The slice classification of categories of coalgebras for comonads (Q5932476)

A functor \(H:{\mathcal H}_1 \to {\mathcal H}_2\) is a slice of a functor \(K: {\mathcal K}_1 \to {\mathcal K}_2\) if there are faithful functors \(\Phi _i : {\mathcal H}_i \to {\mathcal K}_i \;(i=1,2)\) such that the corresponding diagrams of hom-sets are pullbacks in \({\mathcal S}et\). Functors \(H\) and \(K\) are slice-equivalent if \(H\) and \(K\) are slices of one another. Comonads are dual to monads: a comonad \(G=(G,\varepsilon ,\delta)\) over \({\mathcal S}et\) consists of a functor \(G:{\mathcal S}et\to {\mathcal S}et\) and natural transformations \(\varepsilon :G\to I, \delta : G\to G^2\) such that \(\varepsilon G \circ \delta = G \varepsilon \circ \delta = 1_G\) and \(\delta G\circ \delta = G \delta \circ \delta \). A \(G\)-coalgebra for a comonad \(G\) over \({\mathcal S}et\) is a pair \((Y,k)\) where \(Y\) is a set and \(k:Y\to G(Y)\) satisfies \(\varepsilon _Y\circ k = 1_Y\) and \(\delta _Y\circ k = G(k) \circ k\). The main result: The category of coalgebras for a comonad \(G=(G,\varepsilon , \delta)\) belongs to the basket \(A\) (containing the category of all universal algebras of a given type and the category of all compact Hausdorff spaces) iff the comonad \(G\) is non-degenerate (i.e., its functorial part is not a product of the identity functor \(I\) and a constant functor).