From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7795 Path: news.gmane.org!not-for-mail From: Mike Stay Newsgroups: gmane.science.mathematics.categories Subject: Weak monoids and monads in compact bicategories Date: Tue, 9 Jul 2013 23:44:24 -0600 Message-ID: Reply-To: Mike Stay NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1373525967 12250 80.91.229.3 (11 Jul 2013 06:59:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 11 Jul 2013 06:59:27 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Thu Jul 11 08:59:29 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UxAqe-0001kA-AZ for gsmc-categories@m.gmane.org; Thu, 11 Jul 2013 08:59:28 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46028) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UxAov-0007bt-9N; Thu, 11 Jul 2013 03:57:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UxAox-000776-Ah for categories-list@mlist.mta.ca; Thu, 11 Jul 2013 03:57:43 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7795 Archived-At: I'm looking for references to who first published these various algebra-like and category-like constructions, all either weak monoids or monads in compact bicategories: - If C has finite products and pullbacks, a weak monoid in Span(C) is a categorification of an associative algebra, while a monad is a category internal to C. (I think Benabou pointed out the latter.) - In Rel, a weak monoid is a Boolean algebra, while a monad is a preorder. - A 2-rig is a symmetric monoidal category where the tensor product distributes over the colimits. Given a 2-rig R, Mat(R) is the bicategory of natural numbers, matrices of objects of R, and matrices of morphisms of R. A weak monoid in Mat(R) is a categorified finite-dimensional associative algebra, something like a finite field. A monad in Mat(R) is a finite R-enriched category. - A weak monoid in Prof is a promonoidal category. A symmetric monoidal monad in Prof is an "Arrow" in the sense of Hughes and is related to Freyd categories. Is there a name for an arbitrary monad in Prof other than "monad in Prof"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]