Continuous cohesion over sets (Q2927669)

Lawvere proposed to analyze the contrast between cohesion (``space'') andNEWLINEnon-cohesion (``set'') in axiomatic terms: the basic data consist in theNEWLINEtwo categories and a string of four adjoint functors between them: ``set''NEWLINEof components, discrete ``space'', ``set'' of points, codiscrete ``space'' --NEWLINErequired to satisfy certain properties.NEWLINENEWLINEIn terms of such a string, one defines notions like ``Nullstellensatz''NEWLINE(``every component has at last one point''), contractibility, homotopyNEWLINEclasses of maps.NEWLINENEWLINEModels of the axiomatics are best studied in the context of geometricNEWLINEmorphisms between toposes.NEWLINENEWLINEThe paper by Menni under review is a contribution to this analysis, forNEWLINEthe case where the non-cohesive topos is just the category of sets, andNEWLINEthe cohesive one is a presheaf topos, like simplicial sets. It is provedNEWLINEthat for presheaf toposes, certain of the desirable properties of theNEWLINEcohesive/non-cohesive contrast are incompatible. More specifically forNEWLINEpresheaf toposes, the property ``every object embeds in a contractibleNEWLINEobject'' is incompatible with the property ``infinite products ofNEWLINEconnected objects are connected''.NEWLINENEWLINEBut on the other hand, Menni constructs toposes (over sets) where theseNEWLINEtwo properties are compatible, thus the Lawvere axiomatics of cohesionNEWLINEis consistent.