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.

References

[a1]	S. MacLane, "Homology" , Springer (1963) pp. Chapt. IX, §3