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

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.