The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ℂ of Hopf monoids in a symmetric monoidal category ℂ, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ℂ preserve directed colimits one has the following results: (1) If, in ℂ, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ℂ and comonadic over the category of monoids in C. (2) If, in ℂ, extremal monomorphisms are stable under tensor squaring, then Hopf ℂ is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ℂ.