Dear categorists, Is there a good terminology for "relaxed transformations" between 2-functors and the "relaxed adjunctions" where they appear as unit/counit? Generally, I find the Kelly-Street terminology (LNM 420, 1974) for 2-dimensional category theory very convenient: clear, elegant and as simple as possible. However, I have problems in the cases recalled above. 1. Let us agree, with Kelly-Street, that a LAX functor F (between 2-categories, to simplify things) has comparison cells F(g).F(f) --> F(gf), while an OP-LAX one has comparison cells the other way round. (It is quite reasonable to take the first notion as the leading one, since it is related with LIMITS and MONADS, while the opposite is related with COLIMITS and COMONADS). 2. Now, let us say for the moment that an "XXX-transformation" phi: F --> G has comparison cells (corresponding to maps a: A --> B in the domain) phi(a): phi(B).Fa --> Ga.phi(A), while an "op-XXX-transformation" has reversed comparison cells. Let us also speak of an "XXX-adjunction" (between strict 2-functors, to simplify things) to mean that its unit and counit are XXX-transformations; similarly for op-XXX-adjunctions (examples below, point 4). Such notions (and precise definitions) were introduced in the 70's. An XXX-transformation is called: - a "quasi-natural transformation" in Bunge LNM 195 (1971) and Trans. AMS (1974); - a "quasi_d natural transformation" in Gray LNM 391 (1974) ("d" for down); - an "op-lax natural transformation" in Kelly, On clubs and doctrines, LNM 420 (1974). An XXX-adjunction is called a "formal lax adjunction" in Bunge, Trans. AMS. An op-XXX-adjunction is called a "weak quasi-adjunction" in Gray. 3. Here, I find the term "lax" or "op-lax" misleading, since LAX functors can have both XXX- and op-XXX-transformations. Moreover, "lax adjunction" would seem to point to the functors rather than to unit & counit. 4. A few simple examples of XXX- and op-XXX-adjunctions can be of help in clarifying things. Take the strict 2-functors p: RelAb --> 1, i: 1 --> RelAb, sending the one object to the null group 0. We have: (a) an XXX-adjunction p -| i, with XXX unit 1 --> ip, sending an object A to the greatest relation A --> 0 (and trivial counit pi = 1); (b) an op-XXX-adjunction p -| i, with op-XXX unit 1 --> ip, sending an object A to the least relation A --> 0; (c) an XXX-adjunction i -| p, with XXX counit ip --> 1, sending an object A to the least relation 0 --> A; (d) an op-XXX-adjunction i -| p, with XXX unit ip --> 1, sending an object A to the greatest relation 0 --> A. These examples also show (perhaps) some motivation for taking the XXX notion as the leading one: - (a) shows the null group as an XXX-terminal object via the TERMINAL relation to it, - (c) shows the null group as an XXX-initial object via the INITIAL relation from it. 5. Now we are left with finding a good term for "XXX". (a) I would avoid "lax", as already motivated. (b) "weak" should probably be avoided as well, as is often used in the same sense of "pseudo", i.e. with reference to invertible comparisons; moreover, "weak terminal" means a different thing in 1-category theory. (c) "quasi" might be acceptable? It would give "quasi transformation" (or quasi natural transformation") , "op-quasi transformation", "quasi adjunction", "op-quasi adjunction", "quasi terminal",... _________ I would appreciate comments, criticism and suggestions. Best regards to all Marco Grandis Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/