From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7368 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: Alternative closed structure on Cat Date: Thu, 5 Jul 2012 10:38:11 -0300 (ADT) Message-ID: References: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1341515775 29311 80.91.229.3 (5 Jul 2012 19:16:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 5 Jul 2012 19:16:15 +0000 (UTC) Cc: categories@mta.ca (Categories List) To: ondrej.rypacek@gmail.com Original-X-From: majordomo@mlist.mta.ca Thu Jul 05 21:16:14 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SmrXB-0006Ef-WA for gsmc-categories@m.gmane.org; Thu, 05 Jul 2012 21:16:14 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54029) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SmrWJ-0006qo-J5; Thu, 05 Jul 2012 16:15:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SmrWK-0001x4-8V for categories-list@mlist.mta.ca; Thu, 05 Jul 2012 16:15:20 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7368 Archived-At: Dear Ondrej, Power and Robinson state in [1, Section 2] that the tensor you describe is indeed part of a monoidal closed structure: the function category has as objects all functors, and as morphisms the (not necessarily natural) transformations. Moreover, Power and Robinson state that this is the unique other symmetric monoidal closed structure on Cat, i.e., there are no others besides this one and the "usual" one. I have never seen a proof of this last fact. [1] J. Power and E. Robinson. "Premonoidal categories and notions of computation." Mathematical Structures in Computer Science 7(5): 445-452, 1997. (www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps) -- Peter Ondrej Rypacek wrote: > > Dear All > > Is there a name for and what is known about the tensor product of = > ordinary categories which looks like the underlying 1-category of Gray's = > (Gray) tensor product?=20 > Explicitly, roughly:=20 > - objects of C \otimes D are pairs (c,d) , c object of C , d an object = > of D > - arrows alternating lists of arrows from C and D, i.e. they are = > generated by=20 > (f,d) : (c,d) -> (c',d) for f : c -> c', > (c,g) : (c,d) -> (c,d') for g : d -> d' > > and modulo the equations: (f',d) . (f, d) =3D (f'f, d), (c,g') . (c,g) = > =3D (c,g'g), and identities, left and right unit laws and associativity = > in each component separately. > =09 > > And is the category of categories with respect to this tensor closed ?=20= > > > > Thank you! > Ondrej > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]