Giffen, Charles H. Loop spaces for the Q-construction. (English) Zbl 0655.18008 J. Pure Appl. Algebra 52, No. 1-2, 1-30 (1988). 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. Reviewer: R.J.Steiner Cited in 2 Reviews MSC: 18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) 18E10 Abelian categories, Grothendieck categories 55P35 Loop spaces Keywords:exact category; Q-construction; algebraic K-groups; loop space; relative algebraic K-theory of exact functions; cofinal factors PDFBibTeX XMLCite \textit{C. H. Giffen}, J. Pure Appl. Algebra 52, No. 1--2, 1--30 (1988; Zbl 0655.18008) Full Text: DOI References: [1] Giffen, C. H., A looping of the \(Q\)-construction, Talk given at Special Session in Algebraic Topology. Talk given at Special Session in Algebraic Topology, Amer. Math. Soc. Meeting, 784 (October 1980), Notre Dame, IN, March 21, 1981 [2] C.H. Giffen, Unitary algebraic \(K\); C.H. Giffen, Unitary algebraic \(K\) · Zbl 0113.01605 [3] Grayson, D., Higher algebraic \(K\)-theory, II, (Lecture Notes in Mathematics, 551 (1976), Springer: Springer Berlin), 217-240 [4] Quillen, D. G., Higher algebraic \(K\)-theory, I, (Lecture Notes in Mathematics, 341 (1973), Springer: Springer Berlin), 85-147 · Zbl 0292.18004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.