The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with

• simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$
• degeneracies: inserting $0$
• faces: adding consecutive entries or deleting end entries

I'm interested superadditive sequences of natural numbers (in connection with Lusternik-Schnirelmann category of spaces), and I've been led to a construction that takes a superadditive sequence and produces a sub-simplicial set of $C \subseteq N\mathbb{N}$ that has the additional property that it is closed under permutation of tuples.

So my questions:

• have such sub-simplicial sets been studied somewhere?
• are there `obvious' things that are true about them?
• what are their geometric realizations like?

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with

• simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$
• degeneracies: inserting $0$
• faces: adding consecutive entries or deleting end entries

I'm interested superadditive sequences of natural numbers (in connection with Lusternik-Schnirelmann category of spaces), and I've been led to a construction that takes a superadditive sequence and produces a sub-simplicial set   $C \subseteq N\mathbb{N}$ that has the following TWO additional properties:

• $C$ is closed under permutation of tuples
• if $( \bar k_1,\bar\ell) \in C$ and $\sum \bar k_1 = \sum \bar k_2$ then $(\bar k_2,\ell)\in C$ as well.

So my questions:

• have such sub-simplicial sets been studied somewhere?
• are there `obvious' things that are true about them?
• what are their geometric realizations like?

EDIT: Additional constraint on $C$ (substitution rule).