From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10855 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: George Janelidze Newsgroups: gmane.science.mathematics.categories Subject: Re: Direct image functors Date: Thu, 10 Nov 2022 13:03:20 +0200 Message-ID: References: Reply-To: George Janelidze Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="UTF-8"; reply-type=original Content-Transfer-Encoding: 8bit Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="5709"; mail-complaints-to="usenet@ciao.gmane.io" To: categories@mta.ca Original-X-From: majordomo@rr.mta.ca Fri Nov 11 03:04:06 2022 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1otJOo-0001CN-I9 for gsmc-categories@m.gmane-mx.org; Fri, 11 Nov 2022 03:04:06 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:35290) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1otJON-0006Gd-Gc; Thu, 10 Nov 2022 22:03:39 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1otJNy-0005Uj-VK for categories-list@rr.mta.ca; Thu, 10 Nov 2022 22:03:14 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10855 Archived-At: Dear Peter and Steve, If anyone protests against the direct image functor being a right adjoint, I would recall the following simple well-known story: 1. If f : X-->Y is a map of sets, then the inverse image (=pullback) functor f* : Sets/Y --> Sets/X has both left and right adjoint. Let me call them Lf and Rf, respectively. 2. If we replace Sets with an arbitrary category C with pullbacks, then f* and Lf are still there, but Rf disappears, unless C is locally cartesian closed. In particular, there is no Rf (in general) when C = Top is the category of topological spaces. 3. But if I am thinking towards topos theory, I might prefer to consider not f* : Top/Y --> Top/X, but f* : Shv(Y) --> Shv(X), where Shv(?), the category of sheaves (of sets) over "?", is equivalent to the full subcategory of Top/? with objects all local homeomorphisms with codomain "?", and, under this equivalence, the 'new' f* is the restriction of the old one. And then we have f* and Rf but not Lf (in general). Therefore, I might prefer to have name "direct image functor" for the right adjoint! Peter, might this be what you thought of as "geometric aspects"? (Surely, a geometer, considering, say, a manifold X, would be more interested in Shv(X) than in Top/X.) With apologies for trivialities- George -------------------------------------------------- From: "Steve Vickers" Sent: Tuesday, November 8, 2022 4:10 PM To: Cc: Subject: categories: Re: Direct image functors > > Dear Peter, > > I agree the term isn???t likely to change (to ???sections functor??? or anything > else) at this stage. I was partly trying to find out how widely the issue > was recognised, and partly trying to sharpen my discussion of it. > >> I don't think anyone is likely to be deceived into thinking that it's a >> direct >> image in the set-theoretic sense. > > I???m not so sure. I???ve seen how when people start looking more closely at > the points of a topos, and the part they play in topological analogies, > that there is a risk of confusion. I have known a student, learning about > the action of a geometric morphism on points, who wondered if it???s somehow > closely related to the direct image functor. > > By the way, I looked at the Elephant to see what you said there, and I saw > ???we shall see later that, in a sense, f_* ???embodies the geometric aspects??? > of the morphism f???. What did you have in mind for the ???we shall see later???? > > Best wishes, > > Steve. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]