Loop spaces for the Q-construction (Q1109128)

Let \({\mathcal M}\) be an exact category and let Q\({\mathcal M}\) be its Q- construction, so that the algebraic K-groups of \({\mathcal M}\) are given by \(K_ i{\mathcal M}=\pi_{i+1}(Q{\mathcal M})\). The author gives two models, k\({\mathcal M}\) and K\({\mathcal M}\), for the loop space of Q\({\mathcal M}\). Of these K\({\mathcal M}\) is the more complicated, but has the advantage that K\({\mathcal M}\cong K({\mathcal M}^{op})\). The models are related to the previously studied \(S^{-1}S{\mathcal M}\), which gives a model for the loop space of Q\({\mathcal M}\) if exact sequences split in \({\mathcal M}\). There are applications to the relative algebraic K-theory of exact functions, especially cofinal factors.