From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6941 Path: news.gmane.org!not-for-mail From: "Ellis D. Cooper" Newsgroups: gmane.science.mathematics.categories Subject: Natural Functorial Categorical Intuition Date: Sun, 02 Oct 2011 21:48:00 -0400 Message-ID: Reply-To: "Ellis D. Cooper" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Trace: dough.gmane.org 1317644981 8891 80.91.229.12 (3 Oct 2011 12:29:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 3 Oct 2011 12:29:41 +0000 (UTC) To: Categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Oct 03 14:29:37 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RAheJ-0006xe-8q for gsmc-categories@m.gmane.org; Mon, 03 Oct 2011 14:29:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44413) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RAhd4-0002qM-8j; Mon, 03 Oct 2011 09:28:18 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RAhd2-00062X-GO for categories-list@mlist.mta.ca; Mon, 03 Oct 2011 09:28:16 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6941 Archived-At: Many thanks for responses to my initial post on the Subject. It was partly motivated by the seeming variety of kinds of intuition in the cited references. Also, I am utterly in awe of categorical intuitions codified, for example, by adjoint pairs of functors in geometry, algebra, and logic. Until recently I only dreamed of some day having such an intuition. I think I now have one, and would like to know if you agree that it is specifically a categorical intuition. It is a separate question whether there is a rigorous explication and proof. Define a (two-dimensional) shape to be a smooth injection of the circle into the plane. By the Jordan Curve Theorem a shape has an exterior. Given a point in the exterior -- call it a viewpoint -- there exists a finite set of intersections of the lines through the viewpoint -- call them sightlines -- which are either tangent to or pass through an inflection point (with respect to an orthogonal coordinate system in which the sightline is one coordinate). For a sufficiently remote viewpoint (maybe infinitely far away) there exist exactly two of these sightlines tangent to the shape between which the angle is less than pi radians, and such that all other of these sightlines are contained within the sweep of one of the two to the other. I am guessing that if a cover relation (as in Hasse diagrams of finite posets) is defined to be an acyclic irreflexive relation, then the category of finite posets is a reflective subcategory of the category of cover relations, where the adjoint to the inclusion is given by taking the reflexive transitive closure. Is that right? Given a shape and a viewpoint construct a cover relation by declaring that among the elements of the above set of intersections, one element covers another if (1) it is encountered earlier by a sweeping sightline as above, and (2) there exists a segment of the shape connecting the two elements that contains no other intersections. By the aforementioned adjunction this construction leads to a finite poset for any given shape and remote viewpoint. It is my intuitive guess that for a given shape there exists an algebraic structure comprised of the set of all finite posets corresponding to its viewpoints, and that this algebraic structure involves spans of poset maps among those finite posets. I guess moreover that there exists a functor from the isotopy category of shapes to an appropriately defined category of these algebraic structures, which I like to call algebraic models of shapes. Among the aforementioned intersections there are those which are directly "visible" from a viewpoint in the sense that no points of the shape intervene between such a point and the viewpoint. Call the set of directly visible points the partial-view from the viewpoint. It is my intuitive guess that the set of all partial-views of a shape also comprise an algebraic structure. Call it the partial-view model of the shape -- clearly it forgets information contained in the algebraic model. It seems to me that the algebraic model of a shape determines the partial-view model. That is, if multiple intersections lie on the same sightline in the algebraic model, then only the one closest to the viewpoint is in its partial-view. So there exists a functor from a category of algebraic models to a category of partial-view models. That word "closest" is what made my intuition click: this functor has a left adjoint. In other words, I am guessing that partial-views of a shape may be "integrated" to form its algebraic model. If these intuitions can be rigorously worked out maybe there is a mathematical theory that is to "dents" in shapes as algebraic topology is to holes in spaces. Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]