Enriched coalgebras are sometimes comonadic

TL;DR

The paper generalizes coalgebra concepts over operads in enriched categories, establishing conditions for comonadic structures and illustrating with topological space examples.

Contribution

It introduces an enriched framework for coalgebras over operads, constructing associated comonads and linking to known topological coalgebra structures.

Findings

Constructs a V-endofunctor associated to an operad P in a symmetric monoidal V-category C.

Provides conditions under which this endofunctor is a V-comonad.

Recovers known descriptions of coalgebras in topological spaces and relates to Fox's theorem.

Abstract

We introduce an enriched notion of a coalgebra over an operad P in a symmetric monoidal V-category C. When C is semicartesian and P is unital, we construct a V-endofunctor on C associated to P and give conditions under which it is a V-comonad with co-Eilenberg-Moore V-category isomorphic to the V-category of P-coalgebras in C. In many cases, this permits computation of V-categories of coalgebras. The key example is the category of pointed topological spaces with wedge product, enriched over topological spaces with Cartesian product, where this construction recovers the comonadic description of C_n-coalgebras of Moreno-Fern\'andez, Wierstra and the present author. We further recover one direction of Fox's theorem.