From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5883 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: we do meet isomorphisms of categories Date: Sun, 30 May 2010 08:05:41 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1275231855 4642 80.91.229.12 (30 May 2010 15:04:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 30 May 2010 15:04:15 +0000 (UTC) To: "Marco Grandis" , Original-X-From: categories@mta.ca Sun May 30 17:04:14 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OIk3h-0000hq-Q5 for gsmc-categories@m.gmane.org; Sun, 30 May 2010 17:04:13 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIjXW-0006Sd-5b for categories-list@mta.ca; Sun, 30 May 2010 11:30:58 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5883 Archived-At: Dear Marco, Perhaps I can advocate the importance of isomorphism between categories. An equivalence between skeletal categories is necessarly an isomorphism. There are many examples of skeletal categories in mathematics. The category of matrices over a ring is skeletal. The category Delta in homotopy theory is also skeletal. The category Delta(+) of all finite ordinals and order preserving maps is an interesting example because, as everyones know, it is freely generated as a monoidal category by a monoid object. It is not free in the category of strict monoidal functors but free in the category of (strong) monoidal functors. There is of course another category, FatDelta(+), which is=20 freely generated by a monoid object in the category of strict=20 monoidal functors, but it is seldom used. The two categories FatDelta(+) and Delta(+) are equivalent, but the category Delta(+) is simpler because it is skeletal. The category of finite cardinals and all maps is also skeletal. It is freely generated by one object as a category with finite = coproducts. It is also freely generated by a commutative monoid as a symmetric monoidal category. A category C is skeletal iff every equivalence A-->C has a section. This property characterises minimal models in algebraic topology. For example, a Kan complex Y is minimal iff every homotopy equivalence X-->Y, with X a Kan complex, has a section. Minimal models are important in topology. Sullivan's rational homotopy theory is essentially a = technique=20 for constructing minimal models of graded commutative algebras. The = rational=20 homotopy groups of a space can be read directly form its minimal = Sullivan model.=20 Minimal models exists in higher category theory too. Every quasi-category has a minimal model (should I say skeletal?). This not a property shared by all types of (infty,1)-categories. Some are better than others. For example, simplicial categories=20 do not admit minimal models (in general). Strict monoidal categories do not admit minimal models either. This is because strict monoidal structures cannot be transported=20 (in general) along equivalence of categories. Of course, non-strict monoidal structures can. There is an obstruction for transfering a strict monoidal structure=20 to its skeletal model. It is represented by a cohomology class=20 of degree 3 when the category is groupoidal. It is a small miracle of nature that the category Delta(+) is both strict monoidal and skeletal.=20 Similarly for the category of finite cardinals and maps.=20 Best wishes, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]