From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10851 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: Direct image functors Date: 08 Nov 2022 10:46:47 +0000 Message-ID: References: Reply-To: ptj@maths.cam.ac.uk Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="13641"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: Steve Vickers Original-X-From: majordomo@rr.mta.ca Thu Nov 10 03:28:24 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 1osxIm-0003O8-BD for gsmc-categories@m.gmane-mx.org; Thu, 10 Nov 2022 03:28:24 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:35122) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1osxI2-00035m-3M; Wed, 09 Nov 2022 22:27:38 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1osxHT-0000cM-I8 for categories-list@rr.mta.ca; Wed, 09 Nov 2022 22:27:03 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10851 Archived-At: Dear Steve, You are of course right that `direct image functor' was an unfortunate name to choose for the right adjoint part of a geometric morphism. But the term has been around for sixty years now, and it's very well understood; so I think it's too late to change it. I don't think=20 anyone is likely to be deceived into thinking that it's a direct image in the set-theoretic sense. Best regards, Peter On Nov 8 2022, Steve Vickers wrote: > Do others share my discomfort with the phrase =E2=80=9Cdirect image funct= or=E2=80=9D for=20 > the right adjoint f_* in a geometric morphism f: X -> Y? > > It seems to me that a direct image functor should be left adjoint of the= =20 > inverse image, not right adjoint, because in sets and functions, we have= =20 > f(A) subset B iff A subset f^{-1}(B). > > This is clearest in the localic case. If the frame homomorphism f^* has a= =20 > left adjoint g, and moreover a Frobenius condition is satisfied, then=20 > Joyal and Tierney showed that g(U) is indeed the direct image of each=20 > open U: thus f matches the classical characterization of an open map.=20 > (Without Frobenius, g(U) is the up-closure of the direct image.) > > Moving to non-localic toposes, and their sheaves, it gets more=20 > complicated. I wouldn=E2=80=99t suggest that left adjoints are always bes= t=20 > thought of as direct images. For instance, with a locally connected f,=20 > the left adjoint of f^* gives (fibrewise) sets of connected components. > > However, my question is whether the right adjoint deserves that title. In= =20 > the case where Y is 1, it is well known that f_* gives the global=20 > sections. In general f_* is more a _sections_ functor than a direct image= =20 > functor. > > To see why, here=E2=80=99s a pointwise calculation in the notation of typ= e=20 > theory. Suppose U =3D =CE=A3_{x:X} U(x) and V =3D =CE=A3_{y:Y} V(y) are b= undles over X=20 > and Y. (For our topos purposes, calculating sheaves, we take them both to= =20 > be local homeomorphisms, ie the fibres are all discrete spaces.) Then=20 > f^*(V) =3D =CE=A3_x V(f(x)), and a map =CE=B8: f^*(V) -> U has =CE=B8_xy:= V(y) -> U(x) for=20 > each x, y with f(x) =3D y. This is a map > =CE=A3_y V(y) -> =CE=A3_y =CE=A0_{f(x) =3D y} U(x), displaying f_*(U)(y= ) as the set of=20 > sections of U over the fibre of f over y. > > (If you don=E2=80=99t trust these pointwise calculations, think of them a= s=20 > providing intuitions from the case where there are sufficient global=20 > points. But actually they are more generally valid.) > >Steve. > >[For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]