Introduction

Let us recall the essential data required to define functor categories. One requires two arbitrary categories that, in principle, could be large categories, $\mathcal{\A}$ and $\mathcal{C}$ , and also the class $$\textbf{M} = [\mathcal{\A},\mathcal{C}]$$ (alternatively denoted as $\mathcal{C}^{\mathcal{\A}}$ ) of all covariant functors 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 natural transformations from $F$ to $K$ is denoted by $[F, K]$ , (or simply denoted by $K^F$ ). In the particular case when $[F,K]$ is a set one can still define for a small category $\mathcal{\A}$ , the set $Hom(F,K)$ . Thus, (cf. p. 62 in [1]), when $\mathcal{\A}$ is a 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 set so is $[F, K]$ as well. Therefore, with the categorical law of composition of natural transformations of functors, and for $\mathcal{\A}$ being small, ${M} = [\mathcal{\A},\mathcal{C}]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Examples

1. Let us consider $\mathcal{A}b$ to be a small Abelian category and let $\mathbb{G}_{Ab}$ be the category of finite Abelian (or commutative) groups, 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 Abelian functor category.
2. Let $\mathbb{G}_{Ab}$ be a small category of finite Abelian (or commutative) groups and, also let be a small category of group-groupoids, that is, group objects in the category of groupoids. Then, one can show that the imbedding functors ${I}$ : from $\mathbb{G}_{Ab}$ into form a functor category .
3. 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 supercategory defined as any formal interpretation of ETAS with the usual categorical composition law for natural transformations of functors; similarly, one can construct a meta-category called the supercategory of all functor categories.

Bibliography

1

Mitchell, B.: 1965, Theory of Categories, Academic Press: London.

2

Ref.$288$ in the Bibliography of Category Theory and Algebraic Topology.