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%%% Primary Title: functor category
%%% Primary Category Code: 02.
%%% Filename: FunctorCategories.tex
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\begin{document}
\begin{definition}
In order to define the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of {\em functor category}, let us consider for any two \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{\A}$ and $\mathcal{\A'}$, the class
$$\textbf{M} = [\mathcal{\A},\mathcal{\A'}]$$
of all covariant functors from $\mathcal{\A}$ to $\mathcal{\A'}$. For any two such functors $F, K \in [\mathcal{\A}, \mathcal{\A'}]$, $ F: \mathcal{\A} \rightarrow \mathcal{\A'}$ and $ K: \mathcal{\A} \rightarrow \mathcal{\A'}$,
let us denote the class of all \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} from $F$ to $K$ by $[F, K]$. In the particular case when $[F, K]$ is a set one can still define for a \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{\A}$, the set $Hom_{\textbf{M}}(F,K)$. 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, $\textbf{M} = [\mathcal{\A},\mathcal{\A'}]$ {\em satisfies the conditions for the definition of a category}, and it is in fact a {\em functor category}.
\end{definition}
\textbf{Remark}: In the general case when $\mathcal{\A}$ is {\em 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/ETACAxioms.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 {\em meta-category} defined as the {\em supercategory of all functor categories}.
\begin{thebibliography}{99}
\bibitem{Mitchell65}
Mitchell, B.: 1965, \emph{Theory of Categories}, Academic Press: London.
\bibitem{NP1975,ICB71,ICB73,ICB74}
Refs. $[15],[17],[18]$ and $[288]$ in the
Bibliography of Category Theory and Algebraic Topology.
Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic-Topological Quantum Computations., \emph{Proceed. 4th Intl. Congress LMPS}, P. Suppes, Editor (August-Sept. 1971).
\end{thebibliography}
\end{document}
