From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on inbox.vuxu.org X-Spam-Level: * X-Spam-Status: No, score=1.8 required=5.0 tests=DATE_IN_PAST_06_12, LOCALPART_IN_SUBJECT,RCVD_IN_MSPIKE_H2 autolearn=no autolearn_force=no version=3.4.4 Received: (qmail 12412 invoked from network); 30 Jan 2023 20:13:14 -0000 Received: from smtp2.mta.ca (198.164.44.75) by inbox.vuxu.org with ESMTPUTF8; 30 Jan 2023 20:13:14 -0000 Received: from rr.mta.ca ([198.164.44.159]:40460) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pMaWa-0001EW-Jo; Mon, 30 Jan 2023 16:13:08 -0400 Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pMaVt-00060Q-Vs for categories-list@rr.mta.ca; Mon, 30 Jan 2023 16:12:25 -0400 From: Richard Garner To: categories@mta.ca Subject: categories: Sheaves as a localisation of separated presheaves Date: Mon, 30 Jan 2023 22:31:19 +1100 MIME-Version: 1.0 Precedence: bulk Reply-To: Richard Garner Message-Id: Dear all, If (C,J) is a site, then the category Sh(C) can be presented as a category of fractions Psh(C)[D^-1] where D is the class of J-dense monomorphisms. Morphisms in here are equivalence classes of partial maps X --> Y whose domain is dense in X, and where the equivalence relation is generated by the 2-cells of spans. This is all well known. I am wondering if there is a reference for the following fact: if one restricts to separated presheaves, then every equivalence class of morphisms has a maximal representative, given by the dense partial maps X <--< X' ---> Y whose graph X' >---> X*Y is a J-closed monomorphism. So between separated presheaves, no quotienting is necessary---beyond that inherent in the notion of subobject---though now composition is no longer span composition on equivalence classes, but rather span composition followed by J-closure. The proof is rather easy but I am wondering if there is the even easier possibility of citing something. Thanks! Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]