Talk:PlanetPhysics/Examples of Functor Categories

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\begin{document}

 \subsection{Introduction}

Let us recall the essential data required to define \htmladdnormallink{functor categories}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. One requires two arbitrary \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} that, in principle, could be large ones, $\mathcal{\A}$ and $\mathcal{C}$, and also the class
$$\textbf{M} = [\mathcal{\A},\mathcal{C}]$$
(alternatively denoted as $\mathcal{C}^{\mathcal{\A}}$) of all covariant \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} from $\mathcal{\A}$ to $\mathcal{C}$. For any two such functors $F, K \in [\mathcal{\A}, \mathcal{C}]$, $ F: \mathcal{\A} \rightarrow \mathcal{C}$ and $ K: \mathcal{\A} \rightarrow \mathcal{C}$, the class of all \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} from $F$ to $K$ is denoted by $[F, K]$, (or simply denoted by $K^F$). In the particular case when $[F,K]$ is a \textbf{set} one can still define for a \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{\A}$, the set $Hom(F,K)$. Thus, (cf. p. 62 in \cite{Mitchell65}), when $\mathcal{\A}$ is a {\em small} category the class $[F, K]$ of natural transformations from $F$ to $K$ may be viewed as a subclass of the cartesian product $\prod_{A \in \mathcal{\A}}[F(A), K(A)]$, and because the latter is a {\em set} so is $[F, K]$ as well. Therefore, with the categorical law of \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of natural transformations of functors, and for $\mathcal{\A}$ being small, $\textbf{M} = [\mathcal{\A},\mathcal{C}]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.


\subsection{Examples}

\begin{enumerate}
\item Let us consider $\mathcal{A}b$ to be a small \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} and let $\mathbb{G}_{Ab}$ be the category of finite Abelian (or commutative) \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, as well as the set of all covariant functors from $\mathcal{A}b$ to
$\mathbb{G}_{Ab}$. Then, one can show by following the steps defined in the definition of a
functor category that $[\mathcal{A}b,\mathbb{G}_{Ab}]$, or
${\mathbb{G}_{Ab}}^{\mathcal{A}b}$ thus defined is an \emph{Abelian functor category}.

\item Let $\mathbb{G}_{Ab}$ be a small category of finite Abelian (or commutative) groups and, also let $\grp_G$ be a small category of group-groupoids, that is, group \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}. Then, one can show that the imbedding functors
$\textbf{I}$: from $\mathbb{G}_{Ab}$ into $\grp_G$ form a functor category
${\grp_G}^{\mathbb{G}_{Ab}}$.

\item In the general case when $\mathcal{\A}$ is not small, the proper class
$$\textbf{M} = [\mathcal{\A}, \mathcal{\A'}]$$ may be endowed with the structure of a {\em \htmladdnormallink{supercategory}{http://planetphysics.us/encyclopedia/SuperCategory6.html}} defined as any formal interpretation of \htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html} with the usual categorical \htmladdnormallink{composition law}{http://planetphysics.us/encyclopedia/Identity2.html} for natural transformations of functors; similarly, one can construct a meta-category called the \emph{supercategory of all functor categories}.
\end{enumerate}

\begin{thebibliography}{9}

\bibitem{Mitchell65}
Mitchell, B.: 1965, \emph{Theory of Categories}, Academic Press: London.

\bibitem{NP1975}
Ref.$288$ in the
Bibliography of Category Theory and Algebraic Topology.

\end{thebibliography} 

\end{document}