A vague question on weak n-categories
Dear Categorists, A vague kind of question on higher-dimensional analogues of weak categories: basically I was wondering what the current state of play was regarding interchange laws up to isomorphism? Slightly more precisely, given an algebraic description of a natural family of isomorphisms that seem to express 'interchange up to isomorphism' or higher-dimensional analogues of this, what kind of properties should they be expected to have? Obviously, I'm not looking for exact coherence conditions, but I was more wondering whether there is a 'killer application' that rules in or out various possibilities. Peter 17-Jul-2002 16:31:03 -0300,8458;000000000000-0000001c
Peter Hines writes:
A vague kind of question on higher-dimensional analogues of weak categories: basically I was wondering what the current state of play was regarding interchange laws up to isomorphism? Slightly more precisely, given an algebraic description of a natural family of isomorphisms that seem to express 'interchange up to isomorphism' or higher-dimensional analogues of this, what kind of properties should they be expected to have?
A lot is known about this. You could start by looking at "Gray-categories", also known as "semistrict 3-categories". These satisfy the usual interchange law one finds in a 2-category, but only up to isomorphism, and the coherence laws satisfied by these isomorphisms are well-understood. In particular, a semistrict 3-category with one object is the same as a strict braided monoidal category, where the above "up to isomorphism" gives the braiding, and the coherence laws give the usual coherence laws satisfied by the braiding. You can read about this stuff in various places including John Gray's original work, Gordon Power and Street's big paper on tricategories, Kapranov and Voevodsky's big paper on 2-categories and Zamolodchikov tetrahedra equations, my paper with Martin Neuchl on braided monoidal 2-categories, and various papers on teisi by Sjoerd Crans, who pushes these ideas up to higher dimensions. You can find most of these references here: Sjoerd Crans On braidings, syllepses, and symmetries http://www-irma.u-strasbg.fr/~crans/papers/obss.html A crucial thing to keep in mind when fighting your way though this mass of material is that interchange laws and the coherence laws they satisfy have deep relations to geometry and topology, namely the theory of hyperplane arrangements and the theory of iterated loop spaces. One can use this geometry and topology to see if you're getting the right higher coherence laws, though ultimately we'd like to turn the picture around and see the geometry and topology as arising out of the n-categories. 18-Jul-2002 17:05:24 -0300,1933;000000000001-0000001e
Just to add one more reference to John's answer. There is a paper Balteanu C., Fiedorowicz Z., Schw\"{a}nzl R., Vogt R., Iterated Monoidal Categories, preprint, (1998), 1-55, where interchange laws and their interplay in all dimensions are considered. They are, however, not invertible morphisms. This noninvertibility makes this theory interesting. The main result of the paper is: the categories with n (strict) monoidal structures and interchange laws between them satisfying some natural coherence conditions model n-fold loop spaces. If $n=2$ and interchange law is isomorphism the theory collapses to the theory of braided monoidal categories. If $n>2$ but interchanges laws are still isomorphisms we get symmetric monoidal categories. It is also possible to define an n-categorical analogue of this theory, in such a way that one object, one arrow , one 2-arrow ,..., one (n-1)-arrow n-category in this sense is exactly n-fold monoidal category of BFSV. This is different from weak n-category but helps a lot in understanding of higher dimensional structures. I hope to finish a paper about it soon. Michael. 19-Jul-2002 14:32:55 -0300,3321;000000000000-0000001f
participants (3)
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baez＠math.ucr.edu -
Michael Batanin -
P.M.Hines