Since the associahedra (my K_n complexes) exist for all n, can they be regarded as (generating) an \omega-category?? jim =============================
Date: Sat, 26 Oct 91 15:24:58 EDT From: James Stasheff <jds@charlie.math.unc.edu>
Since the associahedra (my K_n complexes) exist for all n, can they be regarded as (generating) an \omega-category?? jim
In retrospect, your associahedra are very closely related to my orientals: something like the geometric realization of their nerve. When writing about orientals, I had categorical coherence and cohomology more clearly in mind than homotopy (except for the vague analogy -which I have been trying to make precise since 1968- that 2-cells are like homotopies). In order to generate an omega-category, one needs to divide the boundary of each cell decisively into two parts. There is a bit of choice about how this is done with your construction, isn't there? I seem to remember that you have more cells than the orientals. Also, do you want inverses to your cells as exist at the homotopy level? Regards, Ross ===========================
two good questions the division into two pieces is relevant to trying to change the geometry into non-abelian cohomology - the boundary as a difference have been looking at some related questions that come up in quasi-bialgebras at the homotopy level, I always have inverses (up to homotopy) - just run the parameter backwards but work in quasi-bialgebra AND in string field theory both suggest imposing stricter inverses e.g. take the category of paths in a space X with endpoint joining of paths as compositions mod out by the relation fg = Id_x and gf=I_y where g is a path from x to y and f is its parameter reverse jim
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