Sweedler Theory for Duoidal Categories

Christina Vasilakopoulou – 12 February 2020

For a symmetric monoidal closed category V, which is locally presentable, there is a tensored and cotensored enrichment of monoids in comonoids via Sweedler's `measuring coalgebras'. In this talk, after we recall the basic narrative, we will extend this result to the context of duoidal categories, where the monoidal closed structure with respect to one tensor provides an enrichment for monoids and comonoids with respect to the other tensor. Given time, we will discuss certain induced enrichment structures for species, with respect to the Hadamard, Cauchy and substitution products: the latter provides an enrichment of operads in cooperads, which is the ultimate goal of this work.