An alternative to hypercovers

Abstract

I introduce a class of diagrams in a Grothendieck site called "atlases" which can be used to study hyperdescent, and show that hypersheaves take atlases to limits using an indexed `nerve' construction that produces hypercovers from atlases. Atlases have the flexibility to be at the same time more explicit and more universal than hypercovers.

26 pages. Keywords: hypercover, higher topos, descent, atlas, test category, locale, Grothendieck topology --- CHANGELOG --- v2 - Corrected Definition 1.1, generally expanded and clarified arguments, some reorganisation (added "Preliminaries" section and split "Index Diagrams" into two). See TeX source for full changelog