On the algebraic \(K\)-theory of higher categories (Q2797321)

The purpose of this paper is to extend Waldhausen's algebraic \(K\)-theory from the setting of categories with cofibrations to a more general framework of what the author calls Waldhausen \(\infty\)-categories. The first part of the paper is devoted to the development of this framework. The first step is to describe pairs of quasicategories, mimicking the structure of a category together with a subcategory of cofibrations. A Waldhausen \(\infty\)-category is defined to be such a pair of quasicategories, satisfying additional axioms which are analogous to the ones used in Waldhausen's original formulation. The collection of all such is also given the structure of a quasicategory.NEWLINENEWLINEThe next major step in this work is to extend the theory of cartesian and cocartesian fibrations into this setting. This portion of the paper is lengthy, but takes time to recall the definitions in the ordinary quasicategorical setting and what their purpose is, which is useful for readers not fluent with these methods. These definitions are then generalized, first to the setting of pairs and then to the Waldhausen context. The first section concludes with the development of the derived \(\infty\)-category of Waldhausen \(\infty\)-categories, whose objects are simplicial presheaves on a category of sufficiently small Waldhausen \(\infty\)-categories and referred to as virtual Waldhausen \(\infty\)-categories. Along the way, many useful properties of the quasicategory of Waldhausen \(\infty\)-categories are established, such as accessibility.NEWLINENEWLINEThe second main section of the paper is concerned with the development of filtered objects in a Waldhausen \(\infty\)-category and then the construction of the analogue of the \(S_\bullet\)-construction in this setting. A key definition is that of a fissile virtual Waldhausen \(\infty\)-category; the sub-quasicategory of these objects is given by a localization, and on it the suspension functor has a nice description which allows for the definition of the \(S_\bullet\)-construction. This localized context allows for an analogue of the additivity theorem and a description of the additivization functor as a Goodwillie derivative. This section concludes with a definition of labeled Waldhausen \(\infty\)-categories, which are the generalization of Waldhausen's categories with cofibrations and weak equivalences.NEWLINENEWLINEFinally, with all the foundations in place, the author treats the algebraic \(K\)-theory functor in the last part. The definition is given via additivization of a functor, as given in the previous part, but representability and universality results are established here. The paper concludes with two specific examples. The first is the algebraic \(K\)-theory of \(E_1\)-algebras, and the second is the algebraic \(K\)-theory of spectral Deligne-Mumford stacks.