Editing Groupoid (section)

== Category of groupoids ==
The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the '''groupoid category''', or the '''category of groupoids''', and is denoted by '''Grpd'''.

The category '''Grpd''' is, like the category of small categories, [[Cartesian closed]]: for any groupoids <math>H,K</math> we can construct a groupoid <math>\operatorname{GPD}(H,K)</math> whose objects are the morphisms <math> H \to K </math> and whose arrows are the natural equivalences of morphisms. Thus if <math> H,K </math> are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids <math> G,H,K </math> there is a natural bijection
: <math>\operatorname{Grpd}(G \times H, K) \cong \operatorname{Grpd}(G, \operatorname{GPD}(H,K)).</math>

This result is of interest even if all the groupoids <math> G,H,K </math> are just groups.

Another important property of '''Grpd''' is that it is both [[Complete category|complete]] and [[Cocomplete category|cocomplete]].

=== Relation to [[Category of small categories|Cat]] ===

The inclusion <math>i : \mathbf{Grpd} \to \mathbf{Cat}</math> has both a left and a right [[Adjoint functors|adjoint]]:
: <math> \hom_{\mathbf{Grpd}}(C[C^{-1}], G) \cong \hom_{\mathbf{Cat}}(C, i(G)) </math>
: <math> \hom_{\mathbf{Cat}}(i(G), C) \cong \hom_{\mathbf{Grpd}}(G, \mathrm{Core}(C)) </math>

Here, <math>C[C^{-1}]</math> denotes the [[localization of a category]] that inverts every morphism, and <math>\mathrm{Core}(C)</math> denotes the subcategory of all isomorphisms.

=== Relation to [[Simplicial set|sSet]] ===

The [[Nerve (category theory)|nerve functor]] <math>N :  \mathbf{Grpd} \to \mathbf{sSet}</math> embeds '''Grpd''' as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a [[Kan complex]].

The nerve has a left adjoint
: <math> \hom_{\mathbf{Grpd}}(\pi_1(X), G) \cong \hom_{\mathbf{sSet}}(X, N(G)) </math>
Here, <math>\pi_1(X)</math> denotes the fundamental groupoid of the simplicial set {{tmath|1= X }}.

=== Groupoids in Grpd ===
{{Main|Double groupoid}}
There is an additional structure which can be derived from groupoids internal to the category of groupoids, '''double-groupoids'''.<ref>{{cite arXiv|last1=Cegarra|first1=Antonio M.|last2=Heredia|first2=Benjamín A.|last3=Remedios|first3=Josué|date=2010-03-19|title=Double groupoids and homotopy 2-types|class=math.AT|eprint=1003.3820}}</ref><ref>{{Cite journal|last=Ehresmann|first=Charles|date=1964|title=Catégories et structures : extraits|url=http://www.numdam.org/item/?id=SE_1964__6__A8_0|journal=Séminaire Ehresmann. Topologie et géométrie différentielle|language=en|volume=6|pages=1–31}}</ref> Because '''Grpd''' is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids <math>\mathcal{G}_1,\mathcal{G}_0</math> with functors<blockquote><math>s,t: \mathcal{G}_1 \to \mathcal{G}_0</math></blockquote>and an embedding given by an identity functor<blockquote><math>i:\mathcal{G}_0 \to\mathcal{G}_1</math></blockquote>One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet
\end{matrix}
</math> and <math>\begin{matrix}
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>with <math>a</math> the same morphism, they can be vertically conjoined giving a diagram<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.