The notion of chain complex has been generalized in various directions. Most recent approaches include Khovanov's Hopfological algebra and Q-shaped homological algebra due to Holm and Jorgensen.

In Hopfological algebra, chain complexes are replaced by modules over a finite dimensional Hopf algebra H. The role of a dg algebra is played by an H module algebra A. An analogue of the category of dg modules over a dg algebra can be constructed by using H-equivariant modules over an H-module algebra. The construction of the derived category of dg modules over a dg algebra can be extended to Hopfological algebra. Furthermore, by work of Ohara and the speaker we may define an Abelian model structure on thecategory of H-equivariant A-modules whose homotopy category is equivalent to the derived category.

On the other hand, Holm and Jorgensen proposed to regard a chain complex as a functor. They found appropriate conditions on a commutative ring k and a small k-linear category Q under which the category of functors Q—> k-Mod has properties analogous to the category of chain complexes. Their framework is wide enough to include the homological algebra of N-complexes, which cannot be handled in Hopfological algebra. They also defined Abelian model structures. However, one of the most serious drawbacks of their approach is the lack of analogues of dg modules over dg algebras.

In this talk, we propose a multiplicative extension of Q-shaped homological algebra by introducing the notion of comonoidal category as a bimonoid object in the duoidal category of k-linear quivers. The category of k-linear functors from a comonoidal category to k-Mod can be made into a monoidal category, from which analogues of dg algebras and dg module can be defined. A finite dimensional graded Hopf algebra H naturally gives rise to a comonoidal category L(H) in such a way that the category of k-linear functors L(H) —> k-Mod is equivalent to the category of left H-modules as monoidal categories. We discuss constructions of Abelian model structures in this framework following Ohara-Tamaki and Holm-Jorgensen.