0.1 Introduction

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

(alternatively denoted as 𝒞^𝒜) of all covariant functors from 𝒜 to 𝒞. For any two such functors F,K ∈ [𝒜,𝒞], F : 𝒜→𝒞 and K : 𝒜→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 𝒜, the set Hom(F,K). Thus, (cf. p. 62 in [1]), 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 ∏ _A∈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 𝒜 being small, M = [𝒜,𝒞] satisfies the conditions for the definition of a category, and it is in fact a functor category.

0.2 Examples

1.

Let us consider 𝒜b to be a small Abelian category and let 𝔾_Ab be the category of finite Abelian (or commutative) 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 objects in the 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

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.

References