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 .