From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10849 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Direct image functors Date: Mon, 7 Nov 2022 10:56:49 +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="18836"; mail-complaints-to="usenet@ciao.gmane.io" To: "categories@mta.ca list" Original-X-From: majordomo@rr.mta.ca Tue Nov 08 03:14:53 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 1osE8b-0004dV-Kl for gsmc-categories@m.gmane-mx.org; Tue, 08 Nov 2022 03:14:53 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:34992) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1osE8E-0000Si-S2; Mon, 07 Nov 2022 22:14:30 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1osE7Z-0004Ze-QE for categories-list@rr.mta.ca; Mon, 07 Nov 2022 22:13:49 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10849 Archived-At: Do others share my discomfort with the phrase =E2=80=9Cdirect image functor=E2= =80=9D for 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 inv= erse image, not right adjoint, because in sets and functions, we have f(A) s= ubset B iff A subset f^{-1}(B). This is clearest in the localic case. If the frame homomorphism f^* has a le= ft adjoint g, and moreover a Frobenius condition is satisfied, then Joyal an= d Tierney showed that g(U) is indeed the direct image of each open U: thus f= matches the classical characterization of an open map. (Without Frobenius, g= (U) is the up-closure of the direct image.) Moving to non-localic toposes, and their sheaves, it gets more complicated. I= wouldn=E2=80=99t suggest that left adjoints are always best thought of as d= irect images. For instance, with a locally connected f, the left adjoint of f= ^* gives (fibrewise) sets of connected components. However, my question is whether the right adjoint deserves that title. In th= e 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. To see why, here=E2=80=99s a pointwise calculation in the notation of type t= heory. Suppose U =3D =CE=A3_{x:X} U(x) and V =3D =CE=A3_{y:Y} V(y) are bundl= es over X and Y. (For our topos purposes, calculating sheaves, we take them b= oth to be local homeomorphisms, ie the fibres are all discrete spaces.) Then= 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.= (If you don=E2=80=99t trust these pointwise calculations, think of them as p= roviding intuitions from the case where there are sufficient global points. B= ut actually they are more generally valid.) Steve.= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]