For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To be completely explicit, $C \in \Omega(n)$ means that $C$ is a subset of $\mathcal{P}([n]) \setminus \{\varnothing\}$ such that $\sigma\in C$ and $\varnothing\neq\sigma'\subseteq\sigma$ implies $\sigma'\in C$.
If $g\colon[n]\to[k]$ is order-preserving and $C \in \Omega(n)$, let me denote $g_!(C)$ the simplicial complex whose elements are the $g(\sigma)$ for $\sigma\in C$.
Let me say that $C \in \Omega(n)$ is a contraction-compatible complex when: for every $k\leq n$, the simplicial complex $g_!(C)$ is the same for every $g\colon[n]\twoheadrightarrow[k]$ order-preserving and surjective. In other words, no matter how you contract the vertices of $[n]$ down to $[k]$ in an order-preserving way, we get the same $g_!(C)$.
Alternatively, we can say that an infinite sequence $(C_n)_{n\in\mathbb{N}}$ of simplicial complexes, with $C_n \in \Omega(n)$ is a contraction-compatible sequence when for any $k\leq n$ and any $g\colon[n]\twoheadrightarrow[k]$ order-preserving and surjective we have $g_!(C_n) = C_k$. (So of course, the $C_n$ must be contraction-compatible in the sense of the previous paragraph, and additionally they are compatible between each other.)
As an example of these notions, if $d\in\mathbb{N}$ is fixed, the set $\mathrm{Skel}^{(d)}_n$ of all subsets having $\leq(d+1)$ elements (i.e. “dimension $\leq d$ simplices”) of $[n]$, is a contraction-compatible complex, and indeed the sequence $(\mathrm{Skel}^{(d)}_n)_{n\in\mathbb{N}}$ is a contraction-compatible sequence. To give further examples, if $C_n = \{\{0\}\}$ (“the first vertex”) then this also defines a contraction-compatible sequence, and the same holds for $C_n = \{\{n\}\}$ (“the last vertex”; but no other value will do); the same is true for $C_n = \{\{0\},\{1\},\{0,1\}\}$ (“the first edge”); or $C_n = \{\{i\} : i\in [n]\} \cup \{\{i,i+1\} : i\in[n-1]\}$.
General questions: Has this notion appeared in the literature before? Have such objects been studied? Can we, for example, compute or estimate the number of contraction-compatible complexes in $\Omega(n)$?
More specific question: If $C \in \Omega(n)$ is a contraction-compatible complex and $m\geq n$, does there always exist a contraction-compatible $C' \in \Omega(m)$ such that $g_!(C') = C$ for some (and hence any) $g\colon[m]\twoheadrightarrow[n]$ order-preserving and surjective? (Equivalently, can we extend any contraction-compatible complex into a contraction-compatible sequence?)
Motivation: contraction-compatible sequences correspond exactly to prenuclei¹ on $\Omega$ in the topos of simplicial sets (i.e., Lawvere-Tierney pretopologies), so they are a fairly natural thing to come up. (See the question “internal logic of the topos of simplicial sets” for a discussion of the subobject classifier $\Omega$ of simplicial sets.) Given a contraction-compatible sequence $(C_n)$ and any simplicial complex $D \in \Omega(n)$, we can define $\widehat{D}$ by “filling in” any simplex when we already have its parts designated by some $C_n$, that is, $\widehat{D}$ is the set of nonempty $\sigma \subseteq [n]$ such that $f^*(D)\supseteq C_n$ where $f\colon[r]\hookrightarrow[n]$ is the unique order-preserving and injective map with image $\sigma$ (here $r+1$ is the number of elements of $\sigma$) and $f^*(D) := \{\tau\subseteq[r] : f(\tau)\in D\}$. For $(C_n)$ to correspond to a nucleus means that (★) $\widehat{\widehat{D}} = \widehat{D}$ for every $D$. (For example, in the case of $(C_n) = (\mathrm{Skel}^{(d)}_n)$ this means that $\widehat{D}$ consists of every simplex whose $d$-dimensional faces are already in $D$, and this is the case for $(\mathrm{Skel}^{(d)}_n)$.)
The above motivation is not really relevant to the question, but it strongly suggests that the structure of these contraction-compatible sequences $(C_n)$, with or without the additional (★) property, should have interesting order-theoretic properties probably worth studying². Please feel free to add the (★) hypothesis if it helps answer the above questions.
A prenucleus on a frame $L$ is a map $j\colon L\to L$ satisfying $j(x)\geq x$ and $j(x\wedge y) = j(x)\wedge j(y)$. (Here $j(D)$ is the $\widehat{D}$ described above.) A prenucleus is said to be a nucleus when additionally it satisfies $j(j(x))=j(x)$.
The set of contraction-compatible sequences $(C_n)$ satisfying the additional (★) property is a Heyting algebra for the order given by reverse inclusion for all $n$. It is to the topos of simplicial sets what is studied in this paper for the effective topos, which certainly suggests that it might be an interesting mathematical object.
