Jacopo Emmenegger: Exact completion and constructive theories of sets

Abstract: In this talk I will present Palmgren's constructive modification (CETCS) of Lawvere's Elementary Theory of the Category of Sets (ETCS), explain how the theory of exact completions tells us that any category satisfing such axioms is a free exact category, and isolate the weaker properties that a projective cover of an exact category C has to satisfy in order for C to be a CETCS category. In particular, Carboni and Rosolini already have a characterisation of local closure in terms of weak closure of the maximal projective cover, but I will explain how some model-theoretic observations strongly suggest that it cannot be applied to our case. Therefore I will introduce an alternative characterisation of closure for free exact categories, which applies to any projective cover. This is joint work with Erik Palmgren.