Sequence category
category of sequences
A particular case of the general construction of functor categories or diagram categories. Let be the set of integers equipped with the usual order relation. Then
can be considered as a small category with integers as objects and all possible pairs
, where
and
, as morphisms. The pair
is the unique morphism from the object
to the object
. Composition of morphisms is defined as follows:
.
For an arbitrary category , the category of functors from
to
is called the category of sequences in
. To define a functor
, it is sufficient to indicate a family of objects from
, indexed by the integers, and for each integer
to choose a morphism
. Then the assignment
,
extends uniquely to a functor
. A natural transformation
from the functor
to a functor
, i.e. a morphism in the category of sequences, is defined by a family of morphisms
such that
for any
.
If is a category with null morphisms, then in the category of sequences in
one can isolate the full subcategory of complexes, i.e. functors
such that
for any
. For any Abelian category
the category of sequences and the subcategory of complexes are Abelian categories.
Instead of the category one can consider its subcategories of non-negative or non-positive numbers. The corresponding diagram categories are also called categories of sequences.
Comments
References
[a1] | S. MacLane, "Homology" , Springer (1963) pp. Chapt. IX, §3 |
Sequence category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sequence_category&oldid=17892