examples of functor categories

Introduction

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

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

(alternatively denoted as $ \mathcal{C}^{\mathcal{\mathcal A}}$ ) of all covariant functors from $ \mathcal{\mathcal A}$ to $ \mathcal{C}$ . For any two such functors $ F, K \in [\mathcal{\mathcal A}, \mathcal{C}]$ , $ F: \mathcal{\mathcal A} \rightarrow \mathcal{C}$ and $ K: \mathcal{\mathcal A} \rightarrow \mathcal{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 $ \mathcal{\mathcal A}$ , the set $ Hom(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{C}]$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Examples

  1. Let us consider $ \mathcal{A}b$ to be a small abelian category and let $ \mathbb{G}_{Ab}$ be the category of finite Abelian (or commutative) groups, as well as the set of all covariant functors from $ \mathcal{A}b$ to $ \mathbb{G}_{Ab}$ . Then, one can show by following the steps defined in the definition of a functor category that $ [\mathcal{A}b,\mathbb{G}_{Ab}]$ , or $ {\mathbb{G}_{Ab}}^{\mathcal{A}b}$ thus defined is an Abelian functor category.

  2. Let $ \mathbb{G}_{Ab}$ be a small category of finite Abelian (or commutative) groups and, also let $ {\mathsf{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 $ \textbf{I}$ : from $ \mathbb{G}_{Ab}$ into $ {\mathsf{G}}_G$ form a functor category $ {{\mathsf{G}}_G}^{\mathbb{G}_{Ab}}$ .

  3. In the general case when $ \mathcal{\mathcal A}$ is not small, the proper class

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

    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 called the supercategory of all functor categories.

Bibliography

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

2
Ref.$ 288$ in the Bibliography of Category Theory and Algebraic Topology.



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