Date: Sun, 19 Jan 92 20:41:52 CST From: kapranov@math.nwu.edu (Mikhail Kapranov)
I would like to be able to place a good reference about representable functors from a weak 2-category (or a bicategory,--this is what Benabou considered) to Cat -the 2-category of all categories. For example the correspondence that takes each object to the corresponding representable 2-functor is a (weak) equivalence of 2-categories (analog of Yoneda's lemma). This, in particular, shows at once that any WEAK 2-category is weakly 2-equivalent to a STRICT one (where associativity for composition of 1-morphisms and the units arer strict). I suppose these things are well known to people, but what would be a good reference?
Mikhail Kapranov
The Yoneda Lemma for bicategories appears in my paper "Fibrations in bicategories" Cahiers top et geom diff 21 (1980) 111-160 [A correction, not to do with the Yoneda Lemma, is in 28 (1987) 53-56.] The application of this result to coherence for bicategories was realised in the last year by Robert Gordon and John Power. It was discussed very openly at the Category Theory Conference in Montreal Mid 1991. I remember knowing of the application of the Yoneda Lemma to homomorphisms [essentially that every fibration is equivalent to a split one, as per Giraud's book] in the late 60's, but I do not recall observing, and know of no old reference for the Gordon-Power realisation. [For readers who missed out in Montreal, the point is that: given a bicategory K, there is a homom of bicategories K --> Hom(K^op,Cat) which is an equivalence on hom-categories; but Hom(K^op,Cat) is a 2-category; so K is biequivalent to the full sub-2-category of Hom(K^op,Cat) consisting of the representables. This gives a different proof of Mac Lane's coherence theorem for monoidal categories and of the Mac Lane-Pare version for bicategories.] A full account of a very slightly modified version of this approach to coherence of monoidal (=tensor) categories appears in Joyal-Street "Braided tensor categories" (still to appear in Advances in Math). We go on to do coherence for tensor functors (=strong monoidal functors) and braided tensor categories. Furthermore, Gordon, Power and I have used a similar technique to prove a coherence theorem for tricategories. This was announced in the Abstracts for the American Math Soc Meeting in Baltimore a week or so ago; Bob Gordon gave a talk at that conference. We have a paper in preparation. [Tricategories are like 3-categories except that composition of 1-cells is only associative up to equivalence and is only a homomorphism of bicategories &c &c.] While every bicategory is biequivalent to a 2-category, not every tricategory is triequivalent to a 3-category; the Gray tensor product is relevant, as expected after the work of Joyal-Tierney on homotopy 3-types. Regards to all, --Ross Street ======================================