“Supercompactness in topos theory”
Mercoledì 29 Ottobre 2025, ore 16:30 - Sala riunioni 7B1 - Roger Morgan (Università Sorbonne Paris Nord)
Abstract
In a presheaf topos, the representable objects have the special property (first shown by Marta Bunge) of being “indecomposable projectives”: any covering family over a representable must contain a split epimorphism. The representables can by recovered, up to retract, as the subcategory of presheaves having this property.
Phrased this way, a natural relaxation is the property that every covering family over an object should contain an epimorphism. Objects with this property are called (by me, at least!) supercompact, by analogy with the topological notion of compactness. In a presheaf category, they are quotients of representables.
In this talk I will discuss the toposes for which the supercompact objects generate, geometric morphisms whose components preserve supercompact objects, and my original motivation for thinking about these things, namely toposes which are categories of actions of (topological) monoids (on sets).
