Date: Thu, 21 Sep 1995 23:46:24 -0300 (ADT) Subject: braided monoidal categories Date: Thu, 21 Sep 1995 12:47:24 -0700 From: john baez I am posting the following for James Dolan, jdolan@math.ucr.edu: is there any well-known way to "turn a braided monoidal category into a simplicial set"? in particular, have you ever heard of anything along the following lines: given a braided monoidal category x, get a simplicial set x# as follows: there's only one 0-simplex. there's only one 1-simplex. the 2-simplexes are the objects of x. the 3-simplexes with 2-faces (in order) a,b,c,d are morphisms a@c -> d@b. the 4-simplexes with 3-faces (in order) k a@d -> g@b l a@e -> h@c m b@f -> i@c n d@f -> j@e o g@i -> j@h are commutative diagrams k@f g@m o@c -> g@b@f -> g@i@c -> a@d@f j@h@c -> a@j@e -> j@a@e -> a@n *@e j@l where "*" is the braiding. the simplexes of dimension 5 and higher are "co-skeletal". the degeneracy maps are, well, i didn't try to work out what they should be yet; maybe it's obvious what all of them should be or maybe it isn't, but at worst you could just have x# be a "simplicial set without degeneracy". there's obviously a relationship between the form of the data constituting a 3- or 4-simplex here and the yang-baxter equation (at least it's obvious if you can decipher my notation), but i'm not quite sure if the relationship is such that what i'm describing is just a completely obvious re-statement of the usual relationship between simplicial geometry and the yang-baxter equation. i'd also like to know of a nice description of some extra structure on the simplicial set x# that would amount to precisely the ability to reconstruct the braided monoidal category x from it. (that's not quite a perfectly well-defined problem the way i just stated it, though.) of course a description of the general relationship between simplicial sets and some sort of "tri-categories" more general than braided monoidal categories would be just as good. Date: Fri, 22 Sep 1995 15:22:29 -0300 (ADT) Subject: Re: braided monoidal categories Date: Fri, 22 Sep 1995 22:16:09 +1000 (EST) From: Ross Street Subject: Re: braided monoidal categories This is a response to the Dolan/Baez question under hard circumstances. I am at Oberwolfach and the delay on the computer via Macquarie University is extreme. In fact, in my talks here at the Descent Theory conference I have described how to construct a descent n-category of a cosimplicial n-category. On the way I defined the nerve of any "file". A 1-file is a category, a 2-file is 2-category, a 3-file is a Gray-category in the sense of Gordon-Power-Street. A 1-object, 1-arrow 3-file is a braided strict monoidal category. It is also possible to define the nerve of a tricategory; a 1 object, 1 arrow tricategory is a general braided monoidal category. Actually, I like to distinguish the braided monoidal category V from the tricategory which I think of as the double suspension of V. The nerve of the double suspension gives the 2-fold delooping of the nerve of V as a mere category. I have written a bunch of notes for this conference where this is all spelt out pretty well. There is also a hint in the published paper JW Duskin The Azumaya complex of a commutative ring SLNM 1348 (1988) 107-117 See page 116 where there is an explanation of a suggestion from me. The point about interpreting the 4-simplex in a tricategory (or Gray-category) is that the pentagonal condition becomes a hexagon since there is a middle-4-interchange needed in the middle of the short path around the hexagon. Since Gray-categories don't admit a unique horizontal composition of 2-cell, an extra edge is introduced pentagon. Bye for now. Ross Date: Mon, 25 Sep 1995 15:43:08 -0300 (ADT) Subject: Re: braided monoidal categories Date: Sat, 23 Sep 95 17:43:48 CDT From: David Yetter Regarding the construction of a simplicial set from a bmc: This is a special case of constructing the nerve of a tri-category. A monoidal category is a bicategory with one object. A monoid in monoidal categories is a braided monoidal category. A braided monoidal category is a tri-category with one object and one map. (Citations anyone? or are these all folk theorems?) David Yetter Date: Sat, 30 Sep 1995 14:25:17 -0300 (ADT) Subject: Braided monoidal 2-categories Date: Fri, 29 Sep 1995 16:27:23 -0700 From: john baez Martin Neuchl and I have written a paper entitled "Higher-dimensional algebra I: braided monoidal 2-categories". In it, we begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d topological quantum field theories and 2-tangles. Then we give concise definitions of semistrict monoidal and braided monoidal 2-categories, and show how these may be unpacked to give definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category as the "center" of a semistrict monoidal 2-category, in a manner analogous to the construction of a braided monoidal category as the center (or "quantum double") of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2-categories. This paper is available by anonymous ftp to math.ucr.edu. It is the file bm2cat in the directory baez. It is in uuencoded, compressed form, because it is 51 pages with lots of pictures. To print it, first save the file as "bm2cat" and do "uudecode bm2cat". This should produce a file "bm2cat.dvi.Z". Then do "uncompress bm2cat.dvi.Z". This should produce a file "bm2cat.dvi," which you can print out in the way you usually print dvi files. John Baez Date: Tue, 3 Oct 1995 20:57:44 -0300 (ADT) Subject: Re: braided monoidal categories Date: Tue, 3 Oct 1995 09:24:12 -0400 From: Stephen Chase This is a belated reply to the query of John Baez regarding the nerve of a braided monoidal category. I'm no expert on these matters and hence may be way off target, but it seems to me that, for a *symmetric* monoidal category, the appropriate (and most useful) analogue of the classifying space should be the spectrum whose homotopy groups yield the algebraic K-theory of the category. There is a simplicial version of spectra which should then provide a suitable nerve. Some of the relevant literature on this is listed below (see especially [4, Section 5] and [5, pp. 1599-1600]). [1] D. Grayson, Higher algebraic K-theory II (after D. Quillen), Springer LNM 551 (1976). [2] M. Jardine, Supercoherence, J. Pure Appl. Alg. 75 (1991), 103-194. [3] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293- 312. [4] R. Thomason, Algebraic K-theory and etale cohomology, Ann. Scient. Ec. Norm. Sup. 13 (1980), 437-552. [5] ____________, First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. in Alg. 19 (1982), 1589-1668. It is natural to ask whether a stable weak n-category (in the sense of Section 5 of the recent preprint by Baez and Dolan, "Higher-dimensional Algebra and Topological Quantum Field Theory") also has a "classifying space" spectrum. Such an object, if it exists, should provide a means of constructing cohomolgy theories in which such n-categories are used as coefficients, in a manner analogous to that in which cochain complexes are used as coefficients in the hypercohomology theories of ordinary homological algebra. In general, I think that algebraic K-theory, which makes heavy use of both category theory and stable homotopy theory, should provide clues regarding what phenomena to expect in higher-dimensional algebra. Steve Chase Date: Tue, 10 Oct 1995 10:03:43 -0300 (ADT) Subject: Re: braided monoidal categories Date: Mon, 09 Oct 1995 10:57:22 -0400 (EDT) From: MTHDUSKN@ubvms.cc.buffalo.edu The question that you raised was exactly the on that I addressed in my talk at the recent Dalhousie conference in July. The abstract follows but if you will send me a snail mail address, I will send the full manscript with the full set of rather pleasing geometric diagrams which , in my view, makes the role of the braiding quite transparent. Abstract: SOME APPLICATIONS OF 2-CATEGORY TECHNIQUES IN THE THEORY OF BRAIDED TENSOR CATEGORIES John Duskin SUNY/Buffalo Two years ago, in a Montreal talk before an audience composed mostly of category theorists, Pierre Cartier suggested that the theory of 2-categories might prove useful in the study of quantum groups and related topics. This work is the result of one such investigation. A strict tensor (=monoidal) category may be equivalently viewed as a monoid object in Cat or a category object in Mon. As a cat-monoid, its nerve is a simplicial monoid whose underlying double complex is that of 2-category with a single object(= 0-cell) with 1-cells provided by the objects of the tensor category and composition of 1-cells provided by the tensor product . The arrows of the tensor category and their composition give the 2-cells and their composition, and the functoriality of the tensor product give the *-composition of 2-cells and the "interchange law". Strict tensor functors between tensor categories become strict 2-functors between 2-categories with a single object, and it is from this entirely equivalent point of view that we wish to pursue the subject of this talk. First, about Joyal-Street "braidings": In any 2-category one can form the double category of "lax-(commutative) squares" which consists squares of composable 1-cells together with a 2-cell connecting the compositions of the 1-cells on each side of the diagonal. There is an obvious "horizontal composition" of such lax-squares as well as a vertical one and they are easily seen to satisfy the interchange law of a double category. Moreover, there is an obvious way to introduce 2-cells between lax-squares which makes the lax-squares into a category object in 2-Cat which plays it play the role of representing "lax functors between 2-categories". Now, among these lax-squares are those which define the "epi-center" of the 2-category, the lax squares whose two vertical 1-cells are identical and whose horizontal 1-cells are as well. This condition forces all of the corners of square to be the same 0-cell and to have its "interior" to be 2-cell of the form A*B=>B*A when the 2-category is that associated to a strict tensor category. We define a (normalized) braiding on a 2-category with a single object C as a bi-functorial (i.e., functorial in each variable) section c of the epi-center of C cA.B:A*B=>B*A is easily seen to satisfy exactly the conditions of the Joyal-Street definition, including naturality, when the associativity isomorphisms are all identities, except that cA.B is not required to be invertible but we require c1.B=id(B) and cA.1=id(A), "normalization". The chief advantage of this 2-category point of view of strict tensor categories is that it gives an easy " geometric" way to handle braidings using the Sydney-School's highly efficient way of dealing with equations and proofs in 2-categories using "pasting diagrams". Now any 2-category C has its own "oriented geometric nerve" which can be defined as the simplicial set which has the 0-cells of C for 0-simplices and the 1-cells for 1-simplices. 2-simplices x are defined as triangles of 1-cells together with a 2-cell x: d1(x)=>d2(x)*d0(x) for interior. 3-simplices consist of the "commutative tetrahedra" each made of four compatible 2-simplices of the foregoing sort for which the unique composition of the odd faces is equal to that of the even faces, a condition easily expressed as an equality of the corresponding pasting diagrams. Degeneracies are equally well supplied, and the full nerve is just the coskeleton of the just defined truncated complex. From a simplicial point of view this nerve just corresponds to the classifying space of the simplicial category defined by the original 2-category which is its fiber. In the case here at hand of a tensor category this geometric nerve is a reduced (i.e., only one 0-simplex) simplicial set and its fiber is the simplicial monoid defined by the strict tensor category. The nerve is thus seen to form the first step in a simplicial spectrum as we will see below. (One equally well has the oppositely oriented version of this nerve where the 2-simplices interior is of the form x:d0(x)*d2(x) =>d1(x). Both are needed in the applications). Now suppose that the tensor category has a braiding c, then the nerve of C has the structure of a simplicial monoid which is trivial in dimension 0, has the composition of 1-cells (tensor product) as multiplication in dimension 1, but uses the braiding to define the product of two arbitrary 2-simplices x and y using their interiors by pasting the square c(x2,y0) between the triangles x and y to form a new 2-simplex whose faces are exactly the products of the corresponding 1-cell faces (so that the face maps become homomorphisms). this product is associative and untitary (using s0(1) as unit) and the product of commutative tetrahedra is commutative. Thus a braiding defines the structure of a simplicial monoid on the classifying space of (C) with C as its loop complex. This allows us to iterate the construction for one more step and obtain a new simplicial set 2(C) which has 1(C) as its fiber.[2(C) is dgenerate in dimensins 0 and 1,2-simplices are provided by the 1-cells of C viewed as the "interiors of 2-simplices all of whose faces are degenerate. 3-simplices are squares of 1-cells numbered in the odd-to -even fashion with 2-cells now forming the interior. 3-simplices are "commutative cubes" made out of the simplicial kernel with the braiding furnishing the "missing square"] 2(C) is supplied the structure of a simplicial monoid if and only if the braiding is symmetric (c2= id), in which case the iteration can be continued indefinitely. (That symmetric monoidal categories are infinite loop spaces was first observed by Peter May, so what we have here is the low dimensional fragment of the symmetric case.) Now given any structure of a simplicial monoid on the nerve of a tensor category viewed in the fashion which has C as its kernel, the product of the degenerate 2-simplices s1(x)s0(y) may be seen to define a 2-cell c(x,y):x*y=>y*x which may be seen to define a braiding on the original tensor category C and we have the Theorem: There is a bijective correspondence between braidings on a strict tensor category and simplicial monoid structures on it nerve. They define spectra if and only if the braiding is symmetric. Note: Most of the objects of interest in braided tensor categories such as algebras, co-algebras, bialgebras, braided-algebras, commutative algebras modules etc. have a pretty geometric picture when put in this 2-category frame, for example, an associative co-algebra is just a commutative tetrahedron in the above sense which has all of its 1-cell faces equal.