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# functor category examples

# 0.1 Introduction

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

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

(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.

# 0.2 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 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

$\textbf{M}=[\mathcal{\mathcal{A}},\mathcal{\mathcal{A}^{{\prime}}}]$ 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*.

# References

- 1
Mitchell, B.: 1965,
*Theory of Categories*, Academic Press: London. - 2 Ref.$288$ in the Bibliography of Category Theory and Algebraic Topology.

## Mathematics Subject Classification

18D05*no label found*18-00

*no label found*18A25

*no label found*

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