## Categories in Algebra, Geometry and Mathematical Physics

### Conference in honour of Ross Street's sixtieth birthday

#### July 11-16 2005, Macquarie University, Sydney

Tue, 12 July: 10:30 - 11:10

##### Iterated wreath product of the simplex category and iterated loop spaces

###### Berger, Clemens (Université de Nice -- Sophia Antipolis)

Certain small categories may serve to establish an explicit link between topological data and algebraic data. In this vein, Graeme Segal constructed infinite loop spaces out of special presheaves on $\Gamma$, while Bob Thomason studied the analogous construction of simple loop spaces out of special presheaves on the simplex category $\Delta$. The forgetful functor which associates to an infinite loop space the underlying simple loop space is induced by a canonical functor from $\Delta$ to $\Gamma$.

I shall interpolate between these two constructions and construct $n$-fold loop spaces out of special presheaves on the $n$-fold wreath product of $\Delta$. Like above, there is a canonical functor from the $n$-fold wreath product of $\Delta$ to $\Gamma$ corresponding to the obvious forgetful functor. Moreover, the inductive limit over $n$ of the $n$-fold wreath products of $\Delta$ is isomorphic to André Joyal's cell category $\Theta$, which is one way of connecting iterated loop spaces to higher categorical structures. The underlying combinatorics are intimately related to the multitude of trees as described by Michael Batanin and Ross Street.