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