Multiple categories and algebraic K-theory
Let \(n\geq 1\) be a natural number and \(P\subset \{1,...,n\}\) an arbitrary subset. The author introduces the notion of a P-exact n-fold exact category as an n-fold category M in the sense of \textit{C. Ehresmann} [Ann. Sci. Éc. Norm. Supér., III. Sér. 80, 349-426 (1963; Zbl 0128.020)], such that the component categories \(M_ p\) are exact categories for all \(p\in P\), and satisfy certain compatibility conditions. The definition insures that \textit{D. Quillen}'s functor Q: (exact categories) \(\to (categories)\) [Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] extends to a functor \(Q_ p:\) (P-exact categories) \(\to ((P-\{p\})\)- exact categories) for each \(p\in P\). For M a small exact category, the n- fold category \(M^{[n]}\) on the set of all commutative n-cubes in M turns out to be \(\{\) 1,...,n\(\}\)-exact and the n-fold category \(Q^ n M^{[n]} =Q_ n...Q_ 2Q_ 1 M^{[n]}\) is defined. This provides a reformulation of \textit{F. Waldhausen}'s iterated Q-construction [Ann. Math., II. Ser. 108, 135-204, 205-256 (1978; Zbl 0397.18012, Zbl 0407.18009)]. The author's main result asserts that \(Q^{n+1} M^{[n+1]}\) deloops \(Q^ n M^{[n]}\); this extends Waldhausen's proof for \(n=1\). As a consequence, a spectrum \({\mathfrak KM}=\{KM_ n; \epsilon_ n\}\) is obtained, such that \(KM_ n\) is the classifying space of \(Q^ n M^{[n]}\) for \(n\geq 1\). There also follows that any natural isomorphism of exact functors induces a homotopy of the associated maps of spectra. The last part of the paper deals with the multiplicative structure of the associated spectra. For each biexact functor \(\psi\) : \(U\times V\to W\) a pairing of spectra \(\psi_{m,n}: KU_ m\wedge KV_ n\to KW_{m+n}\) is constructed; it is shown that any natural isomorphism of biexact functors induces a homotopy of pairings. As a consequence, a generalization of Quillen's projection formula [loc. cit.] is derived.
scientific article; zbMATH DE number 3977224