I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions.


Consider an exact monoidal category and its K- groups as introduced by Quillen.


Do $K\_n(\mathcal C)$ have a ring structure? Clearly for n=0 the Grothendieck groups have a ring structure induced by the tensor monoidal functor of $\mathcal C$. So bassically the question can be reformulated if the same tensor functor induces also a ring structure in the higher K - groups.


Next natural question would be if a monoidal functor $ F: \mathcal C \rightarrow D$ induces a ring homomorphism at any level of the K- groups, $F^\*\_n:K\_n(\mathcal C)\rightarrow K\_n(\mathcal D)$?


I guess these are basic things but unfortunately I don't know any text where they are treated.



K-theory of monoidal categories
(I'm not sure the term *exact monoidal category* is a standard one in the literature, so I'll just assume I know what you mean by it.)


Yes, $K\_n$ commutes with products, as some have mentioned, but the tensor monoidal functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is not exact, so it doesn't induce a map $ K\_n \mathcal C \times K\_n \mathcal C \to K\_n \mathcal C$. Rather, the functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is *bi-exact*, in the sense that it is an exact functor in each of its two variables, separately. (That's what makes $K\_0 \mathcal C$ into a ring.)


The result is that the collection of abelian groups $ K\_n \mathcal C$, indexed by $n$, forms a graded ring, with products $K\_m \mathcal C \otimes K\_n \mathcal C \to K\_{m+n} \mathcal C$. Assuming the tensor product is commutative up to natural isomorphism, the ring is commutative in the graded sense that $ x y = (-1)^{mn} y x$.


The actual construction is intricate, the main idea being to construct a map $K \mathcal C \wedge K \mathcal C \to K \mathcal C$ of spectra. See, for example, Waldhausen's approach in section 9 of Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1978), no. 1, 135–204. It uses the Q-construction of Quillen, but could be simplified by using the S-construction of Segal that Waldhausen developed extensively.


My paper with Gillet, [The loop space of the Q-construction](http://www.math.illinois.edu/~dan/Papers/loopsQ.pdf), Illinois Journal of Mathematics, 31 (1987) 574-597, gives another approach that avoids both spectra (i.e., delooping) and the phony multiplication trap that Steve mentioned.


PS: My paper [Algebraic K-theory via binary complexes](http://www.math.uiuc.edu/K-theory/0988/), Journal of the American Mathematical Society, 25 (2012) 1149-1167, gives an algebraic description of the elements of the higher K-groups that converts the construction of the product into a simple exercise involving tensor product of chain complexes.


