covariant type family

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Directed homotopy type theory
+-- {: .hide}
[[!include directed homotopy type theory - contents]]
=--
=--
=--

\tableofcontents

## Definition

In [[simplicial homotopy type theory]], given a type $A$ and a type family
$$x:A \vdash B(x) \; \mathrm{type}$$
let us define the type family
$$x:A, y:A, f:\mathrm{hom}_A(x, y), u:B(x), v:B(y) \vdash \mathrm{hom}_{B(f)}(u, v) \; \mathrm{type}$$
as
$$\mathrm{hom}_{B(f)}(u, v) \coloneqq \left\langle\prod_{t:\mathbb{2}} C(f(t)) \vert_{[u,v]}^{\partial \Delta^1} \right\rangle$$
where $\mathbb{2}$ is the directed interval primitive, $\Delta^1$ is the $1$-simplex probe shape, and $\partial \Delta^1$ is its boundary.

Then a **covariant type family** over $A$ is a [[type family]]
$$x:A \vdash B(x) \; \mathrm{type}$$
such that given elements $x:A$, $y:A$, $f:\mathrm{hom}_A(x, y)$, and $u:B(x)$, the type
$$\sum_{v:B(y)} \mathrm{hom}_{B(f)}(u, v)$$
is a [[contractible type]].

## Properties

\begin{theorem}
Given a [[Segal type]] $A$, a type family
$$x:A \vdash B(x) \; \mathrm{type}$$
is a covariant type family if and only if for all elements $f:\mathrm{hom}_A(x, y)$, and $u:B(x)$, there is a unique lifting of $f$ that starts at $u$; or equivalently, that the type
$$\sum_{v:B(y)} \left\langle\prod_{t:\mathbb{2}} C(f(t)) \vert_{u}^{0} \right\rangle$$
is contractible.
\end{theorem}

## Related concepts

* [[hom type]]

* [[directed univalence]]

* [[contravariant type family]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$


cover

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

> under construction

#Contents#
* table of contents
{:toc}

## Idea

Generally, for  $X$ an object we think of as a [[space and quantity|space]], a _cover_ of $X$ is some other object $Y$ together with a morphism $\pi : Y \to X$, usually an [[epimorphism]] demanded to be well behaved in certain way.

The idea is that $Y$ provides a "locally resolved" picture of $X$ in that $X$ and $Y$ are "locally the same" but that $Y$ is "more flexible" than $X$.

The archetypical example are ordinary covers of topological spaces $X$ by open subsets $\{U_i\}$: here $Y$ is their [[disjoint union]] $Y := \coprod_i U_i$.

More generally, you might need a cover to be *family* of maps $(\pi_i: Y_i \to X)_i$; if the category has [[coproduct]]s that get along well with the covers, then you can replace these families with single maps as above---see [[superextensive site]].

## Definitions

In the context of [[sheaf and topos theory]] a cover
on an [[object]] $U$ in a [[category]] $C$ is a collection of [[morphism]]s $\{U_i \to U\}_{i \in I}$.

A specification of a collection of covers for each object of the category, subject to some compatibility condition, makes a [[coverage]] on $C$. If the collection of covers in a coverage is being closed under some operations, the result is called a [[Grothendieck topology]]. Equipped with a coverage/Grothendieck topology, the category is called a [[site]]. See there for more details.

Covering families $\{U_i \to U\}$ in $C$ have incarnations as single morphisms in the [[category of presheaves]] $PSh(C)$ over $C$, and these are also sometimes called _covers_ :

the [[Cech nerve]] of the morphism $\coprod_i U_i \to U$ in $PSh(C)$ is a [[simplicial object]] in $PSh(C)$

$$
  C(\{U_i\})
  =
  \left(
     \cdots
   \stackrel{\to}{\stackrel{\to}{\to}}
   \coprod_{i j} U_i \times_U U_j \stackrel{\to}{\to} \coprod_i U_i
  \right)
  \to
  U
  \,.
$$

Its [[colimit]] is the [[local epimorphism]] on $U$ that is the incarnation of the covering family in $C$, now in $PSh(C)$.

In [[higher category theory]], when we do not restrict to presheaves, for instance when we use [[simplicial presheaves]], the full Cech nerve itself $C(\{U_i\}) \to U$ is the "local epimorphism", the covering map.

More generally, given a [[coverage]] one can form [[hypercover]]s in the category of simplicial presheaves, by starting with a Cech nerve and then iteratively refining it in each degree further and further by more covers.

...

## Examples

In the category $C$ =[[Top]] of [[topological space]]s or $C$ = [[Diff]] of [[smooth manifold]]s or similar, one has the notions

* [[open cover]] -- for $X \in C$ a space, an _open cover_ is a collection $\{U_i \subset X\}$ of [[open subset]]s, that cover $X$ in the obvious naive sense of the word, i.e. which are such that their union equals $X$;

* [[good cover]] -- a cover $\{U_i \to X\}$ is called a _good cover_ (or _[[good open cover]]_ if in addition it is an open cover) if all of the $U_i$ and all their finite intersections $U_{i_1} \times_X U_{i_2} \times_X \cdots \times_X U_{i_n}$ are [[contractible]] as topological spaces.

  A parameterized version of this is a [[stacked cover]].

There is also the notion of

* [[covering space]]

of a topological space or manifold. This is a priori an independent notion of cover, but for the standard [[Grothendieck topologies]] on [[Top]], [[Diff]], etc. the projection $\{\hat X \to X\}$ from a covering space is a covering family.

## Related concepts

* [[open cover]]

* [[covering]]

* [[finite cover]], [[locally finite cover]]

[[!redirects covers]]
[[!redirects covering family]]
[[!redirects covering families]]


cover-preserving functor

If $C$ and $D$ are [[sites]], a [[functor]] $F\colon C\to D$ is **cover-preserving** (some people say **continuous**) if whenever $R$ is a [[covering family]] of an object $U\in C$, its image $F(R)$ is a covering family of $F(U)$ in $D$.

If $F$ is also [[flat functor|flat]], then it is called a [[morphism of sites]], and induces a [[geometric morphism]] between [[sheaf toposes]].

[[!redirects cover-preserving functors]]


coverage

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

# Contents
* table of contents
{:toc}

## Idea

A **coverage** on a [[category]] $C$ consists of, for each object $U\in C$, a collection of families $\{f_i:U_i\to U\}_{i\in I}$ of [[morphisms]] with [[target]] $U$ to be thought of as [[cover|covering families]].  The essential characteristic of these covering families is that they be "stable under [[base change|pullback]]."  A number of other "saturation" conditions are frequently also imposed for convenience.  A category equipped with a coverage is called a [[site]].

One of the main purposes of a coverage is that it provides the minimum structure necessary to define a notion of [[sheaf]] (or more generally [[stack]]) on $C$.  A [[Grothendieck topos]] is defined to be the category of sheaves (of sets) on a small site.  From this perspective, the example to keep in mind is the [[partial order|poset]] $O(X)$ of open sets in some [[topological space]] (or [[locale]]) $X$, where a morphism is an inclusion, and a family of inclusions $\{U_i \hookrightarrow U\}$ is a covering family iff $U = \bigcup_i U_i$.

Another perspective on a coverage is that the covering families are "postulated well-behaved quotients."  That is, saying that $\{f_i:U_i\to U\}_{i\in I}$ is a covering family means that we want to think of $U$ as a well-behaved quotient (i.e. colimit) of the $U_i$.  Here "well-behaved" means primarily "stable under pullback."  In general, $U$ may or may not _actually_ be a colimit of the $U_i$; if it always is we call the site _[[subcanonical coverage|subcanonical]]_.  From this perspective, the embedding of $C$ into its category of sheaves is "the free cocompletion of $C$ that takes covering families to well-behaved quotients"; compare how the [[Yoneda embedding]] of an arbitrary category $C$ into its category of [[presheaf|presheaves]] is its [[free cocompletion]], period.

The traditional name for a coverage, with the extra saturation conditions imposed, is a [[Grothendieck topology]], and this is still widely used in mathematics.  Following the [[Elephant]], on this page we use _coverage_ for a pullback-stable system of covering families and _Grothendieck coverage_ if the extra saturation conditions are imposed.  See [[Grothendieck topology]] for a discussion of the objections to that term.

A related notion is that of [[basis for a Grothendieck topology]], which is similar to the notion of coverage, and similarly induces a Grothendieck topology, but assumes existence of [[pullback]]s and closure of covering families under these pullbacks.

## Definition
 {#Definition}

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

A **coverage** on a [[category]] $C$ consists of a [[function]] assigning to each [[object]] $U\in C$ a collection of families of [[morphisms]] $\{f_i:U_i\to U\}_{i\in I}$, called _[[covering]] families_, such that

* If $\{f_i:U_i\to U\}_{i\in I}$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $\{h_j:V_j\to V\}$ such that each composite $g h_j$ factors through some $f_i$.

$$
  \array{
    V_j &\overset{k}{\longrightarrow}& U_i
    \\
    \mathllap{{}^{h_j}}\big\downarrow & & \big\downarrow\mathrlap{^{f_i}}
    \\
    V &\underset{g}{\longrightarrow}& U
  }
  \;\;.
$$

The logic here is: $\forall f, \forall g, \exists h, \forall j, \exists i, \exists k,  =$.

=--

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

A [[site]] is a category equipped with a coverage.

=--

Often [[site]]s are required to be [[small category|small]]; see [[large site]] for complications that may otherwise arise.

### Sheaves on a site

See [[sheaf]], of course, but it seems appropriate to briefly recall the concept here.  If $\{f_i:U_i\to U\}_{i\in I}$ is a family of morphisms with codomain $U$, a [[presheaf]] $X:C^{op}\to Set$ is called a **sheaf** for this family if

* For any collection of elements $x_i \in X(U_i)$ such that, whenever $g:V\to U_i$ and $h:V\to U_j$ are such that $f_i g = f_j h$, we have $X(g)(x_i) = X(h)(x_j)$, then there exists a unique $x\in X(U)$ such that $X(f_i)(x)=x_i$ for all $i$.

If $C$ is a site, a presheaf $X:C^{op}\to Set$ is called a **sheaf** on $C$ if it is a sheaf for every covering family in $C$.  We call a site $C$ **[[subcanonical coverage|subcanonical]]** if every representable functor $C(-,c):C^{op}\to Set$ is a sheaf.

The [[category of sheaves]] $Sh(C)$ is a full [[subcategory]] of the category $[C^{op},Set]$ of presheaves.  If $C$ is [[subcanonical coverage|subcanonical]], then its [[Yoneda embedding]] $C\to [C^{op},Set]$ factors through $Sh(C)$.  If $C$ is small, then $Sh(C)$ is [[reflective subcategory|reflective]] in $[C^{op},Set]$ and a [[Grothendieck topos]].

### Sites with pullbacks
{#cartesian}

If, as is frequently the case, $C$ has [[pullback|pullbacks]], then it is natural to impose the following stronger condition:

* If $\{f_i:U_i\to U\}_{i\in I}$ is a covering family and $g:V\to U$ is a morphism, then the family of pullbacks $\{g^*(f_i):g^*U_i\to V\}$ is a covering family of $V$.

One can also impose the weaker condition that the pullbacks of covering families exist and are covering families, even if not all pullbacks exist in $C$.  The saturation conditions below imply that on a category with pullbacks, every coverage is equivalent to one satisfying this stronger condition, which perhaps we may call a __cartesian coverage__.

Likewise, when $C$ has pullbacks (of covering families), the condition for a presheaf $X$ to be a sheaf for a covering family $\{f_i:U_i\to U\}_{i\in I}$ can be stated more simply (and probably more familiarly, to some readers), as the assertion that the following diagram is an [[equalizer]]:
\[
X(U) \to \prod_{i\in I} X(U_i) \rightrightarrows \prod_{j,k\in I} X(U_j\times_U U_k).
\]
The generalization to [[stack|stacks]] using [[cosimplicial object|cosimplicial objects]] is then straightforward.

### Saturation conditions

The collection of covering families can be "closed up" under a number of convenient operations without changing the notion of sheaf.

1. Any presheaf is a sheaf for the singleton family $\{1_U:U\to U\}$.

1. Any presheaf which is a sheaf for a family $\{f_i:U_i\to U\}_{i\in I}$ and also for some family $\{h_{i j}:U_{i j} \to U_i\}_{j\in J_i}$ for each $i$ is also a sheaf for the family of all composites $\{f_i h_{i j}:U_{i j}\to U\}_{i\in I, j\in J_i}$.

1. Let $C$ be a site and $\{f_i:U_i\to U\}_{i\in I}$ a covering family, and suppose $\{g_j:V_j\to U\}_{j\in J}$ is a family of morphisms such that each $f_i$ factors through some $g_j$.  Then any sheaf $X$ on $C$ is also a sheaf for the family $\{g_j:V_j\to U\}_{j\in J}$.  _(NB: for this condition, it is essential that $\{f_i\}$ be part of a coverage and that $X$ be a sheaf for the entire coverage, not just for $\{f_i\}$.)_

1. For any family $\{f_i:U_i\to U\}_{i\in I}$, the [[sieve]] it generates is the family of all morphisms $g:V\to U$ which factor through some $f_i$.  A presheaf $X$ is a sheaf for $\{f_i\}$ iff it is a sheaf for the sieve it generates.

### Grothendieck coverages

Grothendieck originally considered only coverages that are closed under some or all of the above saturation conditions.

Because of the final condition, we may choose to consider only covering _sieves_.  Incorporating the other saturation conditions as well, we define a **Grothendieck coverage** (commonly called a [[Grothendieck topology]]) to be a collection of sieves called _covering sieves_, satisfying the following pullback-stability and saturation conditions.  (If $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, we define $g^*(R)$ to be the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $g h$ factors through some morphism in $R$.)

* If $R$ is a covering sieve on $U$ and $g:V\to U$ is any morphism, then $g^*(R)$ is a covering sieve on $V$.

* For each $U$ the sieve $M_U$ consisting of _all_ morphisms into $U$ (the sieve generated by the singleton family $\{1_U\}$) is a covering sieve.

* If $R$ is a covering sieve on $U$ and $S$ is an arbitrary sieve on $U$ such that for each $f:V\to U$ in $R$, $f^*(S)$  is a covering sieve on $V$, then $S$ is also a covering sieve on $U$.

One can then show that for every coverage, there is a _unique_ Grothendieck coverage having the same sheaves.  When $C$ is small, then Grothendieck coverages on $C$ are also in bijective correspondence with [[Lawvere-Tierney topology|Lawvere-Tierney topologies]] on its presheaf topos $[C^{op},Set]$, and thus in bijection with [[subtopos|subtoposes]] of $[C^{op},Set]$. For more on this see [[category of sheaves]].

On the other hand, it is often useful to consider only pullback-stable covering families, without needing to close them up into sieves satisfying the saturation conditions.  For instance, in many cases the generating covering families will be finite and easy to describe.  As we saw above, the notion of sheaf can also be defined more explicitly in terms of covering families, especially when $C$ has pullbacks.

Frequently, though, these covering families will satisfy at least some of the saturation conditions.  The name [[Grothendieck pretopology]] or _basis for a Grothendieck topology_ is commonly used for a coverage (often of the stronger sort requiring pullbacks) that also satisfies

* Every isomorphism is a covering family.

* If $\{f_i:U_i\to U\}_{i\in I}$ is a covering family and for each $i$, so is $\{h_{i j}:U_{i j} \to U_i\}_{j\in J_i}$, then $\{f_i h_{i j}:U_{i j}\to U\}_{i\in I, j\in U_i}$ is also a covering family.

## Examples {#Examples}

* For $X$ a [[topological space]] and $Op(X)$ its [[category of open subsets]], the collection of [[open cover]]s is a [[subcanonical coverage|subcanonical]] coverage on $Op(X)$. I.e. a covering family on an open subset $U \subset X$ is a collection of further open subsets $\{U_i \subset X\}$ such that their union (in $X$) is $U$: $\cup_i U_i = U$.

  This is the standard choice of coverage on $Op(X)$. Sheaves for this coverage are the usual notion of sheaf on a topological space.

  A **[[basis for the topology]]** on $X$ is also a coverage on $X$, generating the same Grothendieck topology but in general not being closed under pullbacks (which in $Op(X)$ is intersection of open subsets). Notice that thence a basis for a topology on $X$ is not what is called a [[basis for a Grothendieck topology]] on $Op(X)$.

* Similarly on (any [[small category|small]] version of) the category [[Top]] or [[Diff]] or similar categories of topological spaces possibly with extra [[stuff, structure, property|structure]], [[open cover]]s form a coverage.

  Another choice of coverage is given by taking covering families to
  consist of [[étale map]]s, i.e. of local [[homeomorphism]]s. Notice that
  every open cover $\{U_i \to X\}$ consist of local homeomorphisms and
  in addition gives the local homeomorphism
  out of the [[coproduct]] $\coprod_i U_i \to X$.

* On [[Diff]] also
  [[good open cover]]s form an equivalent coverage.
  While good open covers are not stable under pullback in [[Diff]],
  every pullback of a good open cover gives an open cover that may be
  refined by a good open cover. This is all we need in the definition of
  coverage.

* There are many interesting coverages on the category of [[scheme|schemes]]; it was these examples which originally motivated Grothendieck to consider the notion. See [[fpqc topology]], etc.

* On any [[category]] there is the _[[trivial coverage]]_ which has no covering families at all.  Every presheaf is a sheaf for this coverage (and in particular, it is subcanonical).  The corresponding Grothendieck coverage consists of all sieves that contain a [[split epimorphism]].  (Note that every presheaf is a sheaf for any family containing a split epic.)

* On any [[regular category]] there is a coverage, called the _regular coverage_, whose covering families are the singletons $\{f:V\to U\}$ where $f$ is a [[regular epimorphism]].  It is subcanonical.

* On any [[coherent category]] there is a a coverage, called the _coherent coverage_, whose covering families are the finite families $\{f_i:U_i \to U\}_{1\le i\le n}$ the union of whose images is all of $U$.  It is subcanonical.  Likewise there is a _geometric coverage_ on any infinitary-coherent category.

* On any [[extensive category]] there is a coverage, called the _extensive coverage_, whose covering families are the inclusions into a (finite) coproduct.  It is subcanonical.  The coherent coverage on an extensive coherent category is generated by the union of the regular coverage and the extensive one.

* Any category has a _[[canonical coverage]]_, defined to be the largest subcanonical one.  (Hence the name "subcanonical" = "contained in the canonical coverage.")  The covering sieves for the canonical coverage are precisely those which are **universally effective-epimorphic**, meaning that their target is their colimit and this colimit is preserved by pullback.

* The canonical coverage on a [[Grothendieck topos]] coincides with its geometric coverage, and moreover every sheaf for this coverage is representable.  That is, a Grothendieck topos is a ([[large site|large]]) site which is equivalent to its own category of sheaves.

## Applications

In addition to the construction of [[sheaf|sheaves]] and [[stack|stacks]], other (not unrelated) applications of coverages include:

* The definition of internal [[anafunctor|anafunctors]].

* The construction of [[folk model structure|model structures for internal categories]].

## In higher category theory

The notion of [[site]] and hence that of [[Grothendieck topology]], [[Grothendieck pretopology]] and coverage typically has its straightforward analogs in [[higher category theory]].

in [[(∞,1)-category theory]] the corresponding notion is that of [[(∞,1)-site]]. Such an $(\infty,1)$-site has correspondingly its [[(∞,1)-category of (∞,1)-sheaves]]. A discussion of a [[model category]] [[presentable (∞,1)-category|presentation]] of this in terms of localization at a coverage is at [[model structure on simplicial presheaves]] in the section Localization at a coverage.

## Related concepts

* [[open cover]], [[good open cover]]

* [[closed cover]]

* [[quasi-net]]

## References

* [[Peter Johnstone]], _[[Elephant|Sketches of an Elephant]]_ , especially section C2.1.

* [[Anders Kock]], _[Postulated colimits and left exactness of Kan-extensions](http://home.imf.au.dk/kock/postulated.pdf)_

[[!redirects coverages]]


covering

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

* [[effective epimorphism]] $\Rightarrow$ [[regular epimorphism]] $\Leftrightarrow$ [[covering]]

* [[effective monomorphism]] $\Rightarrow$ [[regular monomorphism]] $\Leftrightarrow$ [[embedding]] .

\linebreak

The term “covering” may refer to:

1. in [[topology]]:

   [[covers]], such as [[open covers]] or [[covering spaces]];

   in which case "covering map" means: "[[deck transformation]]"

1. more generally in [[sheaf and topos theory]]:

   [[covering families]] of a [[coverage]] or [[Grothendieck topology]]

[[!redirects coverings]]

[[!redirects covering map]]
[[!redirects covering maps]]


covering dimension

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[paracompact topological space]] $X$ has **covering dimension** $\leq n \in \mathbb{N}$ if for every [[open cover]] $\{U_i \to X\}$ there exists an open refinement $\{V_i \to X\}$, such that each $(n+1)$-fold intersection of pairwise distinct $V_i$ is empty

$$
 V_{i_1} \cap \cdots \cap V_{i_{n+1}} = \emptyset
  \,.
$$

=--

## Properties

\begin{theorem}

If the [[paracompact topological space]] $X$ has covering dimension $\leq n$, then the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ is an [[(∞,1)-topos]] of [[homotopy dimension]] $\leq n$.

\end{theorem}

This is [[Higher Topos Theory|HTT, theorem 7.2.3.6]].

\begin{remark}
  For [[separable metric spaces]] the notion of covering dimension is particularly well-behaved. See [there](separable+metric+space#PropertiesDimension).
\end{remark}

## Related concepts

[[!include notions of dimension -- contents]]

[[!redirects covering dimensions]]


covering lifting property

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

## Idea

The _covering lifting property_ on a [[functor]] between [[sites]] is a sufficient condition for it to induce a [[geometric morphism]] between the corresponding [[sheaf toposes]] _covariantly_, i.e. with [[direct image]] going in the same direction. Sometimes, one calls such a functor _cocontinuous_, _cover-reflecting_ (e.g., the [[Elephant]]) or a _comorphism of sites_.

(As opposed to a [[morphism of sites]], also known as a _continuous_ functor, which induces a geometric morphism contravariantly, going the other way around.)

## Definition

+-- {: .num_defn #clp}
###### Definition
**(covering lifting property)**

Let $\mathcal{C}$ and $\mathcal{D}$ be [[sites]]. A [[functor]]

$$
  F \;\colon\; \mathcal{C} \to \mathcal{D}
$$

is said to have the _covering lifting property_  if for every object $U \in \mathcal{C}$ and every [[cover]] $\{q_i \colon V_i \to F(U)\}$ of $F(U) \in \mathcal{D}$, there is a cover $\{p_j \colon U_j \to U\}$ of $U \in \mathcal{C}$ such that $\{f(U_j) \overset{f(p_j)}{\to} f(U)\}$ refines $\{ V_i \overset{q_i}{\to} f(U)\}$ (i.e., if every cover of the image of any $U$ under $f$ is refined by the image of a cover of $U$).

=--

([MacLane-Moerdijk, p. 410](#MacLaneMoerdijk))

## Properties

+-- {: .num_prop #Smooth0TypeIsSheavesOnSmoothMfd}
###### Proposition

A functor between sites with covering lifting property (Def. \ref{clp}) induces a [[geometric morphisms]] between the corresponding [[sheaf toposes]] covariantly,

$$
  Sh(\mathcal{C})
    \underoverset
      {\underset{F_{\ast}}{\longrightarrow}}
      {\overset{L \circ F^\ast}{\longleftarrow}}
      {\bot}
  Sh(\mathcal{D})
$$

with [[inverse image]] given by [[composition|pre-composition]] $F^\ast$ with $F$ followed by [[sheafification]] $L \,\colon\, PSh(\mathcal{C}) \longrightarrow Sh(\mathcal{C})$.

=--

([MacLane-Moerdijk, Theorem VII.10.5](#MacLaneMoerdijk), [The Elephant, Proposition C2.3.18](#Elephant)).

## References

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

* {#Elephant} [[Peter Johnstone]], [[Elephant|Sketches of an elephant: a topos theory compendium]]

[[!redirects cover-reflecting]]
[[!redirects comorphism of sites]]


covering of alternating group

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

#Contents#
* table of contents
{:toc}

## Idea

The universal [[group extension]] of [[alternating groups]] $A_n$.

## Properties

These are [[double covers]] except for the alternating groups of degree $n = 6$ and 7 where the covers are 6-fold.

## Examples

* for $A_5$: [[binary icosahedral group]]

## Related concepts

* [symmetric group -- Whitehead tower](permutation#WhietheadTowerAndSupersymmetry)

## Reference

* Wikipedia, _[Covering groups of the alternating and symmetric groups](https://en.wikipedia.org/wiki/Covering_groups_of_the_alternating_and_symmetric_groups)_

[[!redirects coverings of alternating group]]
[[!redirects coverings of alternating groups]]


covering relation

# The covering relation
* table of contents
{: toc}

+-- {: .standout}
Warning: Tentative.
=--

## Idea

The _covering relation_ on a structure (generally already equipped with other relations) is a [[binary relation]] such that $x$ is related to $y$ if and only if $y$ is (in an appropriate sense) an *immediate* (and *only* immediate) successor of $x$.

## In a poset

A pair $(x,y)$ in a [[poset]] satisfies the __covering relation__ if $x \lt y$ but there is no $z$ such that $x \lt z$ and $z \lt y$. In other words, the [[interval]] $[x,y] = \{z \mid x \leq z \leq y\}$ contains exactly two elements $x$ and $y$.  In this case, you would say that "$y$ covers $x$".

## In a directed graph

A pair $(x,y)$ of vertices in a [[directed graph]] or [[quiver]] satisfies the __covering relation__ if there is an edge $x \to y$ but there is no other [[path]] from $x$ to $y$.

## Common generalisation

Given any binary relation $\sim$ on a set $S$, a pair $(x,y)$ of elements of $S$ satisfies the __covering relation__ if the only sequence $x = z_0, \ldots, z_n = y$ such that $x_i \sim x_{i+1}$ satisfies $n = 1$ (so $x \sim y$).  Then the covering relation on a poset is the covering relation of $\leq$, and the covering relation in a directed graph is the covering relation of the adjacency relation of the graph.

## References

* [Wikipedia](http://en.wikipedia.org/wiki/Covering_relation)

[[!redirects covering relation]]
[[!redirects covering relations]]


covering sieve

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

#Contents#
* table of contents
{:toc}

## Definition

A [[sieve]] which is [[covering]].

## Related concepts

* [[local epimorphism]]

[[!redirects covering sieves]]


covering space

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

#Contents#
* table of contents
{:toc}

## Idea

A **covering space** (or **wrapping space**) is a [[bundle]] $p: E \to B$ in which is locally trivial and with [[discrete space|discrete]] fiber.

## Definition

A [[continuous function]] $p: E \to B$ is a covering space over $B$ if for each point $x \in B$, there exists an [[open neighborhood]] $U$ of $x$ which is **evenly covered** by $p$ in that the [[pullback]] of $p$ over $U$ is isomorphic to a product bundle with discrete fiber $E_x = p^{-1}(x)$:

$$\array{
U \times E_x & \cong & p^{-1}(U) & \to & E\\
& \pi \searrow & \downarrow & & \downarrow p\\
& & U & \hookrightarrow & B
}$$

(the square is a pullback and the isomorphism maps $(x, e \in E_x) \mapsto e$).

Covering spaces over $B$ form an evident [[full subcategory]] $Cov/B \hookrightarrow Top/B$ of the [[slice category]] of [[Top]] over $B$, the _[[category of covering spaces]]_. These may be put together to form a [[replete subcategory|replete]], [[wide subcategory|wide]] subcategory $Cov$ of $Top$, so that $Cov/B$ is an [[over category]], just as the notation suggests.

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

Different points in $B$ may have non-isomorphic fibers. However, if open sets $U$ and $V$ are evenly covered by $p: E \to B$ and have nonempty intersection, then there are canonical identifications
$$E_x \cong E_z \cong E_y$$
between typical fibers over $x \in U$, $y \in V$, and $z \in U \cap V$. If $B$ is path-connected, then all the fibers match up to isomorphism (by the unique path-lifting lemma (lemma \ref{CoveringSpacePathLifting} below)).

=--

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

Fibers of a covering space may be empty. Some authors choose to forbid that, adding the condition that $p$ be a surjection, but the resulting category of covering projections over a space $B$ is not as nice as it would be without that condition.

=--

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

The terms "covering space" and "covering projection", while traditional, are certainly not optimal: they mislead by being too close to (open) "[[cover|coverings]]". [[James Dolan]] has suggested "wrapping space" as an alternative (as in the image of thread wrapping around a spool, to evoke the archetypal example of a covering projection, $p: \mathbb{R} \to S^1: x \mapsto \exp(i x)$).

=--

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

There is a generalization to "semi-coverings" ([Brazas12](#Brazas12)). Semicoverings satisfy the "[[2 out of 3]] rule". I.e,, if $f=g h$ and two of $f,g,h$ are semicoverings , then so is the third. This is not true for covering maps.

=--

## Properties

### Basic properties
 {#BasicProperties}

+-- {: .num_prop #CoveringProjectionsAreOpenMaps}
###### Proposition
**(covering projections are [[open maps]])**

If $p \colon E \to X$ is a covering space projection, then $p$ is an [[open map]].

=--

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

By definition of covering space there exists an [[open cover]] $\{U_i \subset X\}_{i \in I}$ and [[homeomorphisms]] $p^{-1}(U_i) \simeq U_i \times Disc(F_i)$ for all $i \in I$. Since the [[projections]] out of a [[product topological space]] are [[open maps]] ([this prop.](product+topological+space#ProjectionsAreOpenMaps)), it follows that $p$ is an open map when restricted to any of the $p^{-1}(U_i)$. But a general open subset $W \subset E$ is the union of its restrictions to these subspaces:

$$
  W = \underset{i \in I}{\cup} (W \cap p^{-1}(U_i))
  \,.
$$

Since images preserve unions ([this prop.](interactions+of+images+and+pre-images+with+unions+and+intersections#PreservationOfUnionsAndIntersectionsOfSets)) it follows that

$$
  p(W) = \underset{i \in I}{\cup} p(W \cap p^{-1}(U_i))
$$

is a union of open sets, and hence itself open.

=--

+-- {: .num_lemma #CoveringSpaceFiberwiseDiagonalOpenAndClosed}
###### Lemma
**([[fiber product|fiber-wise]] [[diagonal]] of covering space is [[open subset|open]] and [[closed subset|closed]])**

Let $E \overset{p}{\to} X$ be a covering space. Consider the [[fiber product]]

$$
  E \times_X E \coloneqq \{ (e_1, e_2) \in E \times E \;\vert\; p(e_1) = p(e_2) \}
$$

hence (by the discussion at [Top - Univeral constructions](Top#UniversalConstructions)) the [[topological subspace]] of the [[product space]] $E \times E$, as shown on the right. By the [[universal property]] of the [[fiber product]], there is the [[diagonal]] [[continuous function]]

$$
  \array{
    E &\longrightarrow& E \times_X E
    \\
    e &\mapsto& (e,e)
  }
  \,.
$$

Then the [[image]] of $E$ under this function is an [[open subset]] and a [[closed subset]]:

$$
  \Delta(E) \subset E \times_X E
  \;\;\;
  \text{is open and closed}
  \,.
$$

=--

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

First to see that it is an open subset. It is sufficient to show that for any $e \in E$ there exists an open neighbourhood of $(e,e) \in E \times_X E$.

Now by definition of covering spaces, there exists an open neighbourhood $U_{p(e)} \subset X$ of $p(e) \in X$ such that

$$
  \array{
    U_{p(e)} \times Disc(p^{-1}(p(e)))
      && \overset{\simeq}{\longrightarrow} && E\vert_{U_{p(e)}}
    \\
    & {}_{\mathllap{pr_1}}\searrow && \swarrow_{\mathrlap{p}}
    \\
    && U_{p(e)}
  }
  \,.
$$

It follows that $U_{p(e)} \times \{e\} \subset E$ is an open neighbourhood. Hence by the nature of the [[product topology]], $U_{p(e)} \times U_{p(e)} \subset E \times E$ is an open neighbourhood of $(e,e)$ in $E \times E$ and hence by the nature of the [[subspace topology]] the restriction

$$
  (E \times_X E) \cap (U_{p(e)} \times U_{p(e)}) \subset E \times_X E
$$

is an open neighbourhood of $(e,e)$ in $E \times_X E$.

Now to see that the diagonal is closed, hence that the complement $(E \times_X E) \setminus \Delta(E)$ is an open subset, it is sufficient to show that every point $(e_1, e_2)$ with $e_1 \neq e_2$ but $p(e_1) = p(e_2)$ has an open neighbourhood in this complement.

As before, there is an open neighbourhood $U \subset X$ of $p(e_1) = p(e_2)$ over which the cover trivializes, and hence $U \times \{e_1\}, U \times \{e_2\} \subset E$ are open neighbourhoods of $e_1$ and $e_2$, respectively. These are disjoint by the assumption that $e_1 \neq e_2$. As above, this means that the intersection

$$
  (E \times_X E) \cap ( (U \times \{e_1\}) \times (U \times \{e_2\}) )
  \subset
  (E \times_X E) \setminus \Delta(E)
$$

is an open subset of the complement of the diagonal in the fiber product.

=--

### Relation to étale spaces

Every covering space (even in the more general sense not requiring any connectedness axiom) is an [[etale space]], but not vice versa:

* for a covering space the inverse image of some open set in the base $B$ needs to be, by the definition, a disjoint union of homeomorphic open sets in $E$; however the 'size' of the neighborhoods over various $e$ in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.

* even if the stalks of the etale space are finite, it need not be locally trivial. For instance the disjoint union $\coprod_i Ui$ of a collection of open subsets of a space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is etale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets $U_i$ that contain this particular point.

### Lifting properties
 {#LiftingProperties}

We discuss [[left lifting properties]] satisfied by covering spaces.

1. [path-lifting property](#CoveringSpacePathLifting),

1. [homotopy-lifting propery](#HomotopyLiftingPropertyOfCoveringSpaces),

1. the [lifting theorem](#TheTheoremLifting).

+-- {: .num_lemma #LiftsOverConnectedSpaceIntoCoveringSpaceAreUniqueRelativePoint}
###### Lemma
**([[lifts]] out of [[connected topological space|connected space]] into [[covering spaces]] are unique relative to any point)**

Let

1. $E \overset{p}{\to} X $ be a covering space,

1. $Y$ a [[connected topological space]]

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

1. $\hat f_1, \hat f_2 \;\colon\; Y \longrightarrow E$ two [[lifts]] of $f$, in that the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       && E
       \\
       & {}^{\mathllap{\hat f_i}}\nearrow & \downarrow^{\mathrlap{p}}
       \\
       Y &\underset{f}{\longrightarrow}& X
     }
   $$

   for $i \in \{1,2\}$.

If there exists $y \in Y$ such that $\hat f_1(y) = \hat f_2(y)$ then the two lifts already agree everywhere: $\hat f_1 = \hat f_2$.

=--

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

By the [[universal property]] of the [[fiber product]]

$$
  E \times_X E \coloneqq \left\{ (e_1, e_2) \in E \times E \;\vert\; p(e_1) = p(e_2) \right\} \subset E \times E
$$

the two lifts determine a single continuous function of the form

$$
  (\hat f_1, \hat f_2)
    \;\colon\;
  Y \longrightarrow E \times_X E
  \,.
$$

Write

$$
  \Delta(E)  \coloneqq \left\{ (e,e) \in E \times_X E \;\vert\; e \in E  \right\}
$$

for the [[diagonal]] on $E$ in the fiber product. By lemma \ref{CoveringSpaceFiberwiseDiagonalOpenAndClosed} this is an open subset and a closed subset of the fiber product space. Hence by continuity of $(\hat f_1, \hat f_2)$ also its pre-image

$$
  (\hat f_1, \hat f_2)^{-1}(\Delta(E)) \subset Y
$$

is both closed and open, hence also its [[complement]] is open in $Y$.

Moreover, the assumption that the functions $\hat f_1$ and $\hat f_2$ agree in at least one point means that the above pre-image is [[inhabited set|non-empty]]. Therefore the assumption that $Y$ is [[connected topological space|connected]] implies that this pre-image coincides with all of $Y$. This is the statement to be proven.

=--

+-- {: .num_lemma #CoveringSpacePathLifting}
###### Lemma
**(path lifting property)**

Let $p \colon E \to X$ be any [[covering space]].  Given

1. $\gamma \colon [0,1] \to X$ a [[path]] in $X$,

1. $\hat x_0 \in E$ be a lift of its starting point, hence such that $p(\hat x_0) = \gamma(0)$

then there exists a unique path $\hat \gamma \colon [0,1] \to E$ such that

1. it is a lift of the original path: $p \circ \hat \gamma = \gamma$;

1. it starts at the given lifted point: $\hat \gamma(0) = \hat x_0$.

In other words, every [[commuting diagram]] in [[Top]] of the form

$$
  \array{
    \{0\} &\overset{\hat x_0}{\longrightarrow}& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    [0,1] &\underset{\gamma}{\longrightarrow}& X
  }
$$

has a unique [[lift]]:

$$
  \array{
    \{0\} &\overset{\hat x_0}{\longrightarrow}& E
    \\
    \downarrow
      &{}^{\mathllap{\hat \gamma}}\nearrow&
    \downarrow^{\mathrlap{p}}
    \\
    [0,1] &\underset{\gamma}{\longrightarrow}& X
  }
  \,.
$$

=--

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

First consider the case that the covering space is trival, hence of the [[Cartesian product]] form

$$
  pr_1 \;\colon\;  X \times Disc(S) \longrightarrow X
  \,.
$$

By the [[universal property]] of the [[product topological spaces]]
in this case a lift $\hat \gamma \colon [0,1] \to X \times Disc(S)$ is equivalently a [[pair]] of continuous functions

$$
  pr_1(\hat \gamma) \colon [0,1] \to X  \phantom{AAAA} pr_2(\hat \gamma) \colon [0,1] \to Disc(S)
  \,,
$$

Now the lifting condition explicitly fixes $pr_1(\hat \gamma) = \gamma$. Moreover, a continuous function into a [[discrete topological space]] $Disc(S)$ is [[locally constant function|locally constant]], and since $[0,1]$ is a [[connected topological space]] this means that $pr_2(\hat \gamma)$ is in fact a [[constant function]] ([this example](connected+space#LocallyConstantFunctionsOnConnectedSpaces)), hence uniquely fixed to be $pr_2(\hat \gamma) = \hat x_0$.

This shows the statement for the case of trivial covering spaces.

Now consider any covering space $p \colon E \to X$. By definition of covering spaces, there exists for every point $x \in X$ a [[open neighbourhood]] $U_x \subset X$ such that the restriction of $E$ to $U_x$ becomes a trivial covering space:

$$
  p^{-1}(U_x) \simeq U_x \times Disc(p^{-1}(x))
  \,.
$$

Consider such a choice

$$
  \{U_x \subset X\}_{x \in X}
  \,.
$$

This is an [[open cover]] of $X$. Accordingly, the [[pre-images]]

$$
  \left\{
    \gamma^{-1}(U_x)
    \subset
    [0,1]
  \right\}_{x \in X}
$$

constitute an open cover of the [[topological interval]] $[0,1]$.

Now the [[closed interval]] is a [[compact topological space]], so that this cover has a finite open subcover. By the [[Euclidean space|Euclidean]] [[metric topology]], each element in this finite subcover is a disjoint union of open intervals. The collection of all these open intervals is an open refinement of the original cover, and by compactness it once more has a finite subcover, now such that each element of the subcover is guaranteed to be a single open interval.

This means that we find a [[finite number]] of points

$$
  t_0 \lt t_1 \lt \cdots \lt_{n+1} \in [0,1]
$$

with $t_0 = 0$ and $t_{n+1} = 1$ such that for all $0 \lt j \leq n$ there is $x_j \in X$ such that the corresponding path segment

$$
  \gamma([t_j, t_{j+1}]) \subset X
$$

is contained in $U_{x_j}$ from above.

Now assume that $\hat \gamma\vert_{[0,t_j]}$ has been found. Then by the triviality of the covering space over $U_{x_j}$ and the first argument above, there is a unique lift of $\gamma\vert_{[t_j, t_{j+1}]}$ to a continuous function $\hat \gamma|_{[t_j,t_{j+1}]}$ with starting point $\hat \gamma(t_j)$. Since $[0,t_{j+1}]$ is the [[pushout]] $[0,t_j] \underset{\{t_j\}}{\sqcup} [t_j,t_{j+1}]$ ([this example](Top#TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself)), it follows that $\hat \gamma|_{[0,t_j]}$ and $\hat \gamma\vert_{[t_j,t_{j+1}]}$ uniquely glue to a continuous function $\hat \gamma\vert_{[0,t_{j+1}]}$ which lifts $\gamma\vert_{[0,t_{j+1}]}$.

By [[induction]] over $j$, this yields the required lift $\hat \gamma$.

Conversely, given any lift, $\hat \gamma$, then its restrictions $\hat \gamma\vert_{[t_j, t_{j+1}]}$ are uniquely fixed by the above inductive argument. Therefore also the total lift is unique. Altrnatively, uniqueness of the lifts is a special case of lemma \ref{LiftsOverConnectedSpaceIntoCoveringSpaceAreUniqueRelativePoint}.

=--

+-- {: .num_prop #HomotopyLiftingPropertyOfCoveringSpaces}
###### Proposition
**([[homotopy lifting property]] of [[covering spaces]] against [[locally connected topological spaces|locally connected spaces]])**

Let

1. $E \overset{p}{\to} X$ be a covering space;

1. $Y$ a topological space.

Then every [[lifting problem]] of the form

$$
  \array{
    Y &\overset{\hat f}{\longrightarrow}& E
    \\
    {}^{\mathllap{ (id_y, const_0)}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    Y \times [0,1] &\underset{\eta}{\longrightarrow}& X
  }
$$

has a unique [[lift]]

$$
  \array{
    Y &\overset{\hat f}{\longrightarrow}& E
    \\
    {}^{\mathllap{ (id_y, const_0)}}\downarrow
     &{}^{\mathllap{\hat \eta}}\nearrow&
    \downarrow^{\mathrlap{p}}
    \\
    Y \times [0,1] &\underset{\eta}{\longrightarrow}& X
  }
  \,.
$$

=--

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

Let $\{U_i \subset X\}_{i \in I}$ be an open cover over which the covering space trivializes. Then the pre-images $\{\eta^{-1}(U_i) \subset Y \times [0,1]\}_{i \in I}$ is an open cover of the product space. By nature of the [[product space|product space topology]] and the [[Euclidean topology]] on $[0,1]$, each of the $\eta^{-1}(U_i)$ is a union of Cartesian products $V_j \times I_j$ with $V_i \subset Y$ an open subset of $Y$ and $I_i \subset [0,1]$ an interval. Hence there is an open cover of the form

$$
  \{
    V_j \times I_j \subset Y \times [0,1]
  \}_{j \in J}
$$

with the property that for each $j$ there exists $i \in I$ with $\eta(V_j \times I_j) \subset U_i$.

Now by the fact that $[0,1]$ is a [[compact topological space]], for each $y \in Y$ there exists a finite set $K_y \subset J$ such that

$$
  \{ V_k \times I_k \}_{k \in K_y \subset K}
$$

still restricts to a cover of $\{y\} \times [I]$. Since $K$ is finite, the intersection

$$
  V_y \;\coloneqq\; \underset{k \in K_y}{\cap}
$$

is still open, and so also

$$
  \{ V_y \times I_k \}_{k \in K_y}
$$

still restricts to a cover of $\{y\} \times [0,1]$.

This means that the same argument as for the path lifting in lemma \ref{CoveringSpacePathLifting} provides a unique lift $\widehat{\eta\vert_{V_y \times [0,1]}}$ for each $y \in Y$.

Moreover, for $y_1, y_2 \in Y$ two points, these lifts clearly have to agree on $V_{y_1} \cap V_{y_2}$.

Since $\{V_y \times [0,1] \subset Y \times [0,1]\}_{y \in Y}$ is an open cover, means that there is a unique function $\hat \eta$ that restricts to all these local lifts ([this prop](Top#ClosedSubspacesGluing)). This is the required lift.

=--

+-- {: .num_remark #CoveringSpacesAreSerreFibrationsButNotInGeneralHurewiczFibrations}
###### Remark
**(covering spaces are [[Hurewicz fibrations]])**

Continuous functions with the [[right lifting property]]
against functions of the form $Y \overset{(id,const_0)}{\longrightarrow} Y \times [0,1]$ are called _[[Hurewicz fibrations]]_.

Hence prop. \ref{HomotopyLiftingPropertyOfCoveringSpaces} says that
covering projections are in particular Hurewicz fibrations.

=--

+-- {: .num_example #CoveringSpacesHomotopyLifting}
###### Example
**(lift homotopy of paths for given lifts of paths)**

Let $p \colon E \to X$ be a [[covering space]]. Then given a [[homotopy]] relative the starting point between two [[paths]] in $X$,

$$
  \eta \;\colon\; \gamma_1 \Rightarrow \gamma_2
$$

there is for every lift $\hat \gamma_1, \hat \gamma_2$ of these two paths to paths in $E$ with the same starting point a unique homotopy

$$
  \hat \eta \;\colon\; \hat \gamma_1 \Rightarrow \hat \gamma_2
$$

between the lifted paths that lifts the given homotopy:

For [[commuting squares]] of the form

$$
  \array{
    ([0,1] \times \{0\}) \cup (\{0,1\} \times [0,1])
      &\overset{(\gamma_1, \gamma_2)}{\longrightarrow}&
    E
    \\
    {}^{\mathllap{}}\downarrow
      &{}^{\mathllap{\hat \eta}}\nearrow&
    \downarrow^{\mathrlap{p}}
    \\
    [0,1] \times [0,1] &\underset{\eta}{\longrightarrow}& X
  }
$$

there is a unique diagonal [[lift]] in the lower diagram, as shown.

Moreover if the homotopy $\eta$ also fixes the endpoint, then so does the lifted homotopy $\hat \eta$.

=--

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

There are horizontal [[homeomorphisms]] such that the following diagram commutes

$$
  \array{
     [0,1]
       &\overset{\simeq}{\longrightarrow}&
     ([0,1] \times \{0\}) \cap ( \{0,1\} \times [0,1] )
     \\
     \downarrow
     &&
     \downarrow
     \\
     [0,1] \times [0,1] &\underset{\simeq}{\longrightarrow}& [0,1] \times [0,1]
  }
  \,.
$$

With this the statement follows from \ref{HomotopyLiftingPropertyOfCoveringSpaces}.

=--

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

Let $(E,e) \overset{p}{\longrightarrow} (X,x)$ be a [[pointed topological space|pointed]] [[covering space]]
and let $f \colon (Y,y) \longrightarrow (X,x)$ be a point-preserving [[continuous function]] such that the image of the [[fundamental group]] of $(Y,y)$ is contained within the image of the fundamental group of $(E,e)$ in that of $(X,x)$:

$$
  f_\ast(\pi_1(Y,y)) \subset p_\ast(\pi_1(E,e))
  \phantom{AA}
  \subset
  \pi_1(X,x)
  \,.
$$

Then for $\ell_Y$ a [[path]] in $(Y,y)$ that happens to be a [[loop]], every lift of its image path $f \circ \ell$ in $(X,x)$ to a path $\widehat{f\circ \ell_Y}$ in $(E,e)$ is also a loop there.

=--

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

By assumption, there is a loop $\ell_E$ in $(E,e)$ and a homotopy fixing the endpoints of the form

$$
  \eta_{X} \;\colon\; p \circ \ell_E \Rightarrow f\circ \ell_Y
  \,.
$$

Then by the homotopy lifting property as in example \ref{CoveringSpacesHomotopyLifting}, there is a homotopy in $(E,e)$ fixing the starting point, of the form

$$
  \eta_{E} \;\colon\; \ell_E \Rightarrow \widehat{f \circ \ell_Y}
$$

and lifting the homotopy $\eta_X$. Since $\eta_X$ in addition fixes the endpoint, the uniqueness of the path lifting (lemma \ref{CoveringSpacePathLifting}) implies that also $\eta_{E}$ fixes the endpoint. Therefore $\eta_E$ is in fact a homotopy between loops, and so $\widehat{f \circ \ell_Y}$ is indeed a loop.

=--

+-- {: .num_prop #TheTheoremLifting}
###### Proposition
**(lifting theorem)**

Let

1. $p \colon E \to X$ be a [[covering space]];

1. $e \in E$ a point, with $x \coloneqq p(e)$ denoting its image,

1. $Y$ be a [[connected topological space|connected]] and [[locally path-connected topological space|locally path-connected]] [[topological space]];

1. $y \in Y$ a point

1. $f \colon (Y,y) \longrightarrow (X,x)$ a [[continuous function]] such that $f(y) = x$.

Then the following are equivalent:

1. There exists a unique lift $\hat f$ in the diagram

   $$
     \array{
       && (E,e)
       \\
       & {}^{\mathllap{\hat f}}\nearrow & \downarrow^{\mathrlap{p}}
       \\
       (Y,y) &\underset{f}{\longrightarrow}& (X,x)
     }
   $$

   of [[pointed topological spaces]].

1. The [[image]] of the [[fundamental group]] of $Y$ under $f$ in that of $X$ is contained in the image of the fundamental group of $E$ under $p$:

   $$
     f_\ast(\pi_1(Y,y)) \subset p_\ast( \pi_1(E,e) )
     \,
   $$

Moreover, if $Y$ is path-connected, then the lift in the first item is unique.

=--

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

The implication $1) \Rightarrow 2)$ is immediate. We need to show that the second statement already implies the first.

Since $Y$ is connected and locally path-connected, it is also a [[path-connected topological space]] ([this prop.](locally+path-connected+space#ForLocallyPathConnectedTheConnectedComponentsCoincideWithThePathConnectedComponents)). Hence for every point $y' \in Y$ there exists a [[path]] $\gamma$ connecting $y$ with $y'$ and hence a path $f \circ \gamma$ connecting $x$ with $f(y')$.  By the path-lifting property (lemma \ref{CoveringSpacePathLifting}) this has a unique lift

$$
  \array{
    \{0\} &\overset{e}{\longrightarrow}& E
    \\
    \downarrow &{}^{\mathllap{\widehat{f \circ \gamma}}}\nearrow& \downarrow^{\mathrlap{p}}
    \\
    [0,1]
     &\underset{f \circ \gamma}{\longrightarrow}&
    X
  }
  \,.
$$

Therefore

$$
  \hat f(y') \coloneqq \widehat{f \circ \gamma}(1)
$$

is a lift of $f(y')$.

We claim now that this pointwise construction is independent of the choice $\gamma$, and that as a function of $y'$ it is indeed continuous. This will prove the claim.

Now by the path lifting lemma \ref{CoveringSpacePathLifting} the lift $\widehat{\f \circ \gamma}$ is unique given $f \circ \gamma$, and hence $\hat f(y')$ depends at most on the choice of $\gamma$.

Hence let $\gamma' \colon [0,1] \to Y$ be another path in $Y$ that connects $y$ with $y'$. We need to show that then $\widehat{f \circ \gamma'} = \widehat{f \circ \gamma}$.

First observe that if $\gamma'$ is related to $\gamma$ by a [[homotopy]], so that then also $f \circ \gamma'$ is related to $f \circ \gamma$ by a homotopy, then this is the statement of the homotopy lifting property as in example \ref{CoveringSpacesHomotopyLifting}.

Next write $\bar\gamma'\cdot \gamma$ for the [[path concatenation]] of the path $\gamma$ with the [[reverse path]] of the path $\gamma'$, hence a loop in $Y$, so that $f \circ (\bar\gamma'\cdot \gamma)$ is a loop in $X$. The assumption that $f_\ast(\pi_1(Y,y)) \subset p_\ast(\pi_1(E,e))$ implies (example \ref{IfFundamentalGroupsIncludeThenfLoopsLiftToLoops}) that the path $\widehat{f \circ (\bar \gamma' \cdot \gamma)}$ which lifts this loop to $E$ is itself a loop in $E$.

By uniqueness of path lifting, this means that the lift of

$$
  f \circ ( \gamma' \cdot (\bar\gamma' \cdot \gamma) )
  =
  (f \circ \gamma') \cdot (f \circ (\bar \gamma' \cdot \gamma) )
$$

coincides
at $1 \in [0,1]$ with that of $f \circ \gamma'$ at 1. But $\gamma' \cdot (\bar \gamma' \cdot \gamma)$ is homotopic (via reparameterization) to just $\gamma$. Hence it follows now with the first statement that the lift of $f \circ \gamma'$ indeed coincides with that of $f \circ \gamma$.

This shows that the above prescription for $\hat f$ is well defined.

It only remains to show that the function $\hat f$ obtained this way is continuous.

Let $y' \in Y$ be a point and $W_{\hat f(y')} \subset E$ an open neighbourhood of its image in $E$. It is sufficient to see that there is an open neighbourhood $V_{y'} \subset Y$ such that $\hat f(V_y) \subset W_{\hat f(y')}$.

Let $U_{f(y')} \subset X$ be an open neighbourhood over which $p$ trivializes. Then the restriction

$$
  p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
    \;\subset\;
  U_{f(y')} \times Disc(p^{-1}(f(y')))
$$

is an open subset of the product space. Consider its further restriction

$$
  \left( U_{f(y')} \times \{\hat f(y')\} \right)
    \cap
  \left(
    p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
  \right)
$$

to the [[leaf]]

$$
  U_{f(y')} \times \{\hat f(y')\} \;\subset\; U_{f(y')} \times p^{-1}(f(y'))
$$

which is itself an open subset. Since $p$ is an [[open map]] ([this prop.](covering+space#CoveringProjectionsAreOpenMaps)), the subset

$$
  p\left(
    \left(
      U_{f(y')} \times \{\hat f(y')\}
    \right)
      \cap
    \left(
      p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
    \right)
  \right)
  \subset X
$$

is open, hence so is its pre-image

$$
  f^{-1}
  \left(
    p
      \left(
        \left(
          U_{f(y')} \times \{\hat f(y')\}
        \right)
          \cap
        \left(
          p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
        \right)
    \right)
  \right)
    \;\subset\;
  Y
  \,.
$$

Since $Y$ is assumed to be [[locally path-connected topological space|locally path-connected]], there exists a path-connected open neighbourhood

$$
  V_{y'}
    \subset
  f^{-1}\left(p\left(
    \left(U_{f(y')} \times \{\hat f(y')\}\right)
      \cap
    \left(
      p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
    \right)
  \right)
  \right)
  \,.
$$

By the uniqueness of pah lifting, the image of that under $\hat f$ is

$$
  \begin{aligned}
    \hat f(V_{y_'})
    & =
    f(V_{y'}) \times \{\hat f(y')\}
    \\
     & \subset
    p\left(
      \left(U_{f(y')} \times \{\hat f(y')\}\right)
        \cap
      \left(
        p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
      \right)
    \right) \times \{\hat f(y')\}
    \\
    & \simeq
      \left( U_{f(y')} \times \{\hat f(y')\} \right)
        \cap
      \left(
        p^{-1}(U_{f(y')}) \cap W_{\hat f(y')}
      \right)
     \\
     & \subset W_{\hat f(y')}
  \end{aligned}
  \,.
$$

This shows that the lifted function is continuous. Finally that this continuous lift is unique is the statement of lemma \ref{LiftsOverConnectedSpaceIntoCoveringSpaceAreUniqueRelativePoint}.

=--

### Monodromy

+-- {: .num_defn #CoveringSpaceMonodromy}
###### Definition
**([[monodromy]] of a covering space)

Let $X$ be a [[topological space]] and $E \overset{p}{\to} X$ a [[covering space]]. Write $\Pi_1(X)$ for the [[fundamental groupoid]] of $X$.

Define a [[functor]]

$$
  Fib_E
    \;\colon\;
  \Pi_1(X)
    \longrightarrow
  Set
$$

to the [[category]] [[Set]] of [[sets]] as follows:

1. to a point $x \in X$ assign the [[fiber]] $p^{-1}(\{x\}) \in Set$;

1. to the [[homotopy class]] of a [[path]] $\gamma$ connecting $x \coloneqq \gamma(0)$ with $y \coloneqq \gamma(1)$ in $X$ assign the function $p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\})$ which takes $\hat x \in p^{-1}(\{x\})$ to the endpoint of a path $\hat \gamma$ in $E$ which lifts $\gamma$ through $p$ with starting point $\hat \gamma(0) = \hat x$

   $$
     \array{
       p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y)
       \\
       (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1)
     }
     \,.
   $$

This construction is well defined for a given representative $\gamma$ due to the unique path-lifting property of covering spaces ([this lemma](covering+space#CoveringSpacePathLifting)) and it is independent of the choice of $\gamma$ in the given homotopy class of paths due to the homotopy-lifting property ([this lemma](covering+space#CoveringSpacesHomotopyLifting)).  Similarly, these two lifting properties give that this construction respects composition in $\Pi_1(X)$ and hence is indeed a [[functor]].

=--

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

Given a [[homomorphism]] between two [[covering spaces]] $E_i \overset{p_i}{\to} X$, hence a [[continuous function]] $f \colon E_1 \to E_2$ which respects [[fibers]] in that the [[diagram]]

$$
  \array{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
$$

[[commuting diagram|commutes]], then the component functions

$$
  f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})
$$

are compatible with the monodromy $Fib_{E}$ (def. \ref{CoveringSpaceMonodromy}) along any [[path]] $\gamma$ between points $x$ and $y$ from def. \ref{CoveringSpaceMonodromy} in that the following [[diagrams]] of [[sets]] [[commuting diagram|commute]]

$$
  \array{
     p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x)
     \\
     {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow
       &&
     \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }}
     \\
     p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\})
  }
  \,.
$$

This means that $f$ induces a [[natural transformation]] between the monodromy functors of $E_1$ and $E_2$, respectively, and hence that constructing monodromy is itself a functor from the [[category]] of [[covering spaces]] of $X$ to that of [[permutation representations]] of the [[fundamental groupoid]] of $X$:

$$
  Fib
   \;\colon\;
  Cov(X)
    \longrightarrow
  Set^{\Pi_1(X)}
  \,.
$$

=--

+-- {: .num_example #CoveringSpaceFundamentalGroupoid}
###### Example
**([[fundamental groupoid]] of covering space)

Let $E \overset{p}{\longrightarrow} X$ be a covering space.

Then the [[fundamental groupoid]] $\Pi_1(E)$ of the total space $E$ is
[[equivalence of categories|equivalently]] the [[Grothendieck construction]]
of the [[monodromy]] functor $Fib_E \;\colon\; \Pi_1(X) \to Set$

$$
  \Pi_1(E)
   \;\simeq\;
  \int_{\Pi_1(X)} Fib_E
$$

whose

* [[objects]] are pairs $(x,\hat x)$ consisting of a point $x \in X$ and en element $\hat x \in Fib_E(x)$;

* [[morphisms]] $[\hat \gamma] \colon (x,\hat x) \to (x', \hat x')$ are morphisms $[\gamma] \colon x \to x'$ in $\Pi_1(X)$ such that $Fib_E([\gamma])(\hat x) = \hat x'$.

=--

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

By the uniqueness of the path-lifting, lemma \ref{CoveringSpacePathLifting} and the very definition of the [[monodromy]] functor.

=--

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

Let $X$ be a [[path-connected topological space]] and let $E \overset{p}{\to} X$ be a [[covering space]]. Then the total space $E$ is

1. [[path-connected topological space|path-connected]] precisely if the [[monodromy]] $Fib_E$ is a [[transitive action]];

1. [[simply connected topological space|simply connected]] precisely if the [[monodromy]] $Fib_E$ is a [[transitive action|transitive]] and [[free action]].

=--

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

By example \ref{CoveringSpaceFundamentalGroupoid}.

=--

### Reconstructing covering spaces from monodromy

Given a [[permutation representation]] of the [[fundamental groupoid]]
of a locally path-connected and semi-locally simply connected space, there is a construction of _[[reconstructing covering spaces from monodromy]]_, which is a [[functor]] of the form

$$
  Rec \;\colon\; Set^{\Pi_1(X)} \longrightarrow Cov(X)
  \,.
$$

### Fundamental theorem of covering spaces
 {#FundamentalTheoremOfCoveringSpaces}

+-- {: .num_theorem #FundamentalTheoremOfCoveringSpaces}
###### Theorem
**([[fundamental theorem of covering spaces]])**

Let $X$ be a [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space]]. Then the operations on

1. extracting the [[monodromy]] $Fib_{E}$ of a [[covering space]] $E$ over $X$

1. [[reconstructing a covering space from monodromy]] $Rec(\rho)$

constitute an [[adjoint equivalence|adjoint]] [[equivalence of categories]]

$$
  Cov(X)
    \underoverset
      {\underset{Fib}{\longrightarrow}}
      {\overset{Rec}{\longleftarrow}}
      {\bot}
  Set^{\Pi_1(X)}
$$

between the [[category of covering spaces]] and the [[permutation representation|permutation]] [[groupoid representations]] of the [[fundamental groupoid]] of $X$.

=--

### Universal covering space

... _[[universal covering space]]_

## Examples
 {#Examples}

+-- {: .num_example #TrivialCoveringSpace}
###### Example
**(trivial covering space)**

For $X$ a [[topological space]] and $S$ a [[set]] with $Disc(S)$ the [[discrete topological space]]
on that set, then the [[projection]] out of the [[product topological space]]

$$
  pr_1 \;\colon\; X \times Disc(S) \longrightarrow X
$$

is a [[covering space]], called the _trivial covering space_ over $X$ with [[fiber]] $Disc(S)$.

If $E \overset{p}{\longrightarrow} X$ is any covering space, then an [[isomorphism]] of covering spaces
of the form

$$
  \array{
    E && \overset{\simeq}{\longrightarrow} && X \times Disc(S)
    \\
    & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{pr_2}}
    \\
    && X
  }
$$

is called a _trivialization_ of $E \overset{p}{\to} X$.

It is in this sense that evry coverin space $E$ is, by definition, locally trvializable.

=--

+-- {: .num_example #nSphereCoveringOverRealProjectiveSpace}
###### Example
**([[n-sphere]] covering over [[real projective space]])**

For $n \in \mathbb{N}$ the canonical projection

$$
  S^n \longrightarrow \mathbb{R}P^n
$$

from the [[n-sphere]] to the [[real projective space]] of [[dimension]] $n$ is a covering space projection. See [this prop.](real+projective+space#nSphereAsCoveringSpaceOverRealProjectiveSpace).

=--

### Universal covering space in terms of homotopy fibers {#HomotopyFibs}

We want to describe here how the universal covering space of $X$ is the [[homotopy fiber]] of the canonical morphism $X \to \Pi_1(X)$, hence the $\Pi_1(X)$-[[principal bundle]] classified by this [[cocycle]].

We place ourselves in the context of [[topological ∞-groupoid]]s and regard both the space $X$ as well as its [[schreiber:path ∞-groupoid]] $\Pi(X)$ and its truncation to the [[fundamental groupoid]] $\Pi_1(X)$ as objects in there.

The canonical morphism $X \to \Pi(X)$ hence $X \to \Pi_1(X)$ given by the inclusion of contant paths may be regarded as a [[cocycle]] for a $\Pi(X)$-[[principal ∞-bundle]], respectively for a $\Pi_1(X)$-principal bundle.

Let $\pi_0(X)$ be the set of connected components of $X$, regarded as a topological $\infty$-groupoid, and choose any [[section]]  $\pi_0(X) \to \Pi(X)$ of the projection $\Pi(X) \to \pi_0(X)$.

Then the $\Pi(X)$-principal $\infty$-bundle classified by the [[cocycle]] $X \to \Pi(X)$ is its [[homotopy fiber]], i.e. the [[homotopy pullback]]

$$
  \array{
    UCov(X) &\to& \pi_0(X)
    \\
    \downarrow && \downarrow
    \\
    X &\to& \Pi(X)
  }
  \,.
$$

We think of this topological $\infty$-groupoid $UCov(X)$ as the **universal covering $\infty$-groupoid** of $X$. To break this down, we check that its [[decategorification]] gives the ordinary universal covering space:

for this we compute the [[homotopy pullback]]

$$
  \array{
    UCov_1(X) &\to& {*}
    \\
    \downarrow && \downarrow^{\mathrlap{x}}
    \\
    X &\to& \Pi_1(X)
  }
  \,,
$$

where we assume $X$ to be connected with chosen baspoint $x$ just to shorten the exposition a little. By the laws of [[homotopy pullback]]s in general and [[homotopy fiber]]s in particular, we may compute this as the ordinary [[pullback]] of a weakly equivalent diagram, where the point $*$ is resolved to the [[generalized universal bundle|universal]] $\Pi_1(X)$-principal bundle

$$
  \mathbf{E}_x \Pi_1(X) = T_x \Pi_1(X)
  \,.
$$

> (More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the _global_ [[model structure on simplicial presheaves]] on [[Top]]. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to $\mathbf{E}_x \Pi_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\infty$-prestacks. Then by left exactness of $\infty$-stackification, the image of that in $\infty$-stacks is still a homotopy pullback. )

The topological groupoid $\mathbf{E}_x \Pi_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$  starting at $x$ and as morphisms $\kappa : \gamma \to \gamma'$ it has commuting triangles

$$
  \array{
    && x
    \\
    &{}^{\mathllap{\gamma}}\swarrow && \searrow^{\mathrlap{\gamma'}}
    \\
    y &&\stackrel{\kappa}{\to}&& y'
  }
$$

in $\Pi_1(X)$. The topology on this can be deduced from thinking of this as the [[pullback]]

$$
  \array{
   \mathbf{E}_x \Pi_1(X) &\to& {*}
   \\
   \downarrow && \downarrow^{\mathrlap{x}}
   \\
   \Pi_1(X)^I &\stackrel{d_0}{\to}& \Pi_1(X)
  }
$$

in simplicial presheaves on [[Top]]. Unwinding what this means we find that the open sets in $Mor(\mathbf{E}_x \Pi_1(X))$ are those where the endpoint evaluation produces an open set in $X$.

Now it is immediate to read off the homotopy pullback as the ordinary pullback

$$
  \array{
    UCov_1(X) &\to& \mathbf{E}_x \Pi_1(X)
    \\
    \downarrow && \downarrow
    \\
    X &\to& \Pi_1(X)
    \,.
  }
$$

Since $X$ is categorically discrete, this simply produces the space of objects of $\mathbf{E}_x \Pi_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection $UCov_1(X) \to X$ being endpoint evaluation.

This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in  ([Dahlke](#Dahlke))

## Related concepts

* [[deck transformation]]

* [[branched covering space]]

## References

A detailed treatment is available in

* [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer, 2016.  [doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8.

Textbook accounts:

* [[William S. Massey]], Chapter 5 of: *Algebraic Topology: An Introduction*, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 ([ISBN:978-0-387-90271-5](https://link.springer.com/book/9780387902715))

* {#tomDieck06} [[Tammo tom Dieck]], sections 2 an 3 of _Algebraic Topology_, EMS 2006 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/diecktop.pdf))

Review

* {#Moller11} [[Jesper Møller]], _The fundamental group and covering spaces_ (2011) ([pdf](http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf))

* [[Ronnie Brown]], chapter 11 of _[Topology and Groupoids](http://pages.bangor.ac.uk/~mas010/topgpds.html)_

* {#Dahlke} Karl Dahlke, _[Covering spaces, Building a universal cover](http://www.mathreference.com/at-cov,build.html)_ , [Math Reference Project](http://www.mathreference.com/)

Some of the problems of generalising covering spaces to deal with  wild spaces are dealt with in:

* {#Brazas12} Jeremy Brazas, _Semicoverings_, Homology, Homotopy and Applications, 14 (2012), No. 1, 33-63.

Discussion of ([[homotopy types]] of) covering spaces via [[homotopy type theory]]:

* [[Kuen-Bang Hou]], *Covering Spaces in Homotopy Type Theory*, extended abstract *Type Theory, Homotopy Theory and Univalent Foundations* (2013) $[$[doi:10.1007/978-3-319-21284-5_15](https://doi.org/10.1007/978-3-319-21284-5_15)$]$

* [[Kuen-Bang Hou]], [[Robert Harper]], *Covering Spaces in Homotopy Type Theory*, Leibniz International Proceedings in Informatics (LIPIcs) **97** (2018) $[$[doi:10.4230/LIPIcs.TYPES.2016.11](https://doi.org/10.4230/LIPIcs.TYPES.2016.11), [pdf](https://drops.dagstuhl.de/opus/volltexte/2018/9851/pdf/LIPIcs-TYPES-2016-11.pdf)$]$

and with emphasis on the [[(n-connected, n-truncated) factorization system|$n$-truncation]]-[[modal homotopy type theory]] involved:

* [[Felix Cherubini]], [[Egbert Rijke]], Def. 8.5 of: *Modal Descent*, Mathematical Structures in Computer Science **31** 4 (2020) $[$[arXiv:2003.09713](https://arxiv.org/abs/2003.09713), [doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201)$]$

[[!redirects covering spaces]]
[[!redirects wrapping space]]
[[!redirects wrapping spaces]]
[[!redirects covering groupoid]]


covert space

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

# Covert space
* table of contents
{: toc}

## Idea

A **covert space** is a [[topological space]] $X$ such that the unique [[map]] $X\to 1$ to the [[terminal object|terminal]] [[point space]] is a [[closed map]].

This should be compared with [[overt spaces]], where $X\to 1$ is an [[open map]], and [[compact spaces]], where $X\to 1$ is a [[proper map]].  Since proper maps are closed, all compact spaces are covert.

In [[classical mathematics]], every topological space is covert, just as every space is overt.  However, in [[constructive mathematics]], and in particular in [[synthetic topology]], covertness can be a strong condition that already carries much of the power of [[compact topological space|compactness]].

## Definition

A [[locale]] $X$ is **covert** if the unique map $X\to 1$ is a [[closed map]] of locales, i.e. the [[adjunction]] $f^* : O(1) \rightleftarrows O(X) : f_*$ satisfies the [[Frobenius reciprocity]] condition $f_*(U\cup f^*(V)) = f_*(U)\cup V$.

If [[excluded middle]] holds, then every locale is covert.

## Covert sets

Constructively, covertness is already interesting when $X$ is a set with the [[discrete topology]].  In this case, since $O(1) = P(1)$ is the poset of [[truth values]] (the [[subobject classifier]]) and $O(X)=P(X)$ is the [[powerset]] of $X$, covertness of $X$ can be expressed more logically as

> if $P$ is a [[proposition]] and $Q$ a [[predicate]] on $X$ such that $\forall x\in X, (P \vee Q(x))$, then $P\vee (\forall x\in X, Q(x))$.

This is always true if $X$ is [[finite set|(Kuratowski) finite]] (in which case the discrete topology on $X$ is not just covert but compact), but in general it can be quite a strong condition.

In [[synthetic topology]], covertness of sets carries much of the strength of compactness.  (Note that here we are talking about covertness of sets --- i.e. "spaces" in the sense of synthetic topology, i.e. basic objects of our foundational system --- equipped with the "discrete [[topology]]" internal to our synthetic-topological formal system; these are quite different from sets whose intrinsic *synthetic* topology is discrete.)  For instance, in various [[gros toposes]] of topological, smooth, or algebraic spaces, the covert spaces are precisely the compact ones; see [Dubuc-Penon](#DubucPenon) (who call covert sets "compact").

Covert sets also satisfy the following property.

+-- {: .un_theorem}
###### Theorem
If $X$ is covert, then for any two predicates $P,Q$ on $X$ such that $\forall x\in X, (P(x) \vee Q(x))$, we have either $\exists x\in X, P(x)$ or $\forall x\in X, Q(x)$.
=--
+-- {: .proof}
###### Proof
Let $P'$ be the proposition $\exists x\in X, P(x)$ in the defining property of covertness for $X$.
=--

Note that if $P$ and $Q$ are incompatible (i.e. $\neg (P(x)\wedge Q(x))$), then this becomes the [[limited principle of omniscience]] for $A$.  Thus, every covert set is [[omniscient set|omniscient]].

## Compact locales are covert

A locale $X$ is defined to be [[compact locale|compact]] if its top element is inaccessible by [[directed joins]], i.e. if $X = \bigvee S$ for some [[directed set]] $S\subseteq O(X)$ implies $X\le U$ for some $U\in S$.  Since the right adjoint part $r_*$ of the locale morphism $r:X\to 1$ is defined by $r_*(U) = (U=X)$ (a [[truth value]], since $O(1) = \Omega$ is the [[subobject classifier]]), this is equivalent to asking that $r_*$ preserve directed joins.

Contrary to a remark in [[Sketches of an Elephant]], this condition implies that $X$ is covert, even constructively (and therefore if $X$ is compact then $X\to 1$ is a [[proper map]], without the need to modify the notion of "compact locale").  For suppose $U\in O(X)$ and $V\in O(1) = \Omega$; we must show that if $r_\ast(U \cup r_\ast(V))$ is true, then so is $r_\ast(U) \cup V$.  As remarked above, $r_\ast(W)$ is the truth value of the statement "$W=X$", while $r^*(P) = \bigcup \{ X \mid P \} $.  Suppose $r_\ast(U \cup r_\ast(V))$, i.e. that $U\cup \bigcup \{ X \mid V \} = X$.  Now consider the set $\{ W\in O(X) \mid V \vee (W\le U) \}$; this is evidently directed, and our supposition in the last sentence says exactly that its union is $X$.  Therefore, if $X$ is compact, there exists a $W$ such that $V \vee (W\le U)$ and $X\le W$.  In other words, either $V$ or $X\le U$, i.e. either $V$ or $r_\ast(U)$, which is what we wanted.  This proof appears to be due to [Vermeulen](#Vermeulen).

## References

* {#DubucPenon} [[Eduardo Dubuc]] and [[Jacques Penon]], *Objets compacts dans les topos*,
  Journal of the Australian Mathematical Society. Series A.
  Pure Mathematics and Statistics, 40(2), 1986, p. 203–217.
  doi:[10.1017/S144678870002718X](https://dx.doi.org/10.1017/S144678870002718X)

* {#Vermeulen} Vermeulen, *Proper maps of locales*, Journal of Pure and Applied Algebra, 92, 1994, p. 79–107. ([pdf (core.ac.uk)](https://core.ac.uk/download/pdf/82053693.pdf))

[[!redirects covert spaces]]
[[!redirects covert sets]]
[[!redirects covert set]]


cowriter comonad

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

#Contents#
* table of contents
{:toc}

## Idea

Let $\mathcal{C}$ be a [[closed monoidal category]] with [[tensor product]] and [[internal hom]] denoted by

\begin{tikzcd}[row sep=4pt]
  \mathcal{C} \times \mathcal{C}
  \ar[r, "{ (\mbox{-})\otimes (\mbox{-})}"]
  &
  \mathcal{C}
  \\
  \mathcal{C}^{\mathrm{op}} \times \mathcal{C}
  \ar[r, "{ [\mbox{-},\,\mbox{-}]}"]
  &
  \mathcal{C}
  \mathrlap{\,,}
\end{tikzcd}

respectively, and consider a [[monoid object|monoid]] [[internalization|internal]] to $\mathcal{C}$:

$$
  A \,\in\, Mon\big(\mathcal{C}\big)
  \,.
$$

Recall that the "$A$-[[action monad]]" induced by $A$ -- often called the "$A$-[[writer monad]]" when considered as a [[monad in computer science]] -- is the [[endofunctor]] of forming the [[tensor product]] with $A$

$$
  A \otimes (\text{-})
  \;\colon\;
  \mathcal{C} \to \mathcal{C}
$$

equipped with unit and multiplication structure morphisms induced, under the tensor product, from the unit and multiplication on $A$, respectively.

The analogous construction but with the tensor product replaced by the [[internal hom]]

$$
  [A,\, \text{-}]
  \;\colon\;
  \mathcal{C} \to \mathcal{C}
$$

which in [[computer science]]/[[type theory]] tends to be written

$$
  (A \to \text{-})
  \;\colon\;
  \mathcal{C} \to \mathcal{C}
  \,,
$$

gives a [[comonad]] which hence deserves to be called the *coaction comonad* or *cowriter comonad* induced by $A$.

Alternatively, one might think to [[formal duality|dualize]] the notion of the [[writer monad]] by considering a [[comonoid]]-object

$$
  B \,\in\, CoMon(\mathcal{C})
$$

and the induced comonad structure on $B \otimes (\text{-})$.

But if $\mathcal{C}$ is in fact a [[compact closed category]], then these two notions agree: In this case the [[internal hom]] out of the [[monoid object]] $A$ is the [[tensor product]] with the [[dual object|dual]] [[comonoid object]]

$$
  [A,\,\text{-}]
  \;\simeq\;
  A^\ast \otimes (\text{-})
  \,.
$$

Finally, if $A$ carries both [[monoid]]- and [[comonoid]]-[[structure]] in a compatible way to make a [[Frobenius monoid]], then the induced (co)writer (co)monad structures are compatible to make a [[Frobenius monad]].

## Related concepts

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

## References

The terminology "cowriter comonad" is used for instance in:

* {#AhmanUustalu17} [[Danel Ahman]], [[Tarmo Uustalu]], p. 5 of: *Taking updates seriously*, Proceedings of the Sixth International Workshop on [Bidirectional Transformations (Bx 2017)](https://ceur-ws.org/Vol-1827/), CEUR Workshop Proceedings **1827** (2017) [[pdf](http://ceur-ws.org/Vol-1827/paper11.pdf)]

* {#Uustalu21} [[Tarmo Uustalu]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):

  {#Uustalu21Lecture3} p. 21 of: *Monads and Interaction Lecture 3* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs3.pdf), [[Uustalu-Monads3.pdf:file]]]

[[!redirects cowriter comonads]]


Coxeter group

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

#Contents#
* table of contents
{:toc}

## Idea

A _Coxeter group_ is a [[group]] determined by a _Coxeter matrix_, and is a combinatorial abstraction of the idea of a [[reflection group]].

## Definition

A _Coxeter matrix_ over an index [[set]] $I$ is a symmetric [[matrix]]

$$M \colon I \times I \to \{1, 2, 3, \ldots, \infty\}$$

such that $M(i, i) = 1$ for all $i \in I$, else $M(i, j) \gt 1$. Writing $m_{i,j} = M(i, j)$, the associated **Coxeter group** $W_M$ is the group presented as having generators $s_i$, $i \in I$, and relations

$$(s_i s_j)^{m_{i, j}} = 1$$

for all $i, j \in I$, whenever $m_{i, j} \neq \infty$. In other words, $m_{i, j}$ is the order of $s_i s_j$ (as is easily shown), and these orders determine the group.

A Coxeter group is usually more properly regarded as a [[group presentation]] rather than as an abstract group, but there is less than perfect consistency on this point in the literature. The $s_i$ are [[involutions]] that play the role of reflections generating the group.

Often Coxeter groups are specified by means of **Coxeter diagrams**. A Coxeter diagram associated with a Coxeter matrix is a [[graph|multigraph]] whose vertices are indexed by $I$, and with $m_{i, j} - 2$ edges between distinct vertices $i, j$. Coxeter diagrams are convenient visual aids; for example, involutions $s_i$, $s_j$ commute precisely when there are no edges between $i$ and $j$, and so the product of two Coxeter groups is specified by the disjoint union of their Coxeter diagrams.

## Examples

### Finite reflection groups

Let $V$ be a finite dimensional real [[Euclidean space]], and suppose $G$ is a finite group generated by linear reflections of $V$ (each pointwise fixing a linear hyperplane and preserving the Euclidean metric). Then $G$ is a Coxeter group. Such finite Coxeter groups are also called **spherical Coxeter groups** (being subgroups of isometry groups of Euclidean spheres).

In this case, each generating reflection $s_i$ is the mirror reflection specified by the fixed hyperplane $H_i$, with $s_i(v_i) = -v_i$ for any $v_i$ orthogonal to $H_i$. It is easy to check that $m_{i, j}$ is the order of $s_i s_j$, where $\pi/m_{i, j}$ is the dihedral angle between $H_i$ and $H_j$. Moreover, the relations $(s_i s_j)^{m_{i, j}} = 1$ suffice to specify the structure of the group, roughly because the dihedral angles uniquely determine a hyperplane arrangement up to an element of the orthogonal group, and the hyperplane arrangement determines the group in kaleidoscopic fashion, using the hyperplanes as mirrors.

Finite reflection groups have been completely classified. Notice that if $G$ and $H$ are finite reflection groups on $V$ and $W$ respectively, then $G \times H$ is a finite reflection group on $V \times W$. Such reflection groups are called **reducible**, and for the purposes of classification it suffices to consider just irreducible reflection groups. Also, if $G$ is a reflection group on $V$, we can regard it as a reflection group on $V \times \mathbb{R}$ in a trivial way, but we ignore such _inessential extensions_ in our descriptions below. Thus it will suffice to consider only irreducible, essential finite reflection groups. The dimension of the Euclidean space on which the group acts will be indicated by a subscript (with mild and fixable exceptions for $E_7$ and $E_6$).

Irreducible essential finite reflection groups fall into four infinite families $A_n$, $B_n = C_n$, $D_n$, $I_2(m)$, together with a small number of exceptional groups:

1. $E_6$, $E_7$, $E_8$,
1. $F_4$,
1. $G_2$,
1. $H_3$, $H_4$.

The Coxeter matrices which specify these groups are often and traditionally encoded in the form of **Coxeter diagrams**, consisting of $n$ dots, and $k-2$ line segments between dots $i$ and $j$ if $m_{i, j} = k$. The case where $k = 2$ (no line segments) means $s_i$ and $s_j$ commute. Coxeter diagrams are highly convenient; for example, there are a number of "coincidences" where various Coxeter groups in different families A-I are isomorphic, and these coincidences are visually apparent by seeing that their Coxeter diagrams are isomorphic. Similarly, the Coxeter diagram of a reducible group $G \times H$ is the disjoint union of the Coxeter diagrams for $G$ and $H$ separately, and so in the irreducible case we are only interested in _connected_ Coxeter diagrams.

#### The family $A_n$

$A_n$ is the isometry group of a regular $n$-simplex, and is identified with the symmetric group $S_{n+1}$, where $s_i$ is identified with the permutation $(i i+1)$. We have $m_{i, i+1} = m_{i+1, i} = 3$, and $m_{i, j} = 2$ if ${|i-j|} \geq 2$. The condition $(s_i s_{i+1})^3 = 1$ may be rewritten as a braid relation
$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$$
since the $s_i$ are involutions.

#### The family $B_n$

$B_n = C_n$ is the isometry group of a regular $n$-cube $\{-1, 1\}^n$, and is identified with a wreath product $S_n \ltimes (\mathbb{Z}_2)^n$. The generators may be given by $s_1, \ldots, s_{n-1}, t_{n-1}$ where $s_i$ is the reflection through the hyperplane $x_i = x_{i+1}$ (i.e., swap the $i^{th}$ and $(i+1)^{st}$ coordinates), and $t_{n-1}$ is the reflection through the hyperplane $x_n = 0$. The Coxeter diagram looks like this:

so that $s_{n-1} t_{n-1}$ is of order 4, but $s_i t_{n-1}$ is otherwise of order 2 (i.e., $s_i$ and $t_{n-1}$ commute).

**Remark:** The distinction between $B_n$ and $C_n$ is not apparent at the level of Coxeter groups, but rather at the level of [[root system|root systems]], used to classify simple complex Lie algebras. In other words, in this case there are two distinct root systems which generate the same reflection group.

#### The family $D_n$

$D_n$ is the linear isometry group on the set of integral vectors in $\mathbb{R}^n$ of length $\sqrt{2}$, of which there are $2n(n-1)$ many. It is not an isometry group of a regular solid, but it is a subgroup of $B_n$ of index 2. In this case there are involutions $s_1, \ldots s_{n-1}, t_{n-1}$ where $s_i$ is as described for the case $B_n$, and $t_{n-1}$ swaps and negates the last two coordinates. The Coxeter diagram looks like this:

We see by examining the Coxeter diagrams that $A_3 \cong D_3$. The case $D_4$ on the other hand admits a symmetry of order 3, called [[triality]].

#### The family $I_2(m)$

The [[dihedral group]] of order $2 n$ is the isometry group of a regular $n$-gon in the plane, or of the set $\{e^{2\pi i j/n}: 1 \leq j \leq n\}$ in $\mathbb{C}^1 \cong \mathbb{R}^2$. This is generated by two reflections: complex conjugation, and reflection through the hyperplane orthogonal to $1 + e^{2\pi i/n}$. The Coxeter diagram has $m-2$ edges between two vertices. There are coincidences $A_2 \cong I_2(3)$, and $B_2 \cong I_2(4)$.

There is also a coincidence $I_2(6) \cong G_2$ (which allows us to elide over the description of the Coxeter group $G_2$!). The distinction between them is again only at the level of root systems, where the root system of $G_2$ consists of the 12 vectors

$$\{e^{2\pi i j/6}\} \cup \{e^{2\pi i j/6} + e^{2\pi i (j+1)/6}\}$$

#### The groups $E_8$, $E_7$, $E_6$

There is a rich and fascinating literature on these structures; we confine ourselves only to a very succinct (and cryptic) description.

$E_8$ is the group of linear isometries of a root system consisting of vectors $x = (x_1, \ldots, x_8)$ in $\mathbb{Z}^8 \cup (\mathbb{Z} + \frac1{2})^8$ such that

$$\sum_i x_i \in 2\mathbb{Z}, \qquad {|x|} = \sqrt{2}.$$

Its Coxeter diagram is:

$E_7$ is the group of linear isometries of the subset of roots used for $E_8$ whose first two coordinates are equal. Its Coxeter diagram is:

$E_6$ is the group of linear isometries of the subset of roots used for $E_8$ whose first three coordinates are equal. Its Coxeter diagram is:

#### The group $F_4$
  {#TheGroupF4}

The group $F_4$ is the [[isometry group]] of the [[24-cell]], which is a [[regular polytope]] in $\mathbb{R}^4$ consisting of the 8 unit [[quaternions]] $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by
$\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$. (These 24 quaternions form a [[group]] under multiplication, and this group is isomorphic to the [[binary tetrahedral group]].)

The Coxeter diagram is:

#### The group $G_2$

See the discussion on $I_2(m)$ above.

#### The groups $H_3$, $H_4$
 {#CoxeterGroupH4}

The group $H_3$ is the isometry group of the regular [[dodecahedron]] or of the regular [[icosahedron]] in $\mathbb{R}^3$, and is abstractly isomorphic to the group $\mathbb{Z}_2 \times Alt_5$, having 120 elements ([[icosahedral group]]). Its Coxeter diagram is:

The group $H_4$ is the isometry group of a [[regular polyhedron]] in $\mathbb{R}^4$ known as the [[120-cell]], or of the dual polyhedron known as the [[600-cell]]. Its Coxeter diagram is:

### Affine reflection groups

### Hyperbolic reflection groups

## Properties of Coxeter groups

+-- {: .num_theorem}
######Theorem
Equality in Coxeter groups is decidable. In other words, there is an algorithm which, given two words $u$, $v$ in the generators $s_i$, determines in finitely many steps whether $u v^{-1}$ belongs to the relator subgroup.
=--

+-- {: .proof}
######Sketch of proof
Consider the equivalence relation on words generated by the relation $u \sim v$ if $v$ is obtained by replacing an alternating substring of the form $s_i s_j s_i \ldots$ of length $m_{i, j}$ by the alternating substring $s_j s_i s_j \ldots$ of the same length. Clearly each equivalence class has only finitely many members since all the word-lengths are the same. Then say that $u$ is _reduced_ if no member of its equivalence class contains a substring of the form $s_i s_i$. If $u$ is unreduced, then $u$ is $\sim$-related to, hence in the same coset of the relator subgroup as, a $v$ which has such a substring $s_i s_i$, which in turn is in the coset of the word obtained by deleting $s_i s_i$, called a _reduction_ of $u$. The algorithm proceeds by enumerating all words in the $\sim$-equivalence class of $u$, passing to the first reduction that arises, and iterating this process until one finally obtains a reduced word $w$; this will be in the same relator coset as $u$. Then, two reduced words are in the same relator coset if and only if they are $\sim$-equivalent, and more generally two words $u$, $v$ are in the same relator coset if and only if the algorithm applied to each produces two reduced words which are $\sim$-equivalent.
=--

Reduced-word expressions for a group element $u$ may be visualized as paths of minimal length from the identity to $u$ in the [[Cayley graph]] given by the group presentation (here, a simple [[graph]] whose vertices are group elements, with an edge between $u$ and $v$ if $s_i u = v$ for some generator $s_i$). Such reduced-word expressions are important in the study of [[building|buildings]] based on the Coxeter group, and also in the related study of [[BN-pair|BN-pairs]].

## References

* N. [[Bourbaki]], _Groupes et Algèbras de Lie_, Chapitres 4-6, Masson, Paris (1981).

* [[Kenneth Brown]], _Buildings_, Springer Monographs in Mathematics, Springer 1989.

* [[James Humphreys]], _Reflection groups and Coxeter groups_, Cambridge Studies in Advanced Mathematics 29, 1990, [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0725.20028&format=complete).

Treatment of Coxeter groups of finite rank via [[M-matrix|M-matrices]]

* Robert B. Howlett, _Coxeter groups and M-matrices_, Bull. London Math. Soc. __14__:2 (1982) 137-141 [doi](https://doi.org/10.1112/blms/14.2.137)

[[!redirects Coxeter groups]]

[[!redirects Coxeter matrix]]
[[!redirects Coxeter matrices]]

[[!redirects Coxeter diagram]]
[[!redirects Coxeter diagrams]]


Coxeter number

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

#Contents#
* table of contents
{:toc}

## Idea

(..)

## Examples

* For [[SU(2)]] the dual Coxeter number is $g^{\vee} = 2$.

* (...)

## Related concepts

* [[root (in representation theory)]]

* [Chern-Simons level renormalization](Chern-Simons+level#LevelRenormalization)

## References

* Ruedi Suter, *Coxeter and dual coxeter numbers*,
Communications in Algebra **26** 1 (1998) $[$[doi:10.1080/00927879808826122](https://doi.org/10.1080/00927879808826122)$]$

[[!redirects Coxeter numbers]]

[[!redirects dual Coxeter number]]
[[!redirects dual Coxeter numbers]]


Coxeter transformation

## Definition

If $W$ is a [[Coxeter group]] of finite rank then a __Coxeter element__, sometimes also called Coxeter transformation, is a product of all simple reflections in any order. Notice that this is not the same as the longest Coxeter element (the maximal element in the Bruhat order).

One talks about __Coxeter transformations__ also in a number of more general situations. We say that a finite rank abelian group $A$ is a bilinear group if it is equipped with a (typically nonsymmetric) biadditive map
$$
\langle -,-\rangle : A\times A\to\mathbf{Z}
$$
and an automorphism $\tau:A\to A$, called the Coxeter transformation, satisfying
$$
\langle y, x\rangle = -\langle x, \tau y\rangle,\,\,\,x,y\in A.
$$
If $\langle -,-\rangle$ is nondegenerate then it follows that $A$ is torsion free hence a lattice, and we say that it is a __bilinear lattice__.

## Examples

...talk about [[Euler bilinear form]] on the [[Grothendieck group]] of a suitable Abelian or triangulated category. In particular, about the Coxeter transformation induced by [[Serre duality]].

...Auslander-Reiten translation induces Coxeter transformation...

## Properties

If $A$ is a bilinear lattice then $ \tau$ belongs to the center of $Aut(A)$.

## Literature

* A. J. Coleman, _Killing and the Coxeter transformation of Kac-Moody algebras_, Invent. Math. 95, no. 3 (1989): 447--477
* N. A’Campo, _Sur les valeurs propres de la transformation de Coxeter_, Invent. Math. 33(1) (1976) 61--67 [doi](https://doi.org/10.1007/BF01425505)
* (around page 71) [[Claus Michael Ringel]], _Tame algebras and integral quadratic forms_, Springer LNM 1099

Chapter VIII.2 in

* [[Maurice Auslander]], [[Idun Reiten]], Sverre O. Smalo, _Representation theory of Artin algebras_, Cambridge University Press 1995 (2010 online [doi](https://doi.org/10.1017/CBO9780511623608))
* Helmut Lenzing, _A K-theoretic study of canonical algebras_, [RG](https://www.researchgate.net/publication/265548116_A_K-theoretic_study_of_canonical_algebras), in book: Representation theory of algebras (Cocoyoc, 1994); CMS Conf. Proc. 18; Eds. R. Bautista, R. Martínez-Villa, J. A. de La Peña
* Andrzej Mróz,  José Antonio de la Peña, _Periodicity in bilinear lattices and the Coxeter formalism_, [doi](https://doi.org/10.1016/j.laa.2015.11.021)
* Xinhong Chen, Ming Lu, _Coxeter transformations of the derived categories of coherent sheaves_, Journal of Algebra __399__ (2014) 79-101 [DOI](https://doi.org/10.1016/j.jalgebra.2013.10.003)
* [[Sefi Ladkani]], _On the periodicity of Coxeter transformations and the non-negativity of their Euler forms_, [pdf](https://www.math.uni-bonn.de/people/sefil/Papers/coxhered.pdf)
*  Michael Barot, Dirk Kussin, Helmut Lenzing, _The Grothendieck group of a cluster category_,
arXiv:[math.RT/0606518](https://arxiv.org/abs/math/0606518)

[[!redirects Coxeter element]]
[[!redirects bilinear lattice]]


Coyoneda lemma > history

< [[Coyoneda lemma]]
[[!redirects Coyoneda lemma -- history]]


CP violation

[[!redirects CP problem]]

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

## Related concepts

* [[theta angle]]

* [[axion]]

* [[baryogenesis]]

* [[flavor problem]]

## References

* [[Roberto Peccei]], _The Strong CP Problem and Axions_, Lect. Notes Phys.741:3-17,2008 ([arXiv:hep-ph/0607268](http://arxiv.org/abs/hep-ph/0607268))

* _Strong CP problem and axion Phenomenology_, Proceedings of the DAE Symp. on Nucl. Phys. 58 (2013) ([pdf](http://sympnp.org/proceedings/58/F14.pdf))

See also

* Wikipedia, _[Strong CP problem](http://en.wikipedia.org/wiki/Strong_CP_problem)_

* Wikipedia, _[CP violation](http://en.wikipedia.org/wiki/CP_violation)_

On [[flavour physics]], [[CP violation]] and [[flavour anomalies]]:

* [[Antonio Pich]], _Flavour Dynamics and Violations of the CP Symmetry_, Lectures at the 2017 and 2019 CERN - Latin-American Schools of High-Energy Physics ([arXiv:1805.08597](https://arxiv.org/abs/1805.08597))

Discussion in the context of [[string theory]] is in

* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], _Axions In String Theory_, JHEP 0606:051,2006 ([arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206))

Discussion specificlly in the context of [[M-theory on G2-manifolds]]/[[G2-MSSM]] is in ([Svrcek-Witten 06, section 6](#SvrcekWitten06)) and

* [[Bobby Acharya]], Konstantin Bobkov, [[Piyush Kumar]], _An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse_, JHEP 1011:105,2010 ([arXiv:1004.5138](http://arxiv.org/abs/1004.5138))

[[!redirects strong CP problem]]

[[!redirects strong CP-problem]]