Spans refinments of the Cech Nerve

DUBUC, EDUARDO J.

Montreal

Conferencia; Category research center; 2010

Mc Gill University

A 2-span is a commutative diagram corresponding in dimension 2 to the span diagrams of dimension 1. Given a cover (epimorphism) U −→ 1 in a topos E −→ Set, it determinesa simplicial object U• , Un = U × U × . . . U n + 1 times. A hypercover X•is a simplicial object together with a morphism X• −→ U• . If the toposis locally connected simplicial objects have a canonical indexing simplicial set γ!(X• ), X• −→ γ ∗ γ!(X• ). For non locally connected topoi the indexing has to be given explicitly as part of the datum. This leads to the notion of simplicial family, X• −→ γ ∗ (S• ) (where S• is a simplicial set) and of indexed hypercover. We discover that the n-simplexes of a simplicial family furnish a notion of n-spans which is intimately related to the coeskeleton functor.We develop this theory and an application to the fundamental progroupoid of a non locally connected topos. In this talk we will not assume familiarity with the Artin-Mazur paper on hypercovers and the coeskeleton functor.