The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the “self-injectivity” of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q.
The talk is based on this arXiv paper with Peter Jørgensen.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
