Algebra Seminar

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.