A rig is a “ring without negatives”. There are many ways to categorify this concept, but here we explain one specific definition that sheds new light on topology and representation theory. Examples of such 2-rigs include categories of group representations, coherent sheaves and vector bundles. We explain some theorems and conjectures about 2-rigs with simple universal properties. For example, the free 2-rig on one generator is called the 2-rig of “Schur functors” because it acts as endofunctors of every 2-rig. It has as objects all finite direct sums of irreducible representations of symmetric groups. Furthermore, the “splitting principle” for vector bundles has a universal formulation in terms of this 2-rig. This is joint work with Joe Moeller and Todd Trimble.
You can see the slides here. Eventually a video should appear. You can see more material on the free 2-rig on one generator here:
Also see our papers: