space, the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(X)$ is a [[locally ∞-connected (∞,1)-topos]].

=--

This is discussed at [[locally ∞-connected (∞,1)-site]].

## Other viewpoints

If one considers [[fundamental ∞-groupoid]]s, the inclusion $U \to X$ being null-homotopic is equivalent to the induced [[(∞,1)-functor]] $\Pi(U) \to \Pi(X)$ being naturally isomorphic to the trivial functor sending everything to a single point.

+--{: .query}
[[David Roberts]]: The following may be straightforwardly obvious, but I have couched it as a conjecture, because I haven't seen it in print.
=--

+-- {: .un_conjecture}
###### Conjecture

If the space $X$ is semi-locally contractible then every locally constant $n$-stack on the site of open sets of $X$ is locally trivial.
=--

See also [[locally ∞-connected (∞,1)-topos]]. There a converse to this conjecture is stated:

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

Let $C$ be a [[site]] coming from a [[coverage]] such that constant [[(∞,1)-presheaf|(∞,1)-presheaves]] satisfy [[descent]] over objects of $C$ with respect to the generating [[covering]] families. Then the [[(∞,1)-category of (∞,1)-sheaves]] $\mathbf{H} = Sh_{(\infty,1)}(C)$ is a [[locally ∞-connected (∞,1)-topos]].
=--

[[!redirects locally contractible space]]
[[!redirects locally contractible spaces]]
[[!redirects locally contractible topological space]]
[[!redirects locally contractible topological spaces]]


> this page is under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _locally contractible topological ∞-groupoid_ is an [[∞-groupoid]] equipped with [[cohesive (∞,1)-topos|cohesion]] in the form of [[locally contractible topological space|locally contractible]] [[topology]].

The collection of all these cohesive $\infty$-groupoids forms a [[cohesive (∞,1)-topos]] $LCTop\infty Grpd$.

This is similar to [[ETop∞Grpd]], which models cohesion in the form of [[Euclidean topology]].

## Definition

Let $CTop$ be some [[small category|small]] version (...details missing...) of the [[site]] of [[locally contractible topological space|locally contractible]] [[contractible topological spaces]] with [[continuous maps]] betwen them and equipped with the standard [[open cover]] [[coverage]].

This is a [[cohesive site]] (for the evident generalization of that definitions where Cech covers are generalized to hypercovers). The key axiom to check is that for $Y \to U$ a [[hypercover]] of $U \in CTop$ degreewise by a [[coproduct]] of contractibles, also the simplicial set $\lim_\to Y$ obtained by sending each contractible to a point is contractible. This follows as pointed out [on MO here](http://mathoverflow.net/a/187891/381).[^ack]

[^ack]:Thanks to [[David Carchedi]] for highlighting this.

Define then

$$
  LCTop\infty Grpd := Sh_{(\infty,1)}(CTop)
$$

to be the [[(∞,1)-category of (∞,1)-sheaves]] on $CTop$.

This is an [[cohesive (∞,1)-topos]].

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : LCTop\infty Grpd \to \infty Grpd
  \,.
$$

## Related concepts

* [[topological infinity-groupoid]]

  * [[Euclidean-topological infinity-groupoid]]

## References

The corresponding 1-[[cohesive topos]] over locally _connected_ topological spaces was considered in

* {#Johnstone11} [[Peter Johnstone]], example 1.5 of _Remarks on punctual local connectedness_, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63 ([web](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

A decent account of the above $\infty$-topos is in prepation by [[David Carchedi]]...

[[!redirects locally contractible topological ∞-groupoid]]
[[!redirects LCTop∞Grpd]]

[[!redirects locally contractible topological infinity-groupoids]]
[[!redirects locally contractible topological ∞-groupoids]]


locally convex topological vector space

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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
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# Contents
* table of contents
{: toc}

## Definition

A [[topological vector space]] is __locally convex__ if it has a [[base of its topology]] consisting of [[convex subsets|convex]] [[open subsets]].  Equivalently, it is a vector space equipped with a [[gauge space|gauge]] consisting of [[seminormed vector space|seminorms]].  As with other topological vector spaces, a locally convex space (LCS or LCTVS) is often assumed to be [[Hausdorff space|Hausdorff]].

Locally convex (topological vector) spaces are the standard setup for much of contemporary [[functional analysis]].

A natural notion of [[smooth map]] between lctvs is given by [[Michal-Bastiani smooth maps]].

## Properties

The category $lctvs$ is a [[symmetric monoidal category]] with the [[inductive tensor product]] and even a symmetric [[closed monoidal category]], where the internal homs are given by the space of continuous linear maps with the topology of pointwise convergence.

### Continuous linear functionals
 {#ContinuousLinearFunctionals}

One reason why locally convex [[topological vector spaces]] are important is that lots of [[continuous linear functionals]] exist on them, at least if one assumes an appropriate choice principle, e.g., [[axiom of choice]] or [[ultrafilter theorem]] (or just [[dependent choice]] for a [[separable space]]). This fact is encapsulated in the [[Hahn-Banach theorem]]; a nice exposition is given in [Tao 09](#Tao09). By way of contrast, a [[topological vector space]] which is _not_ locally convex, such as the [[Lebesgue space]] $L^p([0, 1])$ where $0 \lt p \lt 1$, need not have any (nonzero) [[continuous linear functionals]] at all.

+-- {: .num_defn #DirectedSystemOfSeminorms}
###### Definition
**(directed system of seminorms)**

A family $\{p_q\}_{q \in Q}$ of [[seminorms]] on some [[real vector space]] $V$ is called _directed_ if

$$
  \underset{q_1, q_2 \in Q}{\forall}
  \left(
    \underset{q \in Q}{\exists}
    \left(
      \underset{C \in (0,\infty)}{\exists}
      \left(
        \underset{v \in V}{\forall}
        \left(
          \,
          C p_q(v) \leq max\{ p_{q_1}(v), p_{q_2}(v) \}
          \,
        \right)
      \right)
    \right)
  \right)
  \,.
$$

=--

(e.g. [Infusino 17](#Infusino17) [def. 4.2.15](http://www.math.uni-konstanz.de/~infusino/TVS-SS17/Lect10.pdf))

+-- {: .num_prop #MaximumEnvelopeOfSeminorms}
###### Proposition
**(maximum envelope of seminorms)

Let $V$ be a [[real vector space]] equipped with a set $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$ of [[seminorms]].

Then the [[maxima]] of [[inhabited|non-empty]] [[finite set|finite]] [[subsets]] $J \subset Q$ of these seminorms

$$
  v \mapsto \left( \underset{q \in J}{max} p_q(v) \right)
$$

are themselves seminorms, and the set of them

$$
  \left\{
    \underset{q \subset J}{max} p_q(-)
  \right\}_{ { J \subset Q } \atop {\text{finite, non-empty}} }
$$

generate the same [[topological vector space|topology]] on $V$ as the original $\{p_q\}$ do. Moreover, the system of maximum seminorms evidently form a filtered system according to def. \ref{DirectedSystemOfSeminorms}.

=--

(e.g. [Infusino 17](#Infusino17) [prop.  4.2.13, 4.2.14](http://www.math.uni-konstanz.de/~infusino/TVS-SS17/Lect10.pdf))

+-- {: .num_prop #AlternativeCharacterizationOfContinuityForLinearFunctionals}
###### Proposition
**(characterization of continuity for linear functionals by norm-bounds)

Let $V$ be a [[real vector space]] and $\tau$ a [[topological space|topology]] on $V$ that makes it a locally convex topological vector space, induced from a set of [[seminorms]] $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$.
Consider a [[linear function]]

$$
  L \colon V \to \mathbb{R}
$$

1. (directed system of seminorms) If the system of seminorms $\{p_q\}_{q \in Q}$ is directed (def. \ref{DirectedSystemOfSeminorms}) then $L$ is a [[continuous function]] with respect to $\tau$, hence is a [[continuous linear functional]], precisely if it is $q$-continuous for some $q \in Q$:

   $$
     \left(
       L \,\text{continuous with respect to}\, \tau
     \right)
     \;\Leftrightarrow\;
     \underset{q \in Q}{\exists}
     \left(
       \underset{C \in (0,\infty)}{\exists}
       \left(
         \underset{v \in V}{\forall}
         \left(
           \,
           {\vert L(v)\vert}
            \leq
           C p_q(v)
           \,
         \right)
       \right)
     \right)
     \,.
   $$

1. (general system of seminorms) Together with prop. \ref{MaximumEnvelopeOfSeminorms} this means that for $\{p_q\}_{q \in Q}$ any set of seminorms (not necessarily directed), then $L$ is continuous precisely if there exists a [[inhabited set|inhabited]] [[finite set|finite]] [[subset]] of seminorms such that $L$ is bounded with respect to the [[maximum]] over this set:

   $$
     \left(
       L \,\text{continuous with respect to}\, \tau
     \right)
     \;\Leftrightarrow\;
     \underset{q_1, \cdots, q_n \in Q}{\exists}
     \left(
       \underset{C \in (0,\infty)}{\exists}
       \left(
         \underset{v \in V}{\forall}
         \left(
           \,
           {\vert L(v)\vert}
            \leq
           C \underset{k = 1, \cdots, n}{max} p_{q_n}(v)
           \,
         \right)
       \right)
     \right)
     \,.
   $$

=--

(e.g. [Infusino 17](#Infusino17), [prop. 4.6.1, corollary 4.6.2](http://www.math.uni-konstanz.de/~infusino/Lect12.pdf)), or remark 3-4 4. here: [pdf](https://www.math.ksu.edu/~nagy/func-an-2007-2008/lcvs-5.pdf)

### Co-Probes and curves

The collections of [[continuous linear functionals]] on a LCTVS is used in a way analogous to the collection of [[coordinate]] [[projections]] $pr_i:\mathbb{R}^n\to \mathbb{R}$ out of a [[Cartesian space]]. For example, [[curves]] in a LCTVS over the reals can be composed with functionals to arrive at a collection of functions $\mathbb{R} \to \mathbb{R}$ which are analogous to the 'components' of the curve.

In one respect, a locally convex TVS is a [[nice topological space]] in that there are enough co-probes by maps to the base field.

### Diagram of properties

[[!include diagram of LCTVS properties]]

## Examples

* [[Fréchet spaces]] are locally convex topological vector spaces

## References

* J. L. Taylor, _Notes on locally convex topological vector spaces_ (1995) ([pdf](http://www.math.utah.edu/~taylor/LCS.pdf))

* {#Tao09} [[Terence Tao]], _[Duality and the Hahn-Banach theorem](http://terrytao.wordpress.com/2009/01/26/245b-notes-6-duality-and-the-hahn-banach-theorem/)_, 2009

* {#Infusino17} Maria Infusino, _[Topological vector spaces](http://www.math.uni-konstanz.de/~infusino/TVS-SS17/teachingSS2017.html)_ 2017

Generalizing the [[Tietze extension theorem]] to [[codomains]] which are [[locally convex topological vector spaces]]:

* {#Dugundji51} [[James Dugundji]], *An extension of Tietze's theorem*, Pacific Journal of Mathematics **1** 3 (1951) [[doi:10.2140/pjm.1951.1.353](http://dx.doi.org/10.2140/pjm.1951.1.353), [pdf](https://msp.org/pjm/1951/1-3/pjm-v1-n3-p04-p.pdf)]

category: analysis

[[!redirects locally convex space]]

[[!redirects locally convex]]
[[!redirects locally convex spaces]]
[[!redirects locally convex vector space]]
[[!redirects locally convex vector spaces]]
[[!redirects locally convex topological vector space]]
[[!redirects locally convex topological vector spaces]]
[[!redirects locally convex TVS]]
[[!redirects locally convex TVSs]]
[[!redirects locally convex TVSes]]
[[!redirects LCS]]
[[!redirects LCSs]]
[[!redirects LCSes]]
[[!redirects LCTVS]]
[[!redirects LCTVSs]]
[[!redirects LCTVSes]]

[[!redirects locally convex Hausdorff space]]
[[!redirects locally convex Hausdorff spaces]]
[[!redirects Hausdorff locally convex space]]
[[!redirects Hausdorff locally convex spaces]]
[[!redirects hausdorff locally convex space]]


locally decidable topos

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# Contents
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{:toc}

[[!redirects QD topos]]
[[!redirects locally decidable object]]

##Idea
Loosely speaking a **locally decidable topos** is a topos that is locally Boolean. Whereas in a [[Boolean topos]] every object is decidable, in a locally decidable topos every object is a quotient of a [[decidable object]]. This results in a reasonable approximation to the concept of 'being almost Boolean'.

In Lawvere's approach to cohesion locally decidable toposes are one of the principal classes of **petit toposes** (Lawvere 1991).
In the 1991 paper Lawvere also hypothesizes the discrete base topos $\mathcal{S}$ that he contrasts with the cohesive topos of spaces, to be locally decidable.[^fine]

[^fine]: Lawvere uses the term 'QD'-topos ( _quotient of decidable_) for 'locally
decidable topos' in this paper. He also uses 'SUD'-object (from _separable-unramified-decidable_ to indicate equivalent concepts in algebra-topology-logic) for what we call a locally decidable object. In later papers Lawvere uses also the terms 'locally separable'or 'adequately separable'. The idea to take a locally decidable base topos is probably suggested by [[Galois theory]] where 'locally decidable' corresponds roughly to 'product of separable field extensions' for an algebra over a field $k$. This also connects the notion '[[decidable object|decidable]]' via Grothendieck's Galois theory to the topological notion 'unramified'.

##Definition
An object $X$ in a topos $\mathcal{E}$ is called _locally decidable_ (or, a _quotient of a decidable object_) iff there is an epimorphism $Y\twoheadrightarrow X$ such that $Y$ is a [[decidable object]]. $\mathcal{E}$ is called _locally decidable_ iff every object $X$ is locally decidable.

##Examples

* The [[classifying topos]] $Set[\mathbf{D}]=[FinSet_{mono},Set]$ for the [[theory of decidable objects]] $\mathbf{D}$ is locally decidable. Here $FinSet_{mono}$ is the category of finite sets and monomorphisms, whose opposite category is the carrier of the site for the so called (Myhill-) _[[Schanuel topos]]_.

* Every [[localic topos]] is locally decidable.

* For every [[Grothendieck topos]] $\mathcal{E}$ the full subcategory $\mathcal{E}_{QD}$ of locally decidable objects is a locally decidable topos.

##Properties

* **Proposition**. A topos $\mathcal{E}$ is locally decidable iff there is a [[localic topos|localic]] geometric morphism to a Boolean topos, or iff $\mathcal{E}$ has a site $(\mathcal{C}, J)$ with all morphisms in $\mathcal{C}$ epic.

* Locally decidable (presheaf) toposes are 'co-étendues' in the sense that for a small $\mathcal{C}$ the functor category $[\mathcal{C},Set]$ is locally decidable precisely if $[\mathcal{C}^{op},Set]$ is an [[étendue]]. Also the all-epic-site property is dual to the all-monic-site property of étendues. Both concepts are subsumed under the notion of having a (sub canonical) site representation with no (non-trivial) [[idempotents]] (McLarty 2006, Lawvere 2007,2008).

* A locally decidable topos has a [[localic topos|localic geometric morphism]] to the [[Schanuel topos]] (cf. Johnstone 2002, p.794).

##Related concepts

* [[Boolean topos]]
* [[petit topos]]
* [[étendue]]
* [[decidable object]]
* [[theory of decidable objects]]

##References

* [[Peter Freyd]], _All topoi are localic - or, why permutation models prevail_ , JPAA **46** (1987) pp.49-58.

* [[Simon Henry]], _On toposes generated by finite cardinal objects_ , arxiv:1505.04987 (2015). ([pdf](http://arxiv.org/pdf/1505.04987v1.pdf))

* [[Peter Johnstone]], _Quotients of decidable objects in a topos_ , Math. Proc. Camb. Phil. Soc. **93** (1983) pp.409-419.

* [[Peter Johnstone]], _How general is a generalized space?_ , London Math. Soc. LNS **93** (1985) pp.77-111.

* [[Peter Johnstone]], _Sketches of an Elephant II_, Oxford UP 2002. (sec. C5.4 pp.792-803)

* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* [[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM vol. 1488 (1991).

* [[F. William Lawvere]], _Axiomatic Cohesion_ , TAC **19** no. 3 (2007) pp.41-49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* [[F. William Lawvere]], _Cohesive Toposes: Combinatorial and Infinitesimal Cases_, Como Ms. 2008. ([pdf](http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf))

* [[Colin McLarty]], _Every Grothendieck Topos has a One-Way Site_ , TAC **16** no. 5 (2006) pp.123-126. ([pdf](http://www.tac.mta.ca/tac/volumes/16/5/16-05.pdf))


locally decomposable space

# Locally decomposable spaces
* table of contents
{: toc}

## Idea

*Local decomposability* is a sort of [[separation axiom]], like a weak sort of [[regular space|regularity]], that is trivial in [[classical mathematics]], but interesting in [[constructive mathematics]].

## Definition

A [[topological space]] $X$ is **locally decomposable** if for any open set $U\subseteq X$ and point $x\in X$, there exists an open set $V$ with $x\in V$ such that for all $y\in X$ we have either $y\in U$ or $y\notin V$.  If [[excluded middle]] holds, of course, we can take $V = U$.

For [[point-set apartness spaces]], which are equivalent to certain topological spaces, the condition can be rephrased as: if $x\bowtie A$, then there is a set $B$ such that $x\bowtie B$ and for all $y$ we have either $y\bowtie A$ or $y\in B$.

For [[uniform spaces]], the notion of [[uniformly regular space|uniform regularity]] is really a notion of "uniform local decomposability"; but since in the uniform case it is sufficient to imply full regularity, we generally call it "uniform regularity" instead.  For [[quasi-uniform spaces]] this is no longer true (since after all, there are non-regular quasi-uniform spaces classically), so we should speak of uniform local decomposability instead.

## References

* [[Douglas Bridges]], Peter Schuster, and Luminita Vita, *Apartness, Topology, and Uniformity: a Constructive View*, [doi](http://dx.doi.org/10.1002/1521-3870%28200210%2948:1%2B%3C16::AID-MALQ16%3E3.0.CO;2-7)
 {#BSV}

[[!redirects locally decomposable space]]
[[!redirects locally decomposable spaces]]
[[!redirects locally decomposable topological space]]
[[!redirects locally decomposable topological spaces]]
[[!redirects locally decomposable apartness space]]
[[!redirects locally decomposable apartness spaces]]
[[!redirects local decomposability]]


locally discrete 2-category

A $2$-[[2-category|category]] is __locally discrete__ if each [[hom-category]] is a [[discrete category]].

Just as a discrete category is the same as a [[set]], so a locally discrete $2$-category is the same as a [[category]].  The real value of the concept is to see how a category may be interpreted as a $2$-category.

[[!redirects locally discrete 2-categories]]
[[!redirects locally-discrete 2-category]]
[[!redirects locally-discrete 2-categories]]


locally discrete set of subsets

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#Contents#
* table of contents
{:toc}

## Definition

Given a [[topological space]] $X$, then a set of [[subset]] $\{S_i \subset X\}_{i \in I}$ is _locally discrete_ if every point $x \in X$ has a neighborhood which [[intersection|intersects]] at most one of the $S_i$.

Often one can find a base of a topological space that decomposes into a countable union of locally discrete sets. In this case the base is called [[countably locally discrete set of subsets|$\sigma$-discrete]]. Such a base [[open covers of metric spaces have open countably locally discrete refinements|can be found for any metric space]].

## Related concepts

* [[locally finite cover]]
* [[locally finite set of subsets]]
* [[countably locally discrete set of subsets]]
* [[countably locally finite set of subsets]]

## References

* Ryszard Engelking, _General Topology_, Heldermann Verlag Berlin, 1989.

* K. Kuratowski, _Topology_, 2014, vol. 1.

[[!redirects countably locally discrete sets of subsets]]
[[!redirects sigma-locally discrete sets of subsets]]


locally equi-connected space

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## Definition

A [[topological space]] $X$ is called **locally equi-connected** if the [[diagonal]] map $X \to X \times X$ is a [[Hurewicz cofibration]].

## Properties

Every [[CW-complex]] is locally equi-connected.

[[!redirects locally equi-connected spaces]]


locally finite cell complex

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## Idea

A [[cell complex]] (such as a [[CW-complex]]) is _locally finite_ if every point/vertex is contained only in a [[finite number]] of cells.

## Related concepts

* [[locally finite set of subsets]]

* [[locally finite cover]]

## References

* Ross Geoghegan, _Locally finite CW Complexes and Proper Homotopy_, chapter 10 in: _Topological methods in group theory_, Graduate Texts in Mathematics **243**, Springer 2008 ([doi:10.1007/978-0-387-74614-2_10](https://doi.org/10.1007/978-0-387-74614-2_10))

[[!redirects locally finite cell complexes]]

[[!redirects locally finite CW complex]]
[[!redirects locally finite CW complexes]]

[[!redirects locally finite CW-complex]]
[[!redirects locally finite CW-complexes]]