Every grothendieck topos has a one-way site

Abstract

Lawvere has urged a project of characterizing peut toposes which have the character of generalized spaces and gros toposes which have the character of categories of spaces. Étendues and locally decidable toposes are seemingly petit and have a natural common generalization in sites with all idempotents identities. This note shows every Grothendieck topos has such a site. More, it defines slanted products which take any site to an equivalent one way site, a site where all endomorphisms are identities. On the other hand subcanonical one-way sites are very special. A site criterion for petit toposes will probably require subcanonical sites.