examples of functor categories
Let us recall the essential data required to define functor categories. One requires two arbitrary categories
that, in principle, could be large ones,
and
, and also the class
(alternatively denoted as
) of all covariant functors
from
to
. For any two such functors
,
and
, the class of all natural transformations from
to
is denoted by
, (or simply denoted by
). In the particular case when
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
of natural transformations from
to
may be viewed as a subclass of the cartesian product
, 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
being small,
satisfies the conditions for the definition of a category, and it is in fact a functor category.
- Let us consider
to be a small abelian category
and let
be the category of finite Abelian (or commutative) groups, as well as the set of all covariant functors from
to
. Then, one can show by following the steps defined in the definition of a
functor category that
, or
thus defined is an Abelian functor category.
- Let
be a small category of finite Abelian (or commutative) groups and, also let
be a small category of group-groupoids, that is, group objects in the category of groupoids. Then, one can show that the imbedding functors
: from
into
form a functor category
.
- In the general case when
is not small, the proper class
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.
-
- 1
-
Mitchell, B.: 1965, Theory of Categories, Academic Press: London.
- 2
-
Ref.
in the
Bibliography of Category Theory and Algebraic Topology.
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