From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/628 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Challenge from Harvey Friedman Date: Fri, 30 Jan 1998 15:55:27 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017080 26578 80.91.229.2 (29 Apr 2009 14:58:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:58:00 +0000 (UTC) To: categories Original-X-From: cat-dist Fri Jan 30 15:55:29 1998 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA12166; Fri, 30 Jan 1998 15:55:28 -0400 (AST) Original-Lines: 52 Xref: news.gmane.org gmane.science.mathematics.categories:628 Archived-At: Date: Fri, 30 Jan 1998 10:40:55 -0500 (EST) From: Michael Barr Well the first question has an easy answer. It is just as possible to talk about a category without knowing what a set is as it is to talk about a set without knowing what a set is. Of course, you cannot talk about homsets. It is interesting to read the very first Eilenberg-Mac Lane paper, which did not talk about homsets. A set is an undefined notion and there is a relation, epsilon that may hold between one set and another, subject to certain axioms, one version of which Friedman listed. A category consists of undefined things called arrows and three relations, two functional and the third partially functional (actually better than that, but leave that aside). Friedman's axioms are not coherent, as has been pointed out, while the categorical axioms are. On the other hand, one can state Friedman's axioms, in all their glorious incomprehensibility (I think I could stare at the 8th one from now until the middle of next year without understanding what it says, and the 6th, asserted to be the axiom of infinity is not much clearer) in a couple hundred words, while it is pretty much necessary to interrupt the topos axioms for some definitions (at least monic and subobject) to do the topos axioms. Thus each one looks simpler to its devotees and there is really no point in arguing about it. Michael On Thu, 29 Jan 1998, categories wrote: > Date: Wed, 28 Jan 1998 22:15:27 +0000 > From: Carlos Simpson > > Being a newcomer to the category list, I have a really naive and stupid question > (concerning H. Friedman's challenge). Namely, I was always under the impression > that you had to know what a set was before you could talk about what a > category was (in particular a topos). Is it possible to talk about toposes > without knowing what a set is? > > This seems somewhat related to a question that has been bugging me for some > time, namely how to talk about a ``category'' > which is enhanced over itself, but not necessarily having any functor to or > from Sets. The very first part of the structure would be a class of objects > O > together with a function (x,y)\mapsto H(x,y) from O\times O to O, but I > can't get beyond that. > > ---Carlos Simpson > > >