QUILLEN Q-CONSTRUCTION // HOMOTOPY COLIMIT // K_0 ASSEMBLY MAP
For an exact category F, the Q-construction QF is a category whose objects are those of F and whose morphisms are equivalence classes of diagrams A ← C ↪ B (an admissible epi followed by an admissible mono).
K_n(F) = π_{n+1}(BQF, 0)
The K-theory of a diagram of categories F: I → ExCat is computed via the homotopy colimit: K(hocolim F) ≈ hocolim K(F(-)). This assembly process glues local K-theory data into global invariants.
The assembly map α: H_*(BG; K(R)) → K_*(R[G]) from the homology of BG with K-theory coefficients to the K-theory of the group ring. The Farrell-Jones and Baum-Connes conjectures assert this is an isomorphism.
α: hocolim_{G/H} K(R[H]) → K(R[G])