PlanetPhysics/Functor Categories

In order to define the concept of functor category , let us consider for any two categories Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'}} , the class Failed to parse (unknown function "\A"): {\displaystyle '''M''' = [\mathcal{\A},\mathcal{\A'}]} of all covariant functors from Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} to Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'}} . For any two such functors Failed to parse (unknown function "\A"): {\displaystyle F, K \in [\mathcal{\A}, \mathcal{\A'}]} , Failed to parse (unknown function "\A"): {\displaystyle F: \mathcal{\A} \rightarrow \mathcal{\A'}} and Failed to parse (unknown function "\A"): {\displaystyle K: \mathcal{\A} \rightarrow \mathcal{\A'}} , let us denote the class of all natural transformations from $F$ to $K$ by $[F,K]$. In the particular case when $[F,K]$ is a set one can still define for a small category Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} , the set $Hom_{'''M'''}(F,K)$. Thus, cf. p. 62 in , when Failed to parse (unknown function "\A"): {\displaystyle \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 Failed to parse (unknown function "\A"): {\displaystyle \prod_{A \in \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 Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} being small, Failed to parse (unknown function "\A"): {\displaystyle '''M''' = [\mathcal{\A},\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 Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} is not small , the proper class Failed to parse (unknown function "\A"): {\displaystyle '''M''' = [\mathcal{\A}, \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 .