For $(S_i)_{i \in I}$ a familiy of sets, the set of all prefixes of tuples in $\prod_{i \in I} S_i$ is
$$ \prod_{i\in I} (S_i \dot{\cup} \{\bot\}) \;,$$
which is just a choice of either some element of $S$ or "nothing" (corresponding to $\bot$) for every set $S \in F$ in the family. There is the relation "$(s'_i)_{i\in I}$ is a prefix of $(s_i)_{i \in I}$" iff for all $j \in I$ we have $s'_j = s_j$ or $s'_j = \bot$.
(For example: $(4,\bot,\bot)$ is a prefix of $(4,\bot,b)$, and both of those are a prefix of the tuple $(4,5X+3,b)$. This example is for the set family $S_1 = \mathbb N$, $S_2 = \mathbb Z [X]$, $S_3 = \{a,b,c\}$.)
I want a nice simplicial complex construction of this. This means that I want something with "$\subseteq$" instead of "is a prefix of" as a relation (and we would need to add the empty set to the simplicial complex, corresponding to the tuple with $(\bot)_{i\in I}$).
One possible construction is
$$ \{M \subseteq \dot \bigcup_{S \in F} S \; \mid \; \forall A,B \in M \colon \; (\exists S \in F \colon \; A,B \in S\implies A = B) \} \; . $$
(We see here how we can only take at most one element in $S$ for each $S$)
Here we indeed see how as posets, this construction with the "$\subseteq$" relation is isomorphic to the first construction with the "is prefix of" as a relation for the former construction. In fact, this construction without the empty set is a simplicial complex!
But this construction is too ugly!
Is there a way to make this in a pretty way? Ideally with just a simple kind of categorical limit or colimit.
It reminds me of taking coproducts in the category of monoids: Each element in the coproduct corresponds uniquely to a finite collection of elements in the original monoids, at most one per monoid. So one other ugly way to make the construction I want is via artificially constructing monoids and taking the coproduct and then that identification of having at most one element per monoid, as a set.