My question is basically given in the title: Are there any references for a generalization of algebraic K-theory to the scenario where the domain of the functors consists of commutative semirings (provided this has been done at all)?
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$\begingroup$ I dont know of any references, but at least one can say the obvious generalisation of higher K theory isnt as nice. The forgetful functor U Semiring to Set still has a left adjoint F and thus there is a cotriple FU. So for a semiring there is still a simplicial semiring, and one can apply the GL functor and take simplicial homotopy groups. The problem though is that GL is much less interesting for semirings. $\endgroup$– user1437Commented Aug 15, 2012 at 22:37
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$\begingroup$ A trivial way to do it: define the $K$-theory of a semiring as the $K$-theory of the ring obtained by 'adding' additive inverses. I guess you may have some idea behing your question. Perhapes you should say someting about it in order to possible get interesting answers. $\endgroup$– Fernando MuroCommented Aug 16, 2012 at 0:02
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$\begingroup$ People have certainly been interested in the categorified version of your question (jtopol.oxfordjournals.org/content/4/3/…). Do you have some more context for your question? Is there a particular category of modules or something that you want to understand? $\endgroup$– K.J. MoiCommented Aug 16, 2012 at 6:07
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There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" $\mathbb{F}_1$. Some of the frameworks for $\mathbb{F}_1$ algebra include semirings, so these papers might contain material that covers what you are interested in as well.
The thesis of Nicolai Durov arXiv:0704.2030 describes a setting for algebraic geometry over a class of objects more general than rings. These objects are commutative algebraic theories. Commutative rings, commutative semirings and commutative monoids all form full subcategories of commutative algebraic theories. Among the many things Durov does in his thesis, he includes some discussion of algebraic K-theory in the final chapter.
The "blueprints" of Oliver Lorscheid (arxiv.org/1103.1745 and its sequels) also contain semirings as a full subcategory. According to the abstract, he will eventually get to K-theory of blueprints, which will contain K-theory of semirings as a special case.
answered Aug 16, 2012 at 8:37