Comonad

Summary

A comonad on a category is the dual of a monad. It is an endofunctor $G$ with a counit $\varepsilon \colon G \Rightarrow \mathrm{Id}$ and a comultiplication $\delta \colon G \Rightarrow G G$ that satisfy coassociativity and counit laws. Every adjunction $F \dashv U$ gives a comonad $F U$ on the domain of $U$. Comonads govern coalgebraic structure, descent theory and comonadic resolutions in homological algebra.