Iterated wreath product of the simplex category and iterated loop spaces

Generalising Segal's approach to 1-fold loop spaces, the homotopy theory of $n$-fold loop spaces is shown to be equivalent to the homotopy theory of reduced $\Theta_n$-spaces, where $\Theta_n$ is an iterated wreath product of the simplex category $\Delta$. A sequence of functors from $\Theta_n$ to $\Gamma$ allows for an alternative description of the Segal-spectrum associated to a $\Gamma$-space. In particular, each Eilenberg-MacLane space $K(\pi,n)$ has a canonical reduced $\Theta_n$-set model

Joyal's Suspension Functor on Θ\Theta and Kan's Combinatorial Spectra

In [Joyal] where the category $\Theta$ is first defined it is noted that the dimensional shift on $\Theta$ suggests an elegant presentation of the unreduced suspension on cellular sets. In this note we prove that the reduced suspension associated to that presentation is left Quillen with respect to the Cisinski model category structure presenting the $\left(\infty,1\right)$-category of pointed spaces and enjoys the correct universal property. More, we go on to describe how, in forthcoming work, inspired by the combinatorial spectra described in [Kan], this suspension functor entails a description of spectra which echoes the weaker form of the homotopy hypothesis, we describe the development of a presentation of spectra as locally finite weak $\mathbf{Z}$-groupoids

Involutory reflection groups and their models

A finite subgroup of $GL(n,\mathbb C)$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.Comment: 24 page