PlanetPhysics/Examples of Functor Categories

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 $$M = [\mathcal{\mathcal A},\mathcal{C}]$$ (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 $$F$$ 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 $$\mathcal{\mathcal A}$$, the set $$Hom(F,K)$$. Thus, (cf. p. 62 in ), when $$\mathcal{\mathcal A}$$ 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 $$\prod_{A \in \mathcal{\mathcal A}}[F(A), K(A)]$$, 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, $$M = [\mathcal{\mathcal A},\mathcal{C}]$$ 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 $$\mathsf{G}_G$$ be a small category of group-groupoids, that is, group objects in the category of groupoids. Then, one can show that the imbedding functors $$I$$ : from $$\mathbb{G}_{Ab}$$ into $$\mathsf{G}_G$$ form a functor category $${\mathsf{G}_G}^{\mathbb{G}_{Ab}}$$.
 * 3) In the general case when $$\mathcal{\mathcal A}$$ is not small, the proper class $$M = [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$$ 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.