Multiplicative Structures on Algebraic K-theory

0.1. The kinds of homotopy theories under consideration in this paper are Waldhausen ∞-categories [2, Df. 2.7]. (We employ the quasicategory model of∞-categories for technical convenience.) These are ∞-categories with a zero object and a distinguished class of morphisms (called cofibrations or ingressive morphisms) that satisfies the following conditions. (0.1.1) Any equivalence is ingressive. (0.1.2) Any morphism from the zero object is ingressive. (0.1.3) Any composite of ingressive morphisms is ingressive. (0.1.4) The (homotopy) pushout of an ingressive morphism along any morphism exists and is ingressive. A pushout of a cofibration X Y along the map to the zero object is to be viewed as a cofiber sequence X Y Y/X. Examples of this structure abound: pointed ∞-categories with all finite colimits, exact categories in the sense of Quillen, and many categories with cofibrations and weak equivalences in the sense of Waldhausen all provide examples of Waldhausen ∞-categories. Write Wald∞ for the ∞-category whose objects are Waldhausen ∞-categories and whose morphisms are functors that are exact in the sense that they preserve the cofiber sequences. This is a compactly generated ∞-category [2, Pr. 4.7] that admits direct sums [2, Pr. 4.6].