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

Question

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). This produces a homomorphism $S_n \to \{\pm 1\}$.

I've known the various definitions, the proof of their equivalences and the various applications of them for quite a while, and yet something seems missing. I can't convince myself that any of those definitions is really "natural". Of course "natural" is something rather subjective, but for me at least, it is close in meaning to "categorical". For example, a "natural" presentation of the definition (and basic properties) of addition/multiplication of natural numbers, can be achieved by considering the category of finite sets, where these operations are categorical sum/product.

Since $S_n$ is the automorphism group of a set with $n$ elements, I would say that the (horizontal?) categorification of it is the groupoid of all sets with $n$-elements. This is arguably a more "natural" object. Of course, this groupoid is equivalent to $S_n$ so it is just a matter of perspective. 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? I find it very surprising that this kind of structure associated with plain finite sets, is so well hidden.

I heard that the K-theory of finite sets encodes some information of this sort. If this is so, I would be very happy to hear more about it.

As a final note, one famous neat application of the notion of parity of a permutation is the proof of the impossibility of the 14-15 puzzle. The proof is beautiful, but it applies group-theory techniques to something which is most naturally viewed as a groupoid. This might be completely unrelated to the main question, but it seems that a more natural/groupoidal definition of parity might be applicable to this situation as well.

Answer 1

Here's an answer that might be more like what you were expecting. The collection of all symmetric groups together organize into a symmetric monoidal groupoid, namely the symmetric monoidal groupoid $S$ of finite sets and bijections. This is the free symmetric monoidal groupoid on a point.

Now, given any symmetric monoidal category you can ask what it looks like when you freely adjoin inverses to it, at various levels of sophistication. First, you can just ask for its Grothendieck group; that is, you look at the commutative monoid of isomorphism classes, then adjoin inverses. This gives you $\mathbb{Z}$.

Second, you can ask for a symmetric monoidal groupoid rather than just a group, but one where every object is invertible (I think this is sometimes called a "Picard groupoid"). This gives you a groupoid with $\pi_0 \cong \mathbb{Z}$ and $\pi_1 \cong \mathbb{Z}_2$: that is, the objects under the symmetric monoidal product are just $\mathbb{Z}$ again, but now they have automorphisms of order $2$. The induced map from $S$ sends each permutation to its sign, but also assigns data to isomorphisms between finite sets which we haven't necessarily identified. This Picard groupoid is the free Picard groupoid on a point.

Continuing in this way, you can ask for a symmetric monoidal $n$-groupoid rather than just a groupoid, or in other words a connective $n$-truncated spectrum. The spectrum you get in this way is the $n$-truncation $\tau_{\le n}(\mathbb{S})$ of the sphere spectrum (which is the free spectrum on a point). This is a version of the Barratt-Priddy-Quillen theorem, and it reveals that the $\mathbb{Z}$ above was the zeroth stable homotopy group of spheres, while the $\mathbb{Z}_2$ above was the first. These $n$-truncations, and the natural map from $S$ into them, are candidates for "higher sign characters" as in Ganter-Kapranov.

Answer 2

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 $v\wedge v=0$).

Answer 3

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.

Answer 4

I am not sure at what level of sophistication you are looking for such a "natural" definition but a good candidate is orientability. Thinking of the permutation as permuting vectors in a basis for a finite-dimensional vector space, one can characterize parity of the permutation in terms of preserving or reversing the orientation. The orientation itself can be formalized in terms of the determinant bundle (in this case over a single point) of the vector space. Namely, the orientation is an element of this line bundle, and the permutation induces an action either by $+1$ or by $-1$ on this bundle, corresponding respectively to a even or odd permutation.