ExamplesOfFunctorCategories

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examples of functor categories

(Example)

Introduction

Let us recall the essential data required to define functor categories. One requires two arbitrary categories that, in principle, could be large ones, $\mathcal{\mathcal A}$ and $\mathcal{C}$ , and also the class

$\displaystyle \textbf{M} = [\mathcal{\mathcal A},\mathcal{C}]$

(alternatively denoted as $\mathcal{C}^{\mathcal{\mathcal A}}$ ) of all covariant functors from $\mathcal{\mathcal A}$ to $\mathcal{C}$ . For any two such functors $F, K \in [\mathcal{\mathcal A}, \mathcal{C}]$ , $F: \mathcal{\mathcal A} \rightarrow \mathcal{C}$ and $K: \mathcal{\mathcal A} \rightarrow \mathcal{C}$ , the class of all natural transformations from $F$

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 $\mathcal{\mathcal A}$ , the set $Hom(F,K)$

. Thus, (cf. p. 62 in [1]), when $\mathcal{\mathcal A}$ is a small category the class $[F, K]$

of natural transformations from $F$

may be viewed as a subclass of the cartesian product $\prod_{A \in \mathcal{\mathcal A}}[F(A), K(A)]$ , 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 $\mathcal{\mathcal A}$ being small, $\textbf{M} = [\mathcal{\mathcal A},\mathcal{C}]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Examples

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.
Let $\mathbb{G}_{Ab}$ be a small category of finite Abelian (or commutative) groups and, also let ${\mathsf{G}}_G$ be a small category of group-groupoids, that is, group objects in the category of groupoids. Then, one can show that the imbedding functors $\textbf{I}$ : from $\mathbb{G}_{Ab}$ into ${\mathsf{G}}_G$ form a functor category ${{\mathsf{G}}_G}^{\mathbb{G}_{Ab}}$ .
In the general case when $\mathcal{\mathcal A}$ is not small, the proper class

$\displaystyle \textbf{M} = [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$
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.