# PlanetPhysics/Functor Categories

In order to define the concept of *functor category* , let us consider for any two categories **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
and **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'}}**
, the class
**Failed to parse (unknown function "\A"): {\displaystyle '''M''' = [\mathcal{\A},\mathcal{\A'}]}**
of all covariant functors from **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
to **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'}}**
. For any two such functors **Failed to parse (unknown function "\A"): {\displaystyle F, K \in [\mathcal{\A}, \mathcal{\A'}]}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle F: \mathcal{\A} \rightarrow \mathcal{\A'}}**
and **Failed to parse (unknown function "\A"): {\displaystyle K: \mathcal{\A} \rightarrow \mathcal{\A'}}**
,
let us denote the class of all natural transformations from to by . In the particular case when is a set one can still define for a small category **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
, the set . Thus, cf. p. 62 in ^{[1]}, when **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
is a *small* category the `class' of natural transformations from to may be viewed as a subclass of the cartesian product **Failed to parse (unknown function "\A"): {\displaystyle \prod_{A \in \mathcal{\A}}[F(A), K(A)]}**
, and because the latter is a *set* so is as well. Therefore, with the categorical law of composition of natural transformations of functors, and for **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
being small, **Failed to parse (unknown function "\A"): {\displaystyle '''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 **Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}}**
is *not small* , the proper class **Failed to parse (unknown function "\A"): {\displaystyle '''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* .

## All SourcesEdit

^{[1]}^{[2]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}Mitchell, B.: 1965,*Theory of Categories*, Academic Press: London. - ↑
Refs. and 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).