From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10398 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: (unknown) Date: Fri, 19 Feb 2021 16:50:42 +0100 Message-ID: Reply-To: Marco Grandis Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="30541"; mail-complaints-to="usenet@ciao.gmane.io" To: Original-X-From: majordomo@rr.mta.ca Sat Feb 20 22:37:17 2021 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 1lDZwD-0007oV-GG for gsmc-categories@m.gmane-mx.org; Sat, 20 Feb 2021 22:37:17 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:44894) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1lDZqx-0001Sh-4H; Sat, 20 Feb 2021 17:31:51 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1lDZpU-0004Zg-Lp for categories-list@rr.mta.ca; Sat, 20 Feb 2021 17:30:20 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10398 Archived-At: This book is being published, and scheduled for March 2021: M. Grandis Manifolds and Local Structures. A general theory World Scientific Publishing Co., xii + 361 pages, 2021. - Info at WS:=20 https://www.worldscientific.com/worldscibooks/10.1142/12199 - Downloadable Introduction https://www.worldscientific.com/doi/epdf/10.1142/9789811234002_0001 ________ Local structures are studied as symmetric enriched categories on ordered = categories of partial=20 morphisms; their morphisms are defined as 'compatible profunctors'. This = follows a line=20 presented and developed in 1988-90 (whose main sources are cited in the = Preface, below): - MG, Manifolds as enriched categories, in: 'Categorical Topology', = Prague 1988, pp.358-368,=20 World Scientific Publishing Co., 1989. - MG, Cohesive categories and manifolds, Ann. Mat. Pura Appl. 157 = (1990), 199-244. Downloadable: = https://link.springer.com/article/10.1007/BF01765319 The main basis of enrichment is called a 'cohesive e-category', or = 'e-category' for short. Later, this structure has been re-introduced under the name of = 'restriction category'=20 and equivalent axioms. ________ from the PREFACE Local structures, like differentiable manifolds, fibre bundles, vector = bundles and foliations, can be obtained by gluing together a family=20 of suitable 'elementary spaces', by means of partial homeomorphisms that = fix the gluing conditions and form a sort of 'intrinsic atlas',=20 instead of the more usual system of charts living in an external = framework. An 'intrinsic manifold' is defined here as such an atlas, in a suitable = category of elementary spaces: open euclidean spaces, or trivial=20 bundles, or trivial vector bundles, and so on. This uniform approach allows us to move from one basis to another: for = instance, the elementary tangent bundle of an open Euclidean=20 space is automatically extended to the tangent bundle of any = differentiable manifold. The same holds for tensor calculus. Technically, the goal of this book is to treat these structures as = 'symmetric enriched categories' over a suitable basis, generally an=20 ordered category of partial mappings. This approach to gluing structures is related to Ehresmann's one, based = on inductive pseudogroups and inductive categories. A second=20 source was the theory of enriched categories and Lawvere's unusual view = of interesting mathematical structures as categories enriched=20 over a suitable basis. ________ Marco Grandis= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]