What structure does the alternating group preserve? A common way to define a group is as the group of structure-preserving transformations on some structured set.  For example, the symmetric group on a set $X$ preserves no structure: or, in other words, it preserves only the structure of being a set.  When $X$ is finite, what structure can the alternating group be said to preserve?
As a way of making the question precise, is there a natural definition of a category $C$ equipped with a faithful functor to $\text{FinSet}$ such that the skeleton of the underlying groupoid of $C$ is the groupoid with objects $X_n$ such that $\text{Aut}(X_n) \simeq A_n$?  
Edit:  I've been looking for a purely combinatorial answer, but upon reflection a geometric answer might be more appropriate.  If someone can provide a convincing argument why a geometric answer is more natural than a combinatorial answer I will be happy to accept that answer (or Omar's answer).

The polynomial $p\in K\left[X_1,...,X_n\right]$ given by $$p\left(X_1,...,X_n\right)=\prod_{i<j}\left(X_i-X_j\right)$$
