From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2333 Path: news.gmane.org!not-for-mail From: Tom LEINSTER Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? Date: Wed, 4 Jun 2003 22:07:51 +0200 (CEST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset="US-ASCII" X-Trace: ger.gmane.org 1241018586 3644 80.91.229.2 (29 Apr 2009 15:23:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:06 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jun 5 16:50:31 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 05 Jun 2003 16:50:31 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19O0hs-0004Sy-00 for categories-list@mta.ca; Thu, 05 Jun 2003 16:47:28 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 91 Xref: news.gmane.org gmane.science.mathematics.categories:2333 Archived-At: This may be `mere' pedagogy for ordinary categories, but if you try the same thing for 2-categories then it becomes a `genuine' issue. To put it another way, the two different but equivalent presentations of a concept (natural transformation) become, on categorification, significantly different. First the 1-dimensional situation. As I understand it, Pat Donaly's original point was that given functors F, G: C ---> D between categories, you can either define a natural transformation in the standard way (assigning an arrow of D to each object of C) or in an alternative way (assigning an arrow of D to each arrow of C). With the standard method, vertical composition of transformations is "easy" to define, and horizontal composition is "hard". With the alternative method, horizontal composition is now easy to define (as Pat noted), but vertical composition is "hard". So the situation is reversed. Of course, neither of these "hard"s is really hard, but in both cases you have two evident ways of defining a composite - one left-handed, one right-handed - and if you're going to do anything whatsoever with the definition then you need to show that these two ways give exactly the same result. Now suppose that C and D are 2-categories and F and G are 2-functors. It doesn't matter whether C, D, F and G are strict or weak for the purposes of this discussion. Suppose we're interested in defining weak (=pseudo) transformations F ---> G. The usual way is to say that such a transformation consists of a 1-cell alpha_c : Fc ---> Gc for each c in C, together with an invertible 2-cell inside each naturality square, satisfying axioms. With this definition, vertical composition of transformations is easy to define (and there's only one evident way of doing it), but horizontal composition can be defined in two different ways, which are not equal but canonically isomorphic. An alternative approach is to say that a transformation consists of a 1-cell alpha_f: Fc ---> Gc' for each 1-cell f: c ---> c' in C, together with certain further 2-cells, satisfying axioms. You can guess the rest of this paragraph: with this definition, horizontal composition is now easily (and canonically) defined, but vertical composition can be defined in two different ways, which are not equal but canonically isomorphic. You might think that this isn't a genuine difference or "problem" so far, because everything is the same up to isomorphism. But now suppose that you're interested in *lax* transformations F ---> G (where F and G are still 2-functors, as above). The usual definition is that such a lax transformation alpha consists of a 1-cell alpha_c as above for each object c of C, and then a not-necessarily-invertible 2-cell inside each naturality square (pointing in a direction fixed by convention), satisfying axioms. These lax transformations can still be composed vertically perfectly easily, but horizontal composition is now *impossible* to define. (More accurately, you can define two different horizontal compositions, but they're not isomorphic, only connected by a non-invertible cell; you could of course choose one over the other, but it won't have good properties.) And if you define "lax transformation" according to the alternative method, then horizontal composition is now easy and vertical composition impossible. In summary, if you define transformation of 1- or 2-category in the standard style then vertical composition is always easy, and regarding horizontal composition: - for transformations of categories, it's canonically defined - for weak transformations of 2-categories, it's not canonically defined (you have to choose "left" or "right"), but is canonically defined up to isomorphism - for lax transformations of 2-categories, it's not defined at all. If you use the alternative style then the situation is similar but with "vertical" and "horizontal" reversed. Puzzling. Tom