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 $ F$ to $ K$ by $ [F, K]$ . In the particular case when $ [F, K]$ 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' $ [F, K]$ of natural transformations from $ F$ to $ K$ 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 $ [F, K]$ 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

1
Mitchell, B.: 1965, Theory of Categories, Academic Press: London.

2
Refs. $ [15],[17],[18]$ and $ [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).



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.