# PlanetPhysics/Functor Categories

In order to define the concept of functor category , let us consider for any two categories $\displaystyle \mathcal{\A}$ and $\displaystyle \mathcal{\A'}$ , the class $\displaystyle '''M''' = [\mathcal{\A},\mathcal{\A'}]$ of all covariant functors from $\displaystyle \mathcal{\A}$ to $\displaystyle \mathcal{\A'}$ . For any two such functors $\displaystyle F, K \in [\mathcal{\A}, \mathcal{\A'}]$ , $\displaystyle F: \mathcal{\A} \rightarrow \mathcal{\A'}$ and $\displaystyle K: \mathcal{\A} \rightarrow \mathcal{\A'}$ , let us denote the class of all natural transformations from ${\displaystyle F}$ to ${\displaystyle K}$ by ${\displaystyle [F,K]}$. In the particular case when ${\displaystyle [F,K]}$ is a set one can still define for a small category $\displaystyle \mathcal{\A}$ , the set ${\displaystyle Hom_{'''M'''}(F,K)}$. Thus, cf. p. 62 in [1], when $\displaystyle \mathcal{\A}$ is a small category the `class' ${\displaystyle [F,K]}$ of natural transformations from ${\displaystyle F}$ to ${\displaystyle K}$ may be viewed as a subclass of the cartesian product $\displaystyle \prod_{A \in \mathcal{\A}}[F(A), K(A)]$ , and because the latter is a set so is ${\displaystyle [F,K]}$ as well. Therefore, with the categorical law of composition of natural transformations of functors, and for $\displaystyle \mathcal{\A}$ being small, $\displaystyle '''M''' = [\mathcal{\A},\mathcal{\A'}]$ satisfies the conditions for the definition of a category , and it is in fact a functor category .

Remark : In the general case when $\displaystyle \mathcal{\A}$ is not small , the proper class $\displaystyle '''M''' = [\mathcal{\A}, \mathcal{\A'}]$ 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 .

## References

1. Mitchell, B.: 1965, Theory of Categories , Academic Press: London.
2. Refs. ${\displaystyle [15],[17],[18]}$  and ${\displaystyle [288]}$  in the Bibliography of Category Theory and Algebraic Topology. Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic-Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS , P. Suppes, Editor (August-Sept. 1971).