The homotopy category #
HomotopyCategory V c gives the category of chain complexes of shape
with chain maps identified when they are homotopic.
If two chain maps become equal in the homotopy category, then they are homotopic.
An arbitrarily chosen representation of the image of a chain map in the homotopy category is homotopic to the original chain map.
Homotopy equivalent complexes become isomorphic in the homotopy category.
If two complexes become isomorphic in the homotopy category, then they were homotopy equivalent.
i-th homology, as a functor from the homotopy category.
The homology functor on the homotopy category is just the usual homology functor.
An additive functor induces a functor between homotopy categories.
A natural transformation induces a natural transformation between the induced functors on the homotopy category.