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Pitts monads and a lax descent theorem. (English) Zbl 1350.18008
Summary: A theorem of A. M. Pitts [“Lax descent for essential surjections”, Talk at the Category Theory Conference, Cambridge, July 21, 1986] states that essential surjections of toposes bounded over a base topos $$\mathcal S$$ are of effective lax descent. The symmetric monad $$\mathcal M$$ on the 2-category of toposes bounded over $$\mathcal S$$ is a KZ-monad [M. Bunge and A. Carboni, J. Pure Appl. Algebra 105, No. 3, 233–249 (1995; Zbl 0847.18004)] and the $$\mathcal M$$-maps are precisely the $$\mathcal S$$-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture and then prove the general lax descent theorem that is the subject matter of this paper. By a ‘Pitts KZ-monad’ on a 2-category $$\mathcal K$$ it is meant here a locally fully faithful equivariant KZ-monad $$\mathcal M$$ on $$\mathcal K$$ 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 $$\mathcal M$$ on a 2-category $$\mathcal K$$ (‘of spaces’), every surjective $$\mathcal M$$-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad $$\mathcal N$$. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.
##### MSC:
 18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads 03G10 Logical aspects of lattices and related structures 03G30 Categorical logic, topoi
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