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examples of functor categories (Feature)

Essential data

Let us recall the data required to define functor categories. One requires two arbitrary categories- that in principle (or a priori)- 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 groupoid-groups. 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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See Also: 2-category, axioms of metacategories and supercategories, higher dimensional algebra, categories and supercategories in relational biology, categories and supercategories in relational biology, topic entry on foundations of mathematics, axiomatic theories and categorical foundations of mathematics-II, axiomatics and categorical foundations of mathematical physics, algebra classification, quantum fundamental groupoid, ETAS interpretation, ETAC, n-category, groupoid homomorphism, category theory

Other names:  categories of functors and natural transformations
Also defines:  Abelian functor category, groupoid-group functor category, supercategory of all functor categories
Keywords:  small category, categories of functors and natural transformations
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Cross-references: meta-category, composition, ETAS, interpretation, supercategory, structure, proper class, imbedding, groups, commutative, abelian, finite, abelian category, composition of natural transformations, categorical, Cartesian product, subclass, category, small category, natural transformations, covariant functors, class, a priori, functor categories
There are 5 references to this entry.

This is version 42 of examples of functor categories, born on 2008-07-15, modified 2008-09-25.
Object id is 10792, canonical name is FunctorCategories.
Accessed 889 times total.

Classification:
AMS MSC: 18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories)
 18-00 (Category theory; homological algebra :: General reference works )
 18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations)
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