From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6521 Path: news.gmane.org!not-for-mail From: Francisco Lobo Newsgroups: gmane.science.mathematics.categories Subject: RE: categories with several compositions? Date: Fri, 4 Feb 2011 18:47:08 +0000 Message-ID: References: Reply-To: Francisco Lobo NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1296868346 32095 80.91.229.12 (5 Feb 2011 01:12:26 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 5 Feb 2011 01:12:26 +0000 (UTC) Cc: categories@mta.ca To: John Stell Original-X-From: majordomo@mlist.mta.ca Sat Feb 05 02:12:22 2011 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 1PlWhI-0000tX-9m for gsmc-categories@m.gmane.org; Sat, 05 Feb 2011 02:12:20 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:52694) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PlWh0-0000om-10; Fri, 04 Feb 2011 21:12:02 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PlWgt-0003HD-Bo for categories-list@mlist.mta.ca; Fri, 04 Feb 2011 21:11:55 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6521 Archived-At: Hi! Your permutability axiom for different compositions is reminiscent of the= interchange law, so I wonder if the structures you mean are the n-fold c= ategories introduced by Charles Ehresmann in "Cat=C3=A9gories structur=C3= =A9es" cf. http://www.numdam.org/item?id=3DASENS_1963_3_80_4_349_0 which is possibly the first article on higher-order category theory. An n-fold category C is just a class C equipped with n composition struct= ures (giving composition operations *_0, ..., *_{n-1} on C) that for all = i,jC= , a target operation t:C->C, and a composition operation *: (C x_{s,t} C) -> C where (C x_{s,t} C) is the collection of consecutive arrows with respect = to the source and target operations (i.e. the vertex of the pullback of s= and t), such that for all f,u,g in C 1. s( s(f) ) =3D s(f) =3D t( s(f) ) and s( t(f) ) =3D t(f) =3D t( t(f) = ) 2. (f * u) * g =3D f * (u * g) whenever both sides are defined 3. s(f) * f =3D f =3D f * t(f) The 1st condition says that a fixed point of s is also fixed point of t a= nd vice-versa, and that the range of these operations contains only their= shared fixed points: the objects of the category. The 2nd condition stat= es that * is associative, and the 3rd that the source and target of an ar= row f are respectively left and right units for composition with f (so th= e objects are used as identity arrows). From these axioms it follows that s(f * u) =3D s(f) and t(f * u) =3D t(u) as usual. (Note that f*u means "first do f then u" as is common in semigr= oup theory.) It has already been pointed out that an Eckmann-Hilton argum= ent shows that under the interchange axiom two composition structures i a= nd j will coincide whenever s_i =3D s_j and t_i =3D t_j. Each entity f in an n-fold category C is an arrow in n different ways. Th= is may be written f :_{n-1} s_{n-1}(f) -> t_{n-1)(f) ... :_{0} s_{0}(f) -> t_{0}(f) These are distinct from the cells of (strict) n-categories. The latter no= tion is often defined inductively using enrichment, but its single-sorted= (or arrows only) counterpart is precisely an n-fold category such that f= or all f in C s_i( s_j(f) ) =3D s_i(f) =3D s_i( t_j(f) ) and t_i( s_j(f) ) =3D t_i(f)= =3D t_i( t_j(f) ) whenever i wrote: > Thanks for pointing that out. > I should have been asking for each composition to have its own identity >=20 > John=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]