The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A "rig" is a "ring without negatives", and the free rig on one generator is \(\mathbb{N}[x]\), the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of "symmetric 2-rig", and it turns out the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.
You can see the slides here, and also a video from when I gave the talk at Ohio State:
Here are slides for a pre-talk, to give people a bit of background for the actual talk:
You can also see notes for a more advanced talk I gave at the Edinburgh Category Theory Seminar, and for even more details read our paper: