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functor category
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(Definition)
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Definition 0.1 In order to define functor categories, let us consider for any two categories $\mathcal{\A}$ and $\mathcal{\A'}$ , the class $$\textbf{M} = [\mathcal{\A},\mathcal{\A'}]$$ of all covariant functors from $\mathcal{\A}$ to $\mathcal{\A'}$ . For any two such functors $F, K \in [\mathcal{\A}, \mathcal{\A'}]$ , $ F: \mathcal{\A}
\rightarrow \mathcal{\A'}$ and $ K: \mathcal{\A} \rightarrow \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 $\mathcal{\A}$ , the set $Hom_{{M}}(F,K)$ . Thus, cf. p. 62 in [ 1], when $\mathcal{\A}$ 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 $\prod_{A \in \mathcal{\A}}[F(A), K(A)]$ , 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 $\mathcal{\A}$ being small, ${M} = [\mathcal{\A},\mathcal{\A'}]$ satisfies the conditions for the definition of a category, and it is in fact a Functor Category.
Remark: In the general case when $\mathcal{\A}$ is not small, the proper class ${M} = [\mathcal{\A}, \mathcal{\A'}]$ 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.
- 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).
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"functor category" is owned by bci1.
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See Also: category, functor, 2-category, groupoid categories, supercategory theories, ETAS interpretation, natural transformation, natural equivalence of  and  categories, natural equivalence of categories, generalized Cartesian product, ETAC, class, set, supercategory, supercategory theories
| Other names: |
super-category, meta-category of categories and functors |
| Also defines: |
category of functors, small and large category, supercategory of all functors and natural transformations |
| Keywords: |
functor category, 2-category, supercategory, ETAS, category of functors, small and large category, supercategory of all functors and natural transformations |
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Cross-references: supercategory of all functor categories, meta-category, composition, ETAS, interpretation, supercategory, structure, proper class, composition of natural transformations, categorical, Cartesian product, subclass, small category, natural transformations, covariant functors, class, categories, order
There are 18 references to this entry.
This is version 12 of functor category, born on 2008-09-25, modified 2008-09-25.
Object id is 11090, canonical name is FunctorCategory2.
Accessed 1794 times total.
Classification:
| AMS MSC: | 18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories) | | | 18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations) | | | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) |
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Pending Errata and Addenda
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