This is in reply to a message of James Stasheff and of other participants. I agree that the language of category is more than analogous to homotopy theory, and think that the connection is deep. The work on A-infinity spaces is a striking early illustration of the connection between homotopy coherence and categorical coherence. Homotopy theory is the main guideline for the emergence of higher category theory. For example, the first correct concept of 3-category was discovered as a direct outcome of modelling homotopy 3-types with Gray groupoids. Before this example, all existing 3-categories were equivalent to strict ones. In my opinion, after a first stage of developpement, higher category theory should induce new progress in homotopy theory. This is the real test. In particular, the new theory should help understanding homotopy groups of spheres. I am including an abstract of my talk on a related subject. Regards, Andre Joyal
University of Sydney Algebra Seminar Friday 20th November, 12-1pm, Carslaw 273
Title: The homology of symmetric and braided monoidal categories Abstract: We wish to comment on some aspects of the connection between category theory, topology and homological algebra. The connection is at the root of higher K-theory and it is guiding much of the actual research on higher dimensional categories. We shall concentrate on the relation between monoidal categories and iterated loop spaces. To each category C we can associate a space BC called the (Milgram) classfying space of C. The homology of C is defined to be the homology of BC. The space BC is a monoid when C has a tensor product, and it has the structure of an E-infinity space (resp. of an E-2 space) if the tensor product is symmetric (resp. braided). We shall briefly discuss the work of P. May and F. Cohen on the homology of E-n spaces. It shows that the homology of a symmetric (resp. of a braided) monoidal category is a graded-commutative algebra admitting Dyer-Lashof operations (resp. is a poisson algebra). These structures play a crucial role in determining the homology of the symmetric groups and of the braid groups. The poisson algebra structure also appears in the recent work of Lehrer and Segal on the rational homology of classical regular semisimple varieties.
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Best wishes, Andrew Mathas