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?



Higher categories and semirings
I'm afraid your "unmistakable resemblence between 2-categories and semirings" is mistaken. Distributivity (in the usual sense) means $(a+b)\ast(c+d)=a\ast c+a\ast d+b\ast c+b\ast d$, but you're looking for two structures $+$ and $\ast $ and a comparison map between $(a+b)\ast (c+d)$ and $(a\ast c)+(b\ast d)$, is that correct? Such a thing does not really have a $1$-analog because by the standard argument, if two operations on a set are unital and behave like this, then they are equal to each other and commute. In category theory, however, this is captured in the notion of a "$2$-monoidal category". It is the universal setting in which to define a bimonoid (a.k.a. bialgebra). This is explained very clearly, and with references to earlier work, in Chapter 6 of


Marcelo Aguiar and Swapneel Mahajan. *Monoidal functors, species and Hopf algebras*, volume 29 of *CRM Monograph Series*. American Mathematical Society, Providence, RI, 2010.


On the other hand, if you're really interested in the $2$-analog of semirings, go for Neil's suggestions.


