What kind of K-theory is this?

Suppose I have a triangulated category $T$, say the category of modules over a dg or $A_\infty$-algebra.

Let me write $GL(T)$ for the groupoid whose objects are all finite collections of generators $\{O_1, O_2, \ldots, O_k\}$ for $T$, and where a morphism between $\{O_i\}$ and $\{O'_i\}$ is a bimodule realizing a Morita equivalence between $\bigoplus Hom(O_i, O_j)$ and $\bigoplus Hom(O'_i, O'_j)$.

Let me write $Elem(T)$ for the subgroupoid where the equivalences come from either adding the zero object as one of the generators, or by replacing some $O_i$ with $Cone(O_j \to O_i)$.

I am interested in the quotient $GL(T) / Elem(T)$.

This sounds similar to the construction of $K_1$ of a ring $R$ as the quotient of the general linear group by the elementary matrices which add a multiple of one row to another.

What kind of K-theory is this?

• What kind of answer do you expect? – მამუკა ჯიბლაძე Mar 31 '16 at 20:51
• Eg how is it related to any usual notion of k theory, in particular, is it just equal to one of them. – Vivek Shende Apr 1 '16 at 2:30
• Mostly one defines $K$-theory for objects of a category, less for a whole category. – hänsel Apr 2 '16 at 11:03