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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
 and
 , the class
of all covariant functors from
 to
 . For any two such functors
![$ F, K \in [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img6.png "$ F, K \in [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$") ,
 and
 , let us denote the class of all natural transformations from  to  by ![$ [F, K]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img11.png "$ [F, K]$") . In the particular case when ![$ [F, K]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img12.png "$ [F, K]$") is a set one can still define for a small category
 , the set
 . Thus, cf. p. 62 in [ 1], when
 is a small category the `class' ![$ [F, K]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img16.png "$ [F, K]$") of natural transformations from  to  may be viewed as a subclass of the cartesian product
![$ \prod_{A \in \mathcal{\mathcal A}}[F(A), K(A)]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img19.png "$ \prod_{A \in \mathcal{\mathcal A}}[F(A), K(A)]$") , and because the latter is a set so is ![$ [F, K]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img20.png "$ [F, K]$") as well. Therefore, with the categorical law of composition of natural transformations of functors, and for
 being small,
![$ \textbf{M} = [\mathcal{\mathcal A},\mathcal{\mathcal A'}]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img22.png "$ \textbf{M} = [\mathcal{\mathcal A},\mathcal{\mathcal A'}]$") satisfies the conditions for the definition of a category, and it is in fact a Functor Category.
Remark: In the general case when
 is not small, the proper class
![$ \textbf{M} = [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img24.png "$ \textbf{M} = [\mathcal{\mathcal A}, \mathcal{\mathcal A'}]$") may be endowed with the structure of a supercategory (defined as any formal interpretation of ETAS) with the usual categorical composition law for natural transformations of functors. Similarly, one can construct a meta-category defined as the supercategory of all functor categories.
- 1
- Mitchell, B.: 1965, Theory of Categories, Academic Press: London.
- 2
- Refs.
![$ [15],[17],[18]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img25.png "$ [15],[17],[18]$") and ![$ [288]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img26.png "$ [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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(view preamble | get metadata)
See Also: category, functor, 2-category, groupoid categories,  -category of double groupoids, 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 436 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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![$\displaystyle \textbf{M} = [\mathcal{\mathcal A},\mathcal{\mathcal A'}]$](http://images.planetmath.org:8080/cache/objects/11090/l2h/img3.png "$\displaystyle \textbf{M} = [\mathcal{\mathcal A},\mathcal{\mathcal A'}]$"){kind=small}