A certain kind of simplicial complex

Question

I'm interested in collections $\mathcal{C}$ of tuples $\mathbf{t} = (n_1, n_2, \ldots, n_r)$ of positive integers satsifying

1. if $\mathbf{t}\in \mathcal{C}$ then so is any permutation of $\mathbf{t}$ and
2. if $\mathbf{t}\in \mathcal{C}$ then so is any sub-tuple of $\mathbf{t}$.

If I were talking about sets instead of tuples, then the second condition would mean that such a collection is a simplicial complex. Since I need tuples, I thought that $\mathcal{C}$ could be thought of as a simplicial set. But the reason I want to use tuples is to allow for repetitions (such as $(1,2,2,3,4)$); and the tuples with repeats are in no way degenerate in my context. So maybe I'm thinking about something like a sub-fat-simplicial-set inside the fat nerve (fat meaning "no degeneracies") $B_\mathbf{fat}\mathbb{N}$?

Can someone point me to some useful terminology or results about collections $\mathcal{C}$ like this?

Answer 1

I don't know your context, but maybe it will be useful to think of your structure as a semi-simplicial set http://ncatlab.org/nlab/show/semi-simplicial+set

Semi-simplicial set is a presheaf on the semi-simplicial category, which is like the simplicial category without degeneracy maps. Every simplicial category generates a simplicial set.

Your structure forms a semi-simplicial set as it seems implicit in the question. Maybe you want the simplicial set generated by this semi-simplicial set.