
School of Mathematics and Statistics
André Joyal
University of Quebec
The homology of symmetric and braided monoidal categories
Friday 20th November, 121pm, Carslaw 273.
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 Ktheory 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 Einfinity
space (resp. of an E2 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
En spaces. It shows that the homology of a symmetric
(resp. of a braided) monoidal category is a gradedcommutative
algebra admitting DyerLashof 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.
