Is it possible to model a subcollection as a function? A subset of X can be viewed as injection $f :S \rightarrow X$, where $S$ is a structure whose only relation is equality.
If we allow $f$ to be an arbitrary function, we get the notion of a multiset.
If we furthermore assert that $S$ is a well-ordered set, we get the notion of a sequence in $X$ (potentially transfinite).
Question: is it possible to model the notion of a subcollection $\mathcal{K} \subseteq \mathcal P(X)$ as a function $f:S \rightarrow X$?
 A: It may be possible, but I think any method would be highly "unnatural".
Here is an alternative that you may like, though: the power set $\mathcal{P}(X)$ of $X$ has a natural interpretation as the collection $\{0,1\}^X$ of all functions $X\to\{0,1\}$, via the correspondence
$$g\in\{0,1\}^X\quad\longleftrightarrow\quad g^{-1}(0)\subseteq X,$$
and then you could say that a $\mathcal{K}\subseteq\mathcal{P}(X)$ can be interpreted as an injective map $f:S\to \{0,1\}^X$.
A: I don't usually answer my own questions.... but, here's an idea.
Let $S$ be a structure consisting of two "sorts," call them "elements" and "subsets." Equip $S$ with a relation $\in$ between elements (on the left) and subsets (on the right). Let us write $f : S \rightarrow X$ to mean that $f$ is a function that is defined for every element of $S$.
Then whenever $\in$ is extensional, we can say that a bijection $f : S \rightarrow X$ is a "subcollection" of $X$. If $\in$ fails to be extensional, we can still call $f$ a "multisubcollection." If the subsets of $S$ are well-ordered, call $f$ "a (potentially transfinite) sequence of subsets of $X$."
Just an idea.
