Pitts monads and a lax descent theorem

Abstract

A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos $\mathscr{S}$ are of effective lax descent. The symmetric monad $\mathscr{M}$ on the 2-category of toposes bounded over $\mathscr{S}$ is a KZ-monad (Bunge-Carboni 1995) and the $\mathscr{M}$-maps are precisely the $\mathscr{S}$-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture¹ and then prove² the general lax descent theorem that is the subject matter of this paper. By a ‘Pitts KZ-monad’ on a 2-category $\mathscr{K}$ it is meant here a locally fully faithful equivariant KZ-monad $\mathscr{M}$ on $\mathscr{K}$ that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad $\mathscr{M}$ on a 2-category $\mathscr{K}$ (‘of spaces’), every surjective $\mathscr{M}$-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad $\mathscr{N}$. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.