Date: Fri, 18 Oct 1996 11:02:48 -0300 (ADT) Subject: *-autonomous 2-categories Date: Fri, 18 Oct 1996 15:05:35 +0200 From: Gian Luca Cattani I am a PhD student at the Computer Science Department of Aarhus University. I would like to ask, whether anybody knows of extensions of the notion of *-autonomous category to 2-categories, i.e., what does it mean for a 2-category to be *-autonomous. Regards, Luca Cattani -- +-------------------------------------------------------------+ | Gian Luca Cattani (luca@brics.dk) | +-------------------------------------------------------------+ | ========= BRICS: Basic Research In Computer Science | | ==== ==== Computer Science Department | | ========= Aarhus University | | ==== ==== Ny Munkegade Bldg. 540 | | ==== ==== 8000 Aarhus C, Denmark | | ==== ==== Phone : (+45) 8942 3472 Fax : (+45) 8942 3255 | +-------------------------------------------------------------+ Date: Sun, 20 Oct 1996 19:46:07 -0300 (ADT) Subject: Re: *-autonomous 2-categories Date: Fri, 18 Oct 1996 11:01:43 -0700 (PDT) From: john baez Luca Cattani writes: > I am a PhD student at the Computer Science Department of Aarhus University. > I would like to ask, whether anybody knows of extensions of the notion of > *-autonomous category to 2-categories, i.e., what does it mean for a > 2-category to be *-autonomous. I'd like to know this too. What Laurel Langford and I have are notions of: 2-category with duals, monoidal 2-category with duals, and braided monoidal 2-category with duals. Here "with duals" refers to the fact that objects, morphisms, and 2-morphisms all have "duals". This may be stronger than what you are thinking of, since one can consider situations where only the j-morphisms for certain j have duals. One may be able to remove clauses and get a definition of *-autonomous 2-category. If you like, I could send you our definition. It is probably not the ultimate definitive definition but it is good enough for our present purposes. It may help if I explain what our present purposes are. In the paper "Higher-dimensional algebra and topological quantum field theory" (Jour. Math. Phys. 36 (1995), 6073-6105), James Dolan and I proposed a general conjecture relating the topology of k-tangles in n dimensions to n-category theory. In a special case this says roughly the following: isotopy classes of framed 2-dimensional surfaces embedded in R^n are in 1-1 correspondence with certain 2-morphisms in a certain special 2-category. In the case n = 2 this would be the "free 2-category with duals on one object"; for n = 3 this would be the "free monoidal 2-category with duals on one object", and for n = 4 this would be the "free braided monoidal 2-category with duals on one object." In "Higher-dimensional algebra I: braided monoidal 2-categories" (Adv. Math. 121 (1996), 196-244) Martin Neuchl and I gave a precise definition of braided monoidal 2-categories (which corrects the definition of Kapranov and Voevodsky, and has subsequently been given a more elegant formulation by Day and Street). In "HDA IV: 2-tangles", Laurel Langford and I prove the theorem relating surfaces in 4-dimensional space to the free braided monoidal 2-category with duals on one object. We are still writing this up. A nice example of a braided monoidal 2-category with duals is the 2-category of (finite-dimensional) 2-Hilbert spaces, discussed in "HDA II: 2-Hilbert spaces". This is available at http://math.ucr.edu/home/baez/ I think this should be a *-autonomous 2-category by any reasonable definition thereof. Best, John Baez Date: Wed, 23 Oct 1996 11:37:37 -0300 (ADT) Subject: Re: *-autonomous 2-categories Date: Wed, 23 Oct 1996 11:15:10 +1000 From: Ross Street Response to Gian Luca Cattani Perhaps I can add some remarks to John Baez's helpful reply. The correct notion of monoidal bicategory seems to be tricategory with one object (see Memoirs AMS #558 Sept 1995). There is a coherence theorem (loc cit) which says each monoidal bicategory is appropriately equivalent to a Gray monoid M (also called semistrict monoidal 2-category); and these are easier to work with (although the tensor product # : M x M --> M is only a special kind of pseudofunctor, not a 2-functor in general). The paper by Brian Day and me that John Baez mentions (called "Monoidal bicategories and Hopf algebroids" - submitted) also defines closed Gray monoid. There are no surprises here: a right hom in M is an object [a,b] together with a pseudonatural equivalence of categories M(a#c,b) <--> M(c,[a,b]). We write [a,b]' for a left hom: M(c#a,b) <--> M(c,[a,b]'). We call M closed when each pair of objects has a left and right hom. Of course, if M is braided then [a,b]' = [a,b]. We do not discuss *-autonomous monoidal bicategories (although we considered including *-autonomous objects in M since we have some examples). What I suggest is that a monoidal bicategory is *-autonomous when it is closed and has a dualising object d. By this last I mean that the canonical a --> [[a,d]',d] should be an equivalence. Hope this is of some use. --Ross