If a category V is locally presentable and symmetric monoidal closed, then so is the category Coalg(V) of coalgebras in V. Moreover, the category Alg(V) of algebras in V is enriched tensored and cotensored over Coalg(V). A similar result holds for operads and cooperads enriched over V. We will sketch applications to the bar-cobar duality.
Remark: Our theory of coalgebras is not entirely new. There are important contribution by Sweedler, Barr, Fox, Prost and more recently by Vasilakopoulou.
Joint work with Mathieu Anel.