Commutative 2-algebra

A "2-rig" arises as a categorification of the notion of a semiring. One possible definition is that it is a monoidal category where the A ⊗ _ functor preserves colimits for each object A. While this is not the only possible definition, it plays well with the goal of categorifying commutative algebra. Basic theorems in this area, such as existence and properties of polynomial rings, can be explored within this framework.

The theory becomes particularly intriguing when one categorifies *differential* rings and studies their properties. This approach illuminates certain properties of the category of combinatorial species, which is the analogue of polynomial/power series rings in the 2-category of 2-rigs. I will survey these topics, which form an ongoing project with Todd Trimble.