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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).

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    $\begingroup$ Your simplicial set is actually a $\Gamma$-set (in the sense of Segal): the $\Gamma$-structure in particular encodes the symmetric group actions. So perhaps you are really interested in sub-$\Gamma$-sets? (Not that I have anything to say about those either.) $\endgroup$
    – Charles Rezk
    Commented Nov 5, 2015 at 18:20
  • $\begingroup$ I'll look into it! $\endgroup$
    – Jeff Strom
    Commented Nov 5, 2015 at 18:36

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