From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/798 Path: news.gmane.org!not-for-mail From: street@mpce.mq.edu.au (Ross Street) Newsgroups: gmane.science.mathematics.categories Subject: re: co- Date: Tue, 7 Jul 1998 10:49:43 +1000 Message-ID: <199807070048.KAA19559@macadam.mpce.mq.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017186 27414 80.91.229.2 (29 Apr 2009 14:59:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:59:46 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Tue Jul 7 12:36:32 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA11151 for categories-list; Tue, 7 Jul 1998 11:09:02 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: street@macadam.mpce.mq.edu.au Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:798 Archived-At: While our insecurities about "co-" are being aired, I thought I should admit to even more worries in the case of 2-categories (or bicategories)! In these terminological matters, I have given up on linguistic correctness and have also almost given up worrying about mathematical consistency. Here is the difficulty. Motivation for 2-category theory comes from (at least) two different directions which often lead to the same basic concepts yet with different suggestions for terminology for the three other dual concepts. Each concept has a co-, op-, and coop-version but the good choice of op or co is not clear at all. First motivation: We can take the view that our 2-category is foremost a category with the 2-cells as extra structure (like homotopies in Top). Then, for example, as pointed out by John Gray in the La Jolla 1965 volume, Grothendieck was wrong in using "cofibration" for the *2-cell*-reversing dual of fibration. Compare the situation in Top where cofibrations are the *arrow*-reversing dual of fibrations. So this leads to "opfibration" for the *2-cell*-reversing dual of "fibration" (this is unnecessary in Top since homotopies are invertible). However, Grothendieck's terminology has stuck in some literature. Using this first motivation, we define products and coproducts of objects in a 2-category as we would in a category plus an extra 2-cell condition. Second motivation: We think of our 2-category K as a place to develop category theory so that arrows f : U --> A into an object A of K are thought of as generalised objects of A, and 2-cells into A are generalised arrows of A. Take a notion such as monad on A. From this motivation, reversing *2-cells* in K, we should get the notion of "comonad". This terminology is in conflict with the doctrine developed on the basis of the first motivation. Of course, "monad" is invariant under *arrow*-reversal, but there are other concepts which are not. --Ross