Coalgebras in a category of classes (Q876384)

The author extends the familiar result that if \(C\) is a topos and \(G\) is a cartesian comonad of \(C,\) then the category of coalgebras \(C_{G} \) is also a topos in the setting of algebraic set theory [\textit{A. Joyal} and \textit{I. Moerdijk}, ``Algebraic set theory'', Lond. Math. Soc. Lect. Note Series 220 (1995; Zbl 0847.03025)]. In particular, it is shown that, under suitable assumptions, several kinds of categories of classes are stable under the formation of coalgebras for cartesian comonad, internal presheaves and comma categories.