PlanetPhysics/Examples of Functor Categories

Introduction

Let us recall the essential data required to define [[../TrivialGroupoid/|functor categories]]. One requires two arbitrary [[../Cod/|categories]] that, in principle, could be large ones, $𝒜$ and $𝒞$, and also the class $M^{‴}=[𝒜,𝒞]$ (alternatively denoted as ${𝒞}^{𝒜}$) of all covariant [[../TrivialGroupoid/|functors]] from $𝒜$ to $𝒞$. For any two such functors $F,K\in [𝒜,𝒞]$, $F:𝒜\to 𝒞$ and $K:𝒜\to 𝒞$, the class of all [[../VariableCategory2/|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 [[../Cod/|small category]] $𝒜$, the set $Hom(F,K)$. Thus, (cf. p. 62 in ), when $𝒜$ 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 $\prod _{A\in 𝒜}[F(A),K(A)]$, and because the latter is a set so is $[F,K]$ as well. Therefore, with the categorical law of [[../Cod/|composition]] of natural transformations of functors, and for $𝒜$ being small, $M^{‴}=[𝒜,𝒞]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Examples

1. Let us consider $𝒜b$ to be a small [[../AbelianCategory2/|abelian category]] and let ${𝔾}_{Ab}$ be the category of finite Abelian (or commutative) [[../TrivialGroupoid/|groups]], as well as the set of all covariant functors from $𝒜b$ to ${𝔾}_{Ab}$. Then, one can show by following the steps defined in the definition of a functor category that $[𝒜b,{𝔾}_{Ab}]$, or ${{𝔾}_{Ab}}^{𝒜b}$ thus defined is an Abelian functor category .
2. Let ${𝔾}_{Ab}$ be a small category of finite Abelian (or commutative) groups and, also let ${𝖦}_G$ be a small category of group-groupoids, that is, group [[../TrivialGroupoid/|objects]] in the [[../GroupoidCategory/|category of groupoids]]. Then, one can show that the imbedding functors $I$ : from ${𝔾}_{Ab}$ into ${𝖦}_G$ form a functor category ${{𝖦}_G}^{{𝔾}_{Ab}}$.
3. In the general case when $𝒜$ is not small, the proper class $M=[𝒜,{𝒜}^{\prime }]$ may be endowed with the structure of a [[../SuperCategory6/|supercategory]] defined as any formal interpretation of [[../ETACAxioms/|ETAS]] with the usual categorical [[../Identity2/|composition law]] for natural transformations of functors; similarly, one can construct a meta-category called the supercategory of all functor categories .