Free and co-free constructions for Hopf categories

Abstract

We show that under mild conditions on the monoidal base category $\mathcal{V}$, the category $\mathsf{VHopf}$ of Hopf $\mathcal{V}$-categories is locally presentable and deduce the existence of free and cofree Hopf categories. We also provide an explicit description of the free and cofree Hopf categories over a semi-Hopf category. One of the conditions on the base category $\mathcal{V}$, states that endofunctors obtained by tensoring with a fixed object preserve jointly monic families, which leads us to the notion of ``very flat monoidal product'', which we investigate in particular for module categories.

Several corrections with respect to previous version. Final version as accepted for publication in Journal of Pure and Applied Algebra