From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9454 Path: news.gmane.org!.POSTED!not-for-mail From: Neil Barton Newsgroups: gmane.science.mathematics.categories Subject: Re: How analogous are categorial and material set theories? Date: Sun, 3 Dec 2017 17:12:27 +0100 Message-ID: References: Reply-To: Neil Barton NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1512482540 14927 195.159.176.226 (5 Dec 2017 14:02:20 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 5 Dec 2017 14:02:20 +0000 (UTC) Cc: categories@mta.ca To: Steve Vickers , shulman@sandiego.edu, peklund@cs.umu.se Original-X-From: majordomo@mlist.mta.ca Tue Dec 05 15:02:03 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eMDnO-00038x-Ll for gsmc-categories@m.gmane.org; Tue, 05 Dec 2017 15:02:02 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:39418) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eMDnT-00048r-Ou; Tue, 05 Dec 2017 10:02:07 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eMDm0-00043O-Gj for categories-list@mlist.mta.ca; Tue, 05 Dec 2017 10:00:36 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9454 Archived-At: Dear All, Thanks so much for your kind and patient responses, and apologies for the slow reply (I wanted to check out some of Michael's recommendations before replying). @Patrik: Thanks for the nice examples and applications. I certainly don't want to deny that category theory has applications where material set theory would be inappropriate, but rather to specifically see if there were any applications to which material set theory was more suited (and how this can then be incorporated into the structural setting). However, your examples are useful to see just how different the two perspectives are (I agree that systematising certain subject matters in material set theory would be a fruitless project). @Michael: Thanks very much for the reference. I think I see the proof: One recovers a model of ZFC not by considering the membership relation of ETCS (given by various f: 1--> X), but rather by finding the `membership graph' for the relevant sets in a category satisfying ETCS (with the replacement stack axiom added). I do have a question here though---here we are expected to take the *class* of all well-founded extensional accessible pointed graphs, and note that ZFC holds within this class. Whilst I'm happy that the lemmas showing that the required graphs exist in the category Set are categorial, it seems to me that to isolate all these and talk about the *class* of all of them requires some material-set-theoretic machinery. Is there a *purely categorial* way of talking about this `collection' of subgraphs in Set? Or have I missed something? I suppose this is equivalent to the requirement of asking for a (set) model of ZFC in material set theory. So could one simply state an extra axiom to be added to ETCS+: ``Set contains an well-founded extensional accessible pointed graphs such that...[list the APG ZFC axioms].''. Is this acceptable in a *pure* categorial framework, or do you think that presupposes some material set theory? I suppose there is also the question of how to recover the cumulative hierarchy in this framework---in the paper you sent me this is done with a non-categorial theory of ordinals (possibly given by material set theory). (This relates to a more general question I have concerning categorial foundations: Are there people who claim we should do foundational research *solely* in the language of category theory, or does almost everyone accept that the `external' (possibly material set-theoretic) perspective is also allowed? So, for example, when considering the category of sheaves over a topological space, whilst I could take a purely categorial outlook, nonetheless sometimes I might want to just look at the equivalence class of an open set U relative to a point i in the topological space defined in material set theory. The two perspectives seem to complement rather than contradict each other, but I wonder what the general feeling is concerning the interellation of the two foundational systems.) (This in turn relates to the wider question: How can material and structural set theories inform one another? It *seems* to me (without any deep arguments for the claim) that material set theory is just better suited to certain roles (such as the formulation of large cardinal hypotheses) whilst structural set theory better suited to systematising the algebraic roles we want sets to perform. This is despite the fact that we can simulate one perspective within the other; for example just because you *can* simulate talk of categories with, say, Grothendieck universes, doesn't mean that it's a particularly *natural* interpretation.) @Steve. You ask: When you look at making set theory more categorical, are you just looking for a categorical way to do essentially the same thing, or are you trying more deeply to expose possible limitations of set theory? I don't think I'm clearly aiming at either (though I am interested in these questions). I'm trying to understand more clearly what purposes each foundation is best suited to, and how we can relate the two. I suppose it's a mixture of the two---the present paper I'm currently writing looks to modify material set theory to get something more `structure respecting', but nonetheless facilitating the combinatorial power and conceptual simplicity it offers (allowing us to easily work with notions such as cardinality). I'm trying to get a better picture of the landscape though, and doing this requires understanding the other direction (i.e. how one can mimic material set-theoretic claims in the structural setting). Thanks again! Best Wishes, Neil On 27 November 2017 at 17:49, Steve Vickers wro= te: > Dear Neil, > > This is not an answer to your question, so please ignore it if you're not= so > interested in these broader issues. > > My broad question is this. When you look at making set theory more > categorical, are you just looking for a categorical way to do essentially > the same thing, or are you trying more deeply to expose possible limitati= ons > of set theory? > > One thing shared by ETCS and ZFC is the well-pointedness: that the object= is > determined by its global elements (morphisms from 1). > > That can seem obvious if what you're trying to capture is some idea of > collection, but in fact it breaks down when the collection has topologica= l > structure. The cohesion between points goes beyond what can be explained = in > terms of the global points themselves, and in point-free topology we see > non-trivial spaces with no global points at all. This is not necessarily = a > pathology of point-free topology but can be related to topological facts > such as the existence of principal bundles with no continuous global > sections. It also feeds back into "sets" as discrete spaces, with > non-well-pointed toposes of sheaves (=3D local homeomorphisms =3D fibrewi= se > discrete bundles). > ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]