From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7856 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: A category internal to itself Date: Thu, 5 Sep 2013 07:46:13 -0400 Message-ID: References: Reply-To: Colin McLarty NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1378386026 16252 80.91.229.3 (5 Sep 2013 13:00:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 5 Sep 2013 13:00:26 +0000 (UTC) Cc: categories list To: Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Thu Sep 05 15:00:29 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1VHZAi-0006JH-UH for gsmc-categories@m.gmane.org; Thu, 05 Sep 2013 15:00:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36156) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VHZ9W-0000J3-FN; Thu, 05 Sep 2013 09:59:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VHZ9V-0003da-BJ for categories-list@mlist.mta.ca; Thu, 05 Sep 2013 09:59:13 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7856 Archived-At: Perhaps this is not really what you want, though. What I would like to know, related to this, is can there be a non-trivial category internally fibred over itself? This question, like your question, leaves some room to ask just what it means. If we just try to Grothendieck-fiber this category of all categories in NF over itself we meet type problems. A type-level notion of pair will not let us form the set of all pairs of a category and an object in it. Maybe a routine NF trick gets around this, maybe not. I wanted to share this thought, and clarify it for myself, before plunging into the world of NF tricks. best, Colin On Wed, Sep 4, 2013 at 6:11 PM, Colin McLarty wrote: > > In the set theory New Foundations (NF) using Quine's type-level pairing (so > a pair has the same type in a stratification as its components) you can > define small categories and small functors the usual way. Then, just as > there is a set of all sets, there is a small category of all small > categories. This is not a tautology. You have to verify a few things. > > Notably, in this context there is a set of all small functors because > there is a set of all functions (yet the category of sets is not > cartesian closed, because it lacks evaluation functions). Since > a function is stratified at the same level as its domain and codomain > sets there is no problem defining domain, codomain, composition, and > identity-assigning functors for this category. > > This category is internal to itself. This example is even left exact. > But it is not cartesian closed. > > Of course the consistency of NF is not settled. But I think everyone > supposes it is equiconsistent with some more usual set theory (likely with > ETCS). > > best, Colin > > > > > On Wed, Sep 4, 2013 at 5:23 AM, Andrej Bauer wrote: > >> Chatting at a conference, the question came up why there is no >> (non-trivial) category which is "internal to itself" (interpret this >> in some sensible sense). And over coffee we thought this must be well >> known, but not to us. Can somene shed some light on the matter? >> >> With kind regards, >> >> Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]