From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9466 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: Sat, 9 Dec 2017 01:20:38 +0000 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 1512923997 24404 195.159.176.226 (10 Dec 2017 16:39:57 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 10 Dec 2017 16:39:57 +0000 (UTC) Cc: Patrik Eklund , categories@mta.ca To: Steve Vickers , cory.m.knapp@gmail.com Original-X-From: majordomo@mlist.mta.ca Sun Dec 10 17:39:48 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 1eO4dn-0005u1-VU for gsmc-categories@m.gmane.org; Sun, 10 Dec 2017 17:39:48 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40995) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eO4ef-0002je-N1; Sun, 10 Dec 2017 12:40:41 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eO4dC-0005em-5R for categories-list@mlist.mta.ca; Sun, 10 Dec 2017 12:39:10 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9466 Archived-At: Dear All, Thanks once again for the responses---I appreciate it! @Cory. Thanks! This was helpful to see that the two perspectives respond to different `needs' within mathematics, and that for many applications the algebraic perspective suffices. I did just want to pick up on the point about large cardinals. I suspect that those sorts of `arithmetical definability' characterisations will only work for cardinals you can characterise from below (i.e. by stating that there's a fixed point for some sort of describability property) and thus are likely to be consistent with V=3DL. However, there are really interesting categorial characterisations of *very* strong large cardinals, some of them dating back to the 1960s (see Brooke-Taylor and Bagaria, `On Colimits and Elementary Embeddings'). But I'm not really on top of this material, so don't have a feel for how the proof works yet (other than that the embeddability in the category-theoretic setting somehow transfers to the existence of large cardinal embeddings in set theory). @Mike. Thanks for the points about the meta-theory---all clear to me now. Similarly for ordinals. I was talking about the etale space. My claim wasn't meant to be that it *couldn't* be done structurally, just that sometimes the set-theoretic perspective is useful for seeing what's going on in a particular construction (but maybe this is just a holdover of how I first came across sheaves---then there was a lot of quotienting in material set theory). You're right of course about the large cardinals (as I mentioned to Cory). This strikes me as very interesting stuff that I'm just not on top of yet. The example of L is also super-nice...thanks! I will have a think about this. @Patrik: I'm also unsure how category theory is not a `theory'. Obviously were interested in many and varied categories but the core axiomatisation of what a category is is still first-order. Similarly in material set theory, even though ZFC is the `core' set theory (for many people anyway) we still consider lots of different theories. So I don't see how the phrase `is category theory a theory? I think not.' wouldn't apply mutatis mutandis to material set theory. I *do* think however there's an important philosophical difference: Set theory aims at an intended interpretation (the cumulative hierarchy), whereas category theory doesn't (the whole *point* of it is to apply across diverse contexts). But maybe things are more subtle than I realise [plus I'm maybe straying into off-list philosophical territory with this claim]. Best Wishes, Neil On 8 December 2017 at 06:49, Steve Vickers wrot= e: > Dear Patrik, > > The theory of categories is a first order theory, so what exactly are you= denying here? > > Steve. > >> On 7 Dec 2017, at 18:58, peklund@cs.umu.se wrote: >> >> Is Category Theory a Theory? I think not. At least not in a logical >> sense. >> > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] --=20 Dr. Neil Barton Postdoctoral Research Fellow Kurt G=C3=B6del Research Center for Mathematical Logic University of Vienna [For admin and other information see: http://www.mta.ca/~cat-dist/ ]