From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8786 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: The PROP for commutative monoids Date: Sat, 9 Jan 2016 14:34:56 +0100 Message-ID: Reply-To: Marco Grandis NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1452347923 29173 80.91.229.3 (9 Jan 2016 13:58:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 9 Jan 2016 13:58:43 +0000 (UTC) To: John Baez , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Jan 09 14:58:35 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1aHu2M-0003za-C2 for gsmc-categories@m.gmane.org; Sat, 09 Jan 2016 14:58:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59770) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1aHu1g-0008Mk-Bu; Sat, 09 Jan 2016 09:57:52 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1aHu1b-0003yb-VP for categories-list@mlist.mta.ca; Sat, 09 Jan 2016 09:57:47 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8786 Archived-At: Dear John, This result is also proved in my paper below, published in 2001. M. Grandis, Finite sets and symmetric simplicial sets, Theory Appl. = Categ. 8 (2001), No. 8, 244-252. Abstract. The category of finite cardinals (or equivalently, of finite = sets) is the symmetric analogue of the category of finite ordinals, and = the ground category of a relevant category of presheaves, the augmented = symmetric simplicial sets. We prove here that this ground category has = characterisations similar to the classical ones for the category of = finite ordinals, by the existence of a universal symmetric monoid, or by = generators and relations. The latter provides a definition of symmetric = simplicial sets by faces, degeneracies and transpositions, under = suitable relations. Best wishes, Marco On 05/gen/2016, at 00.13, John Baez wrote: > Dear Categorists - >=20 > A student of mine is wondering who first noticed this fact: if you > take a skeleton of the category of finite sets and make it into a > strict symmetric monoidal category using cartesian product, it's the > "free strict symmetric monoidal category on a commutative monoid > object". Or in other words, it's the PROP for commutative monoids. >=20 > He noticed that in 2001, Teimuraz Pirashvili wrote a paper "On the > PROP corresponding to bialgebras": >=20 > http://arxiv.org/abs/math/0110014 >=20 > Pirashvili says this fact is "well known", and gives a proof, but no = reference. >=20 > Can you help us dig deeper? It's just a matter of getting the history = right. >=20 > Best, > jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]