PlanetPhysics: functor category

Definition 0.1 In order to define the concept of functor category, let us consider for any two categories $\mathcal{\mathcal A}$ and $\mathcal{\mathcal A'}$ , the class

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

of all covariant functors from $\mathcal{\mathcal A}$ to $\mathcal{\mathcal A'}$ . For any two such functors $F, K \in [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$ , $F: \mathcal{\mathcal A} \rightarrow \mathcal{\mathcal A'}$ and $K: \mathcal{\mathcal A} \rightarrow \mathcal{\mathcal A'}$ , let us denote the class of all natural transformations from

. In the particular case when

is a set one can still define for a small category $\mathcal{\mathcal A}$ , the set $Hom_{\textbf{M}}(F,K)$ . Thus, cf. p. 62 in [1], when $\mathcal{\mathcal A}$ is a small category the `class'

of natural transformations from

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

as well. Therefore, with the categorical law of composition of natural transformations of functors, and for $\mathcal{\mathcal A}$ being small, $\textbf{M} = [\mathcal{\mathcal A},\mathcal{\mathcal A'}]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Remark: In the general case when $\mathcal{\mathcal A}$ is not small, the proper class $\textbf{M} = [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$ 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 defined as the supercategory of all functor categories.

Bibliography