Higher-dimensional categories
I'm very interested in the work of Louis Crane, Dan Freed and others on how extending TQFTs to higher dimensions or higher codimensions can be nicely phrased in the language of n-categories. Luckily James Dolan is here at UCR now and is educating me in such matters. He and I are beginning to struggle towards a nice concept of "weak n-categories," or even better "weak omega-categories," or perhaps even better, "weak Z-categories" (in some sense a homotopical analog of chain complexes). I am interested in these things for doing physics, but I dimly realize that lots of people have struggled with these concepts, and I want to get a better grasp of what the key achievements and problems in this subject are. I have read Kapranov and Voevodsky's massive preprint on Braided Monoidal 2-categories, 2-vector spaces and Zamolodchikov tetrahedra equations, and soon I should receive a copy of the new paper on coherence in 3-categories by Gordon, Powell, and Street. What are the other main things I should find out about? How come all you categorists haven't yet invented a notion of "weak Z-categories," in which there are n-morphisms for all integers n, and all relations between n-morphisms are expressed in terms of (n+1)-morphisms? Regards, John Baez ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I've been working for several months on an approach to higher dimensional composition and the accompanying higher dim category theory (functors, natural transformations, limits, etc...). (It may be related to Vogt's work, cited in Barr's reply, but I'm unfamiliar with that work.) I'd be happy to send a "work-in-progress" report (in Latex format) giving details on what I've discovered thus far. The "dimension 1" cases are ordinary categories. In the dimension n case: objects are (n-1)-simplexes, maps are n-simplexes and composites of maps are (n+1)-simplexes of a simplicial set C. As Barr described in his response, the partial binary operation which defines composition in an ordinary category, associates a map (= 1-simplex) to a pair of maps whose source and target match appropriately. In the n-dim case there is a partial (n+1)-ary operation which, given n+1 n-simplexes whose faces match appropriately, produces their "composite", another n-simplex whose faces match the given ones according to the simplicial identities. One has to specify *which* of the faces of the n+1 simplex is the composite of the others. It can be the face opposite vertex i for any i = 0,..., n+1. The most efficient way to specify all this is by saying that the standard function from C_{n+1} ( = the n+1 simplexes of C) to the set of open i-boxes of C of dimension n+1 is an isomorphism. Examples (though special cases) of such structures are already known: n-dimensional hypergroupoids. These include Eilenberg-MacLane spaces and also arise from the singular complex of any topological space. Rreferences to hypergroupoids include my 1982 paper ("Realization of Cohomology Classes in Arbitrary Exact Categories", Jour. of Pure and Appl. Alg. 25 (1982) 33-105) and papers by Jack Duskin. There seems to be a reasonable "category theory" for this kind of higher-dimensional composition. For example, functors C --> C' and natural transformations of such functors occur as simplexes in the function complex C'^C (of dimensions n-1 and n respectively). If C and C' are higher dim categories in the sense I described above, then so is C'^C. Paul Glenn Dept. of Mathematics Catholic University of America Washington, DC 20064 Internet: glenn@cua.edu Phone: 202 319 5221 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I am very interested in seeing the paper of Paul Glenn, but in the meantime, perhaps relevant is Conjecture 5.3 of my "The algebra of oriented simplexes" JPAA 49 (1987) 283-335. This is a slightly more explicit form of a conjecture ( pre 1978 ) of John E. Roberts who motivated my work. Dominic Verity has proved this conjecture (it was announced in Bangor early this year and at the MSRI Conference last July). Verity's preprint on this will be available very soon. This work characterizes n-categories as simplicial sets with certain elements at each dimension distinguished (and called "hollow" in loc cit, but now we call them "thin" in accord with the Welsh School). The kinds of operations Barr and Glenn hint at are expressed as UNIQUE horn filler conditions. Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (3)
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baez＠ucrmath.ucr.edu -
GLENN＠CUA.EDU -
street＠macadam.mpce.mq.edu.au