Tensor Hopf endofunctor

Abstract:
An approach to A_¥-categories presented at the book "Closed multicategory of pretriangulated A_¥-categories" (joint with Volodymyr Lyubashenko and Oleksandr Manzyuk) involves construction of Kleisli multicategory for free cocategory comonad T ^{³ 1} on the Monoidal category of augmented graded k-linear quivers.
The talk is devoted to an observation concerning Hopf algebra structure on the endofunctor T ^{³ 1}. We explain how bialgebra projection appear in the construction of A_¥-analog of Verdier factor-category. Finally, we discus some new constructions from higher dimensional algebra that concern the matter.