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=1.2 required=5.0 tests=DATE_IN_PAST_24_48, LOCALPART_IN_SUBJECT,RCVD_IN_MSPIKE_H2 autolearn=no autolearn_force=no version=3.4.4 Received: (qmail 7776 invoked from network); 1 Feb 2023 01:11:43 -0000 Received: from smtp2.mta.ca (198.164.44.75) by inbox.vuxu.org with ESMTPUTF8; 1 Feb 2023 01:11:43 -0000 Received: from rr.mta.ca ([198.164.44.159]:40602) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pN1f4-00051n-Uj; Tue, 31 Jan 2023 21:11:42 -0400 Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pN1eB-0002vD-HV for categories-list@rr.mta.ca; Tue, 31 Jan 2023 21:10:47 -0400 From: Richard Garner To: categories@mta.ca Subject: categories: Re: Sheaves as a localisation of separated presheaves In-Reply-To: Date: Tue, 31 Jan 2023 09:34:37 +1100 MIME-Version: 1.0 Precedence: bulk Reply-To: Richard Garner Message-Id: Hi 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. Looking more carefully, the proof of this seems to work fine if only Y is separated. So another way of saying the above is that the plus-construction on separated presheaves can be described without taking a quotient. However, thinking about this a bit more I should also fess up that my original statement is in error. Psh(C)[D^-1] is NOT the category of sheaves. It is the result of universally inverting the dense monos, but the category so obtained is not cocomplete. To get a cocomplete category, we need to invert the class of all bidense morphisms. How do we know Psh(C)[D^-1] isn't cocomplete? Well, if it were, then the localisation functor Psh(C) --> Psh(C)[D^-1], which obviously preserves colimits, would necessarily be sheafification Psh(C) --> Sh(C) (by the universal property of the latter), with as right adjoint the singular functor of C --> Psh(C) --> Psh(C)[D^-1]. But the monad induced by this adjunction on Psh(C) would then be the single plus construction, which is well-known not to be the reflector into the category of sheaves. (This had me worried for a while, as it seemed I had come up with an interesting proof of _|_.) On the other hand, it seems to be totally fine to describe Sh(C) as SepPsh(C)[D^-1], because the latter is the Kleisli category for the single-plus construction on SepPsh(C); and then the above quotient-free description does pertain. I realise I am basically talking to myself at this point but figure I should try to set the record straight! Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]