Pitts Monads and a Lax Descent Theorem

Abstract

For *M* the symmetric monad (M.Bunge and A.Carboni, 1995) on the
2-category *B* of toposes bounded over a base topos *S* with
a natural
numbers object, the *M*-maps (M.Bunge and J.Funk, 2006) are the
*S*-essential geometric morphisms. By a "KZ monad" we mean a
Kock-Zoeberlein monad (A. Kock, 1975). A "Pitts KZ monad" on a
2-category *B* is, roughly speaking, a KZ monad *M*
on *B* that satisfies the
analogue of Pitts' theorem (A.M.Pitts, 1996) on comma squares along
*S*-essential geometric morphisms. There is a dual notion of
Pitts co-KZ
monad *N* on any 2-category *B*. The purpose of this talk is
to state
and prove the following general lax descent theorem: If *M* is a
Pitts KZ
monad on a 2-category *B* of categories with pullbacks and stable
finite
colimits, such that *B* has an objects classifier which is
an *M*-algebra,
then every surjective *M*-map is of effective lax descent. There
is a
dual version for Pitts co-KZ monads. These theorems have several
applications for morphisms of toposes and of locales. The Pitts monads
involved in them are the symmetric monad, the lower and upper power
locale monads, and a "coherent monad" introduced for the intended
application (M.Zawadowski 1995, I. Moerdijk and J.J.C.Vermeulen 2000).

Continued 11 November 2014