The [Grothendieck Group](http://en.wikipedia.org/wiki/Grothendieck_group) construction seems very similar to the Int construction over [traced monoidal categories](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf) (pdf).


[It may help to either rewrite the group construction multiplicatively, or to use additive notation for the monoidal structure to make this more obvious]. Roughly speaking the trace is what gives the needed invariance with respect to a common factor on both sides, which is the same thing expressed through quotienting via an equivalence relation in the group construction.


So I must be missing some fundamental difference between these constructions (or I have not read the right paper which already makes this observation). Which is it?



Int construction (traced monoidal categories) and Grothendieck Group
I believe that the Grothendieck group of a commutative monoid can be viewed as a special case of the Int construction. Namely, you can think of a commutative monoid $M$ as a discrete symmetric monoidal category (i.e., a symmetric monoidal category with no non-identity morphisms). This category admits a unique trace, and taking $\operatorname{Int} M$ just gives the Grothendieck group of $M$, thought of as a discrete symmetric monoidal category.


