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\begin{document}

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\section*{groupoid representation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contentsrepre}{}\section*{{Contentsrepre}}\label{contentsrepre}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{groupoid representation} is a [[representation]] of a [[groupoid]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{GroupoidRepresentation}\hypertarget{GroupoidRepresentation}{}
\textbf{([[groupoid representation]])}

Let $\mathcal{G}$ be a [[groupoid]]. Then:

A [[linear representation]] of $\mathcal{G}$ is a groupoid homomorphism ([[functor]])

\begin{displaymath}
\rho \;\colon\; \mathcal{G} \longrightarrow Core(Vect)
\end{displaymath}
to the groupoid [[core]] of the category [[Vect]] of [[vector spaces]] (\href{groupoid#CoreGroupoid}{this example}). Hence this is

\begin{enumerate}%
\item For each object $x$ of $\mathcal{G}$ a [[vector space]] $V_x$;


\item for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a [[linear map]] $\rho(f) \;\colon\; V_x \to V_y$



\end{enumerate}
such that

\begin{enumerate}%
\item (respect for composition) for all composable morphisms $x \overset{f}{\to}y \overset{g}{\to} z$ in the groupoid we have an [[equality]]

\begin{displaymath}
\rho(g) \circ \rho(f) = \rho(g \circ f)
\end{displaymath}

\item (respect for identities) for each object $x$ of the groupoid we have an equality

\begin{displaymath}
\rho(id_x) = id_{V_x}
  \,.
\end{displaymath}


\end{enumerate}
Similarly a \emph{[[permutation representation]]} of $\mathcal{G}$ is a groupoid homomorphism ([[functor]])

\begin{displaymath}
\rho \;\colon\; \mathcal{G} \longrightarrow Core(Set)
\end{displaymath}
to the groupoid core of [[Set]]. Hence this is

\begin{enumerate}%
\item For each object $x$ of $\mathcal{G}$ a [[set]] $S_x$;


\item for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a [[function]] $\rho(f) \;\colon\; S_x \to S_y$



\end{enumerate}
such that composition and identities are respected, as above.

For $\rho_1$ and $\rho_2$ two such representations, then a homomorphism of representations

\begin{displaymath}
\phi \;\colon\; \rho_1 \longrightarrow \rho_2
\end{displaymath}
is a [[natural transformation]] between these functors, hence is

\begin{itemize}%
\item for each object $x$ of the groupoid a (linear) function

\begin{displaymath}
(V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x
\end{displaymath}

\item such that for all morphisms $x \overset{f}{\longrightarrow} y$ we have

\begin{displaymath}
\phi(y) \circ \rho_1(f) = \rho_2(x) \circ \phi(x)
  \phantom{AAAAAA}
  \itexarray{
    (V_1)_x &\overset{\phi(x)}{\longrightarrow}& (V_2)_x
    \\
    {}^{\mathllap{\rho_1(f)}}\downarrow && \downarrow^{\mathrlap{\phi_2(f)}}
    \\
    (V_1)_y &\underset{\phi(y)}{\longrightarrow}& (V_2)_y
  }
\end{displaymath}


\end{itemize}
A permutation representation of $\mathcal{G}$ is often called a ``$\mathcal{G}$-set'' (see at \emph{[[G-set]]}) and the category of permutation representations is also often denoted

\begin{displaymath}
\mathcal{G}Set
  \phantom{AAAAA}
  \text{or}
  \phantom{AAAAA}
  Set^{\mathcal{G}}
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{cor}
\label{GroupoidRepresentationsAreProductsOfGroupRepresentations}\hypertarget{GroupoidRepresentationsAreProductsOfGroupRepresentations}{}
\textbf{([[groupoid representations]] are [[product category|products]] of [[group representations]])}

Assuming the [[axiom of choice]] then the following holds:

Let $\mathcal{G}$ be a [[groupoid]]. Then its [[category of representations|category of]] groupoid representations is [[equivalence of categories|equivalent]] to the [[product category]] indexed by the set of [[connected components]] $\pi_0(\mathcal{G})$ (\href{groupoid#GroupoidConnectedComponents}{this def.}) of [[group representations]] of the [[automorphism group]] $G_i \coloneqq Aut_{\mathcal{G}}(x_i)$ (\href{groupoid#InGrupoidAutomorphismGroup}{this def.}) for $x_i$ any object in the $i$th connected component:

\begin{displaymath}
Rep_{Grpd}(\mathcal{G})
   \;\simeq\;
  \underset{i \in \pi_0(\mathcal{G})}{\prod} Rep(G_i)
  \,.
\end{displaymath}
\end{cor}
\begin{proof}
Let $\mathcal{C}$ be the category that the representation is on. Then by definition

\begin{displaymath}
Rep_{Grpd}(\mathcal{G}) = Hom( \mathcal{G} , \mathcal{C}  )
  \,.
\end{displaymath}
Consider the injection functor of the [[skeleton]] (from \href{groupoid#DeloopingGroupoidEquivalence}{this lemma})

\begin{displaymath}
inc
  \;\colon\;
  \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i
  \overset{}{\longrightarrow}
  \mathcal{G}
  \,.
\end{displaymath}
By \href{groupoid#HmotopiesWithMorphismsHorizontaComposition}{this lemma} the pre-composition with this constitutes a functor

\begin{displaymath}
inc^\ast \;\colon\; Hom( \mathcal{G}, \mathcal{C} ) \longrightarrow Hom( \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i, \mathcal{C} )
\end{displaymath}
and by combining \href{groupoid#DeloopingGroupoidEquivalence}{this lemma} with \href{groupoid#HorizontalCompositionWithHomotopyIsNaturalTransformation}{this lemma} this is an [[equivalence of categories]]. Finally, by \href{groupoid#GroupoidRepresentationOfDeloopingGroupoid}{this example} the category on the right is the product of group representation categories as claimed.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{GroupoidRepresentationOfDeloopingGroupoid}\hypertarget{GroupoidRepresentationOfDeloopingGroupoid}{}
\textbf{([[groupoid representation]] of [[delooping]] groupoid is [[group representation]])}

If $B G$ is the [[delooping]] groupoid of a [[group]] $G$ (\href{groupoid#GroupoidFromDelooping}{this example}), then a [[groupoid representation]] of $B G$ according to def. \ref{GroupoidRepresentation} is equivalently a [[group representation]] of the group $G$:

\begin{displaymath}
Rep_{Grpd}(B G) \simeq Rep(G)
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{([[fundamental theorem of covering spaces]])}

For $X$ a [[topological space]] then forming [[monodromy]] is a [[functor]] from the [[category of covering spaces]] over $X$ to that of [[permutation representations]] of the [[fundamental groupoid]] of $X$:

\begin{displaymath}
Fib \;\colon\; Cov(C)  \longrightarrow Set^{\Pi_1(X)}
  \,.
\end{displaymath}
If $X$ is [[locally path-connected topological space|locally path connected]] and [[semi-locally simply connected topological space|semi-locally simply connected]], then this is an [[equivalence of categories]]. See at \emph{[[fundamental theorem of covering spaces]]} for details.

\end{example}
[[!redirects groupoid representations]]

[[!redirects representation of groupoids]] [[!redirects representations of groupoids]]



\end{document}