From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6397 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Severe Strict Monoidal Category Naivete Date: Fri, 3 Dec 2010 13:54:21 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1291427002 4211 80.91.229.12 (4 Dec 2010 01:43:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Dec 2010 01:43:22 +0000 (UTC) Cc: categories@mta.ca To: "Ellis D. Cooper" Original-X-From: majordomo@mlist.mta.ca Sat Dec 04 02:43:18 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1POh9f-0007vZ-Mt for gsmc-categories@m.gmane.org; Sat, 04 Dec 2010 02:43:15 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:50157) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POh9H-0005ed-Rd; Fri, 03 Dec 2010 21:42:51 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POh9F-0002LE-O8 for categories-list@mlist.mta.ca; Fri, 03 Dec 2010 21:42:49 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6397 Archived-At: Dear Ellis, On 03/12/2010, at 1:55 AM, Ellis D. Cooper wrote: > (1) Is strict monoidal category the same as monoid in category of = categories? Yes.=20 > (2) Is it not true that in a strict monoidal category if > $X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g=3D g\circ f$? =20 If I understand correctly, you have arrows f:X->Y and g:Y->Z and you=20 are comparing the tensor products f@g:X@Y->Y@Z and g@f:Y@X->Z@Y. They have different domain and codomain, so cannot be equal. If you considered a commutative monoid in the category of categories, = then these arrows would be equal. But such commutative monoids are very = rare.=20 > (3) Is the pentagon axiom automatically satisfied in a strict > monoidal category? >=20 Yes. In that case it asserts that two identity arrows with the same = domain and codomain are equal.=20 Steve Lack. > Many thanks for your patience and pointers. >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]