From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10853 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Direct image functors Date: Tue, 8 Nov 2022 14:10:21 +0000 Message-ID: Reply-To: Steve Vickers Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="21362"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: Original-X-From: majordomo@rr.mta.ca Thu Nov 10 03:30:15 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 1osxKY-0005Qh-Sm for gsmc-categories@m.gmane-mx.org; Thu, 10 Nov 2022 03:30:15 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:35162) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1osxK9-0003bb-5V; Wed, 09 Nov 2022 22:29:49 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1osxJf-0000eX-KL for categories-list@rr.mta.ca; Wed, 09 Nov 2022 22:29:19 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10853 Archived-At: =EF=BB=BF Dear Peter, I agree the term isn=E2=80=99t likely to change (to =E2=80=9Csections functo= r=E2=80=9D or anything else) at this stage. I was partly trying to find out h= ow widely the issue was recognised, and partly trying to sharpen my discussi= on of it. > I don't think anyone is likely to be deceived into thinking that it's a di= rect > image in the set-theoretic sense. I=E2=80=99m not so sure. I=E2=80=99ve seen how when people start looking mor= e closely at the points of a topos, and the part they play in topological an= alogies, 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=E2=80= =99s 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 =E2= =80=9Cwe shall see later that, in a sense, f_* =E2=80=98embodies the geometr= ic aspects=E2=80=99 of the morphism f=E2=80=9D. What did you have in mind fo= r the =E2=80=9Cwe shall see later=E2=80=9D? Best wishes, Steve. > On 8 Nov 2022, at 10:50, ptj@maths.cam.ac.uk wrote: > =EF=BB=BFDear Steve, >=20 > 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 anyone is= likely to be deceived into thinking that it's a direct > image in the set-theoretic sense. >=20 > Best regards, > Peter >=20 > On Nov 8 2022, Steve Vickers wrote: >=20 >> Do others share my discomfort with the phrase =E2=80=9Cdirect image funct= or=E2=80=9D for the right adjoint f_* in a geometric morphism f: X -> Y? >>=20 >> It seems to me that a direct image functor should be left adjoint of the i= nverse image, not right adjoint, because in sets and functions, we have f(A)= subset B iff A subset f^{-1}(B). >>=20 >> This is clearest in the localic case. If the frame homomorphism f^* has a= left adjoint g, and moreover a Frobenius condition is satisfied, then Joyal= and Tierney showed that g(U) is indeed the direct image of each open U: thu= s f matches the classical characterization of an open map. (Without Frobeniu= s, g(U) is the up-closure of the direct image.) >>=20 >> Moving to non-localic toposes, and their sheaves, it gets more complicate= d. I wouldn=E2=80=99t suggest that left adjoints are always best thought of a= s direct images. For instance, with a locally connected f, the left adjoint o= f f^* gives (fibrewise) sets of connected components. >>=20 >> However, my question is whether the right adjoint deserves that title. In= the case where Y is 1, it is well known that f_* gives the global sections.= In general f_* is more a _sections_ functor than a direct image functor. >>=20 >> To see why, here=E2=80=99s a pointwise calculation in the notation of typ= e theory. Suppose U =3D =CE=A3_{x:X} U(x) and V =3D =CE=A3_{y:Y} V(y) are bu= ndles over X and Y. (For our topos purposes, calculating sheaves, we take th= em both to be local homeomorphisms, ie the fibres are all discrete spaces.) T= hen 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 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 sections of U over the fibre of f over y. >>=20 >> (If you don=E2=80=99t trust these pointwise calculations, think of them a= s providing intuitions from the case where there are sufficient global point= s. But actually they are more generally valid.) >>=20 >> Steve. >>=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]