Maybe my thinking here is completely wrong headed, but this seems like something that ought to have been answered before. Here is my question:
What is the (n - )categorical analogue of a semiring?
As a starting point here is a common bit of folklore: categories act as a generalization of monoids, where the latter is simply a special case consisting of the collection of endomorphisms for a single object. Similarly, semirings can be thought of as the collection of all endomorphisms of a commutative monoid.
Now there is an unmistakable resemblence between 2-categories and semirings. Specifically, in a 2-category, we get 2 operations for composing 2-morphisms, vertical and horizontal composition, and these operations distribute over each other via the interchange law; just like the addition and multiplication within a semiring. However, I am having difficulty making this correspondence sharp (if it is even possible).
Here is a sketch of something that doesn't work. First, take all 2-endomorphisms of the identity morphism of any object in a 2-category to be the elements of the semiring. Then map vertical composition to addition and horizontal composition to multiplication. However, this creates a problem since there identity element for addition and multiplication correspond to the same 2-morphism, so the semiring must be trivial.
Is there a way to salvage this idea or am I just not understanding something obvious?