Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So globes (which are not any convex polyhedra but still are cell complexes), simplices and cubes (which are) are examples, giving, respectively, globular, simplicial and cubical sets, but there are many others (to mention just one more, several people (Baues, Fiedorowicz, Loday, Saneblidze, ...) initiated working with permutohedral sets; although permutohedra are not faces of each other, still each face is a product of permutohedra).
Now look at a particular task. $P_1^n$ must have decomposition into a bunch of $P_n$'s; the symmetric group $\Sigma_n$ acts on $P_1^n$ (as it does on the $n$th power of any object in any category with finite products), so it must permute these $P_n$'s. In this way one gets a permutation representation of $\Sigma_n$. For cubes, this is the trivial representation since $\Box_1^n$ is just $\Box_n$. For simplices, it is the representation of $\Sigma_n$ on itself via (either left or right) multiplication since $\Delta_1^n$ is the union of $n!$ copies of $\Delta_n$. Globes I don't even know how to handle since they do not readily come with any preferred decompositions of products of globes into globes.
What I want is this: is there a system $P_*$ as above such that $P_1^n$ decomposes into $n$ copies of $P_n$ and the symmetric group acts as it does standardly on an $n$-element set? I started to construct the system, it might be called "pyramidal sets" since $P_n$ seems to be forced to be taken the cone over the $(n-1)$-cube, but I had problems enumerating the needed face/degeneracy maps. Has anybody encountered this system anywhere? Is there a nice category (like finite linear orders in the simplicial case) which would represent the whole thing?
Maybe I'll add another (probably much simpler, but I still have no idea about an answer) question: the same for the sign representation, that is, action of $\Sigma_n$ on a two element set realized as $\Sigma_n/A_n$ (quotient by the alternating group). In other words, here $P_1^n$ must decompose into two copies of $P_n$, with odd permutations interchanging these copies and even permutations leaving them alone. This strongly suggests trying the globes but as I said I don't know how to decompose them.
