The intersection of algebra and coalgebra
The aim of this paper is to prove that if a category is a variety of \(F\)-algebras and as well as a covariety of \(G\)-algebras for endofunctors \(F\) and \(G\) of many-sorted sets, then it is a presheaf category (shortly, algebra \(\cap \) coalgebra = presheaves). The main result is that all coalgebras on a set functor \(H\) form a presheaf category iff \(H\) is a reduction of a polynomial functor. The present paper is an expanded version of \textit{J. Adámek} [Lect. Notes Comput. Sci. 3629, 67--82 (2005; Zbl 1151.18004)] presented at the conference on algebra and coalgebra in computer science, CALCO 2005, in Swansea.
scientific article; zbMATH DE number 5083520