On braided monoidal categories and double loop spaces
Clemens Berger – 5 March 1997
I first introduced the little squares operad of Boardman-Vogt and mentionned that an operad-action of the little squares on a topological space X induced a double delooping of X. The geometric idea behind this is the realization of the little squares as a subspace of the space of pointed maps of the 2-sphere into a bunch of 2-spheres.
Then I outlined a proof of Fiedorowicz's theorem that the geometric realization of a braided monoidal category admits a double delooping. By Joyal-Street's Coherence Theorem for braided monoidal categories there is actually a categorical non-$\Sigma$-operad action of the weak Bruhat orders on each (lax) braided (strict) monoidal category. After taking nerves, this operad-action extends canonically to a simplicial $\Sigma$-operad-action. The proof that this simplicial operad is equivalent to the little squares operad depends on a canonically defined filtration of the standard simplicial $E_\infty$-operad.
I finally discussed a categorical double delooping construction which assigns to a braided monoidal category a lax 3-category (i.e. a category enriched over the category of 2-categories with its closed structure given by Gray's tensor product). The assignment is just shifting up dimension twice, the braidings transform into Gray composition.
The nerve of a lax 3-category is defined by means of a cosimplicial object in the category of lax 3-categories whose strictification gives exactly Street's (3-categorical) orientals. The 2-category between initial and terminal object of the lax n-th oriental is a 2-categorical (n-1)-cube, whose 2-cells have an interpretation in terms of the weak Bruhat orders. Gray composition induces geometrically the cartesian product of cubes.