Let be an additive category.
Call a chain complex
bounded below if there is such that ;
bounded above if there is such that ;
bounded if it is bounded below and bounded above. We have
This is the category of chain complexes in .
Several variants of this category are of relevance.
Write for the full subcategory on the chain complexes which are, respectively, bounded above, bounded below or bounded.
Write for the category obtained from by identifying homotopic chain maps.
Accordingly denotes the full subcategory on the chain complexes bounded above, bounded below or bounded, respectively.
This is sometimes called the homotopy category of chain complexes. But see the warning on terminology there, as this term is also appropriate for the category in the following remark.
For an abelian category also the category of chain complexes is again an abelian category.
We discuss the ingredients that go into this statement.
For a chain map,
A sequence of chain complexes is a short exact sequence in precisely if each component is a short exact sequence in .
Equipped with the standard tensor product of chain complexes the category of chain complexes is a monoidal category . The unit object is the chain complex concentrated in degree 0 on the tensor unit of .
A basic introduction is in chapter 1 of