Lawvere has urged a project of characterizing petit toposes which have the character of generalized spaces and gros toposes which have the character of categories of spaces. Etendues 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.
Keywords: Grothendieck topos, petit topos, locale, one way site
2000 MSC: 18B25
Theory and Applications of Categories,
Vol. 16, 2006,
No. 5, pp 123-126.