From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10474 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steve Awodey Newsgroups: gmane.science.mathematics.categories Subject: =?utf-8?Q?Bohemian_L=26P_Caf=C3=A9=3A_Michael_Shulman=2C_May_4?= Date: Wed, 28 Apr 2021 22:23:26 -0400 Message-ID: Reply-To: Steve Awodey Mime-Version: 1.0 (Mac OS X Mail 11.5 \(3445.9.7\)) 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="6846"; mail-complaints-to="usenet@ciao.gmane.io" To: categories net Original-X-From: majordomo@rr.mta.ca Fri Apr 30 03:45:18 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 1lcIDW-0001gH-69 for gsmc-categories@m.gmane-mx.org; Fri, 30 Apr 2021 03:45:18 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:55346) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1lcIBn-0000o5-1S; Thu, 29 Apr 2021 22:43:31 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1lcI66-0007Nv-Se for categories-list@rr.mta.ca; Thu, 29 Apr 2021 22:37:38 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10474 Archived-At: ~~~ Bohemian Logical & Philosophical Caf=C3=A9 ~~~ Speaker: Michael Shulman Title: The derivator of setoids. May 4 at 16.00 Prague time (GMT+1) The zoom link is on the page: https://bohemianlpc.github.io Abstract: Can homotopy theory be developed in constructive mathematics, or even in = ZF set theory without the axiom of choice? Recent work inspired by = homotopy type theory has yielded two constructive homotopy theories = (simplicial sets and equivariant cubical sets) that are classically = equivalent to that of spaces, but it is unknown if they are = constructively equivalent to each other. If they are not, then which one = is correct? Or are they both correct, or both incorrect? What do these = questions even mean? I will propose one criterion for the correctness of a constructive = homotopy theory of spaces: as a derivator, it should be the free = cocompletion of a point. (A derivator is an enhancement of the homotopy = category that remains 1-categorical, but can still express universal = properties of this sort.) Then I will give some evidence that any such = homotopy theory must have the curious property that its 1-truncation = contains, not only sets, but also setoids. Specifically, the ex/lex = completion of the category of sets defines a derivator that satisfies a = relative version of the aforementioned universal property; thus it = should be a localization of any derivator satisfying the absolute = condition. This suggests that either setoids are an unavoidable aspect = of constructive homotopy theory, or the criterion needs to be modified. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]