Higher categories and semirings 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?
 A: A category with two symmetric monoidal structures, one distributing over the other, is called a symmetric bimonoidal category.  These have been studied extensively in stable homotopy theory, because one can apply a K-theory construction to such a category to get an $E_\infty$ ring spectrum.  There are complicated coherence conditions for the interaction between the distributivity isomorphisms $A\otimes (B\oplus C)\to(A\otimes B)\oplus(A\otimes C)$ with the rest of the structure; these are best encoded using some kind of operadic formalism, or more recently, using Lawvere-type theories in an $\infty$-category setting.  This paper would be one entry point into the literature:
 \bib{MR1361586}{article}{
   author={Dunn, Gerald},
   title={$K$-theory of braided tensor ring categories with higher
   commutativity},
   journal={$K$-Theory},
   volume={9},
   date={1995},
   number={6},
   pages={591--605},
   issn={0920-3036},
   review={\MR{1361586 (97b:18003)}},
   doi={10.1007/BF00998132},
 }

Things are easier if you do not want $\otimes$ to be symmetric, but I think that the coherence conditions are still fairly complex.
A: 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.
