Date: Sat, 23 Sep 1995 10:55:55 -0300 (ADT) Subject: pseudonatural transformations Date: Fri, 22 Sep 95 16:57:49 BST From: john baez Is there a good reference somewhere for the definition of a "pseudonatural transformation" between strict 2-functors between 2-categories, and of a "modification" between such pseudonatural transformations? In Springer Lecture Notes 420, Kelly and Street define lax natural transformations between strict 2-functors, and (apparently following Benabou) modifications between strict natural transformations. The definitions I want are hopefully straightforward modifications of these, but I don't actually know if anyone has given them anywhere. Perhaps it's up to me? John Baez Date: Wed, 27 Sep 1995 07:23:56 -0300 (ADT) Subject: RE: pseudonatural transformations Date: Wed, 27 Sep 1995 09:35:08 +0100 From: Marco Grandis As John Baez points out, a "pseudonatural transformation between strict 2-functors between 2-categories" is just the obvious particular case of a lax natural transformation; the paper by Kelly-Street he refers too, in SLNM 420, provides a general, coherent terminology for relaxed 2-categorical notions, as well as references to their earlier appearences. I think it already provides the reference he is looking for. The following paper might contain things connected with his interests. A definition of the precise cases he is interested in, pseudonatural transformations and their modifications, is written out in S 0.6 R. Betti - M. Grandis, Complete theories in 2-categories, Cahiers Top. Geom. Diff. Cat. 29 (1988), 9-57. Marco Grandis. Date: Wed, 27 Sep 1995 14:38:14 -0300 (ADT) Subject: pseudonatural transformations Date: Wed, 27 Sep 1995 11:58:13 +0100 From: Prof R. Brown In the groupoid case a theory of higher homotopies is written out in terms of the monoidal closed structures on the categories of \omega-groupoids (= ``cubical'' \omega-groupoids with connections, in some other terminology) and the equivalent category of crossed complexes in the paper R. BROWN and P.J. HIGGINS, ``Tensor products and homotopies for $\omega$-groupoids and crossed complexes'', {\em J. Pure Appl. Alg.}, 47 (1987) 1-33. In these categories, we easily deal with ``lax equivalences'' as ``homotopies'', i.e. maps C \otimes I \to D, where I is the usual unit interval groupoid. Since these categories are equivalent to that of ``globular'' infinity- (omega?)-groupoids, (Brown and Higgins, CTGDC, 1981) one gets a corresponding theory there, but without such explicit formulae. So I suppose it is not quite so clear this theory extends the lax 2-groupoid theory, but it seems highly likely. There are quite a few recent applications of these methods, for example to coherent and to equivariant homotopy theory. See for example R. Brown, M. Golasinski, T.Porter and A.P.Tonks), ``On function spaces of equivariant maps and the equivariant homotopy theory of crossed complexes''. A.P. Tonks, ``Theory and applications of crossed complexes'', Bangor PhD Thesis, 1993. I expect that recent work of Crans, Steiner, Street et al (not all together) gives a category version of these groupoid formulations for the omega-category case, but for the cubical version (preferably with connections?) it all seems clear, since one uses cubes to define tensor. Ronnie Brown Prof R. Brown School of Mathematics Dean St University of Wales Bangor Gwynedd LL57 1UT UK Tel: (direct) +44 1248 382474 (office) +44 1248 382475 Fax: +44 1248 355881 email: mas010@bangor.ac.uk wwweb for maths: http: //www.bangor.ac.uk/ma