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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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