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?
