Is there a "natural" / "categorical" definition of the "parity" of a permutation?

Now, we can define the quotient groupoid for which the hom-sets are the two element sets of equivalence classes of isomorphisms, where two are equivalent if there quotient is an even permutation. This is cheating of course. The question is, can we define this quotient in a «natural» way?

Fix a field $k$ (of $\operatorname{char}\ne2$). There is a functor $\det\colon S\mapsto\Lambda^{top}(k[S])$ from our groupoid to the category of vector spaces. Now we can define the quotient groupoid ($f\sim g\iff\det f=\det g$).

Whether this definition is natural enough, can be debated, of course. At least it's natural in the sense that one doesn't have to identify a set with $\{1,\ldots,n\}$ etc.

P.S. The exterior algebra can be defined w/o permutations (as the quotient of the free algebra by relations $v\wedge v=0$).

This is the abelianization map $S_n \to S_n/[S_n, S_n]$. It's universal with respect to maps from $S_n$ to abelian groups.