Every Grothendieck Topos Has a One-way Site

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. 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. Lawvere has urged a project of distinguishing which toposes have the character of (generalized) spaces and which the character of categories of spaces (Lawvere 1986). Special cases of such a distinction had been called petit and gros by Giraud and Grothendieck in topology and in étale schemes. Lawvere calls for a general distinction, emphasizing among other things that the gros topos of a point itself amounts to the definition of a branch of geometry. The two should be defined both intrinsically and in terms of their sites. Localic toposes are paradigmatically petit and two results in this direction concern site descriptions for generalizations which are still seemingly petit. Johnstone showed a Grothendieck topos is locally decidable iff it has some small site with all arrows epic (Johnstone 2002, C5.4.4). Kock and Moerdijk proved a topos is an étendue if and only it it has some small site with all arrows monic (Kock & Moerdijk 1991) or (Johnstone 2002, C5.2.5). Lawvere asked about a further generalization: sites without idempotents in the sense that all idempotents are identities (equivalently all are epic, or all are monic). This note shows every Grothendieck topos has such a site, and in fact has a one way site, a site where all endomorphisms are identities. On the other hand we notice that subcanonical one-way sites are very special. This suggests that a site criterion for petit toposes will probably require subcanonical sites. The definitions and proofs are formulated for small categories and Grothendieck toposes over Set. But every step through the proof of Theorem 2.1 applies when Set is replaced by any elementary topos with a natural number object, reasoning internally to that topos. The proof of Theorem 2.2 uses excluded middle. Received by the editors 2006-01-21. Transmitted by F. W. Lawvere. Published on 2006-02-10. 2000 Mathematics Subject Classification: 18B25.