On the algebraic K-theory of higher categories

Abstract

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.

107 pages. Numerous corrections, thanks to an exceptional referee. Final preprint version; accepted at the Journal of Topology

References in corpus (6)

• Higher Topos Theory
• On the Unicity of the Homotopy Theory of Higher Categories
• On exact infinity-categories and the Theorem of the Heart
• The localization sequence for the algebraic K-theory of topological K-theory
• Multiplicative structures on algebraic K-theory
• Approximation in K-theory for Waldhausen Quasicategories

Cited by in corpus (47)

• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• On exact infinity-categories and the Theorem of the Heart
• Enhancing the filtered derived category
• Modules over algebraic cobordism
• Spectral Mackey functors and equivariant algebraic K-theory (II)
• The homotopy theory of type theories
• On the Q construction for exact quasicategories
• External triangulation of the homotopy category of exact quasi-category
• K-theory and G-theory of derived algebraic stacks
• Multiplicative structures on algebraic K-theory
• Topological Hochschild Homology and Higher Characteristics
• Adjoint functor theorems for $\infty$-categories
• Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
• 2-Segal objects and the Waldhausen construction
• Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces
• A multiplicative comparison of Waldhausen and Segal K-theory
• Weight structures and the algebraic $K$-theory of stable $\infty$-categories
• Controlled objects in left-exact $\infty$-categories and the Novikov conjecture
• A^1-homotopy invariance in spectral algebraic geometry
• Derived $\ell$-adic zeta functions
• Comparison of Waldhausen constructions
• K-theory of endomorphisms via noncommutative motives
• Hodge-Iwasawa Theory I
• K-theory and polynomial functors
• Stable moduli spaces of hermitian forms
• Locally Cartesian Closed Quasicategories from Type Theory
• On nilpotent extensions of $\infty$-categories and the cyclotomic trace
• Integral Models for Spaces via the Higher Frobenius
• Approximation in K-theory for Waldhausen Quasicategories
• On The Algebraic $K$-Theory of Double Points
• Deformation theory of perfect complexes and traces
• Shortening binary complexes and commutativity of $K$-theory with infinite products
• Real K-theory for Waldhausen infinity categories with genuine duality
• Determinant map for the prestack of Tate objects
• On the universal property of Waldhausen's K-theory
• Model $\infty$-categories III: the fundamental theorem
• Enriched infinity categories I: enriched presheaves
• Homotopy invariance of higher K-theory for abelian categories
• The Fundamental Theorem of Localizing Invariants
• A Whirlwind Tour of the World of $(\infty ,1)$-categories
• On lax limits in infinity categories
• A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras
• On exact categories and their stable envelopes
• A variant of algebraic K-theory
• A universal characterization of noncommutative motives and secondary algebraic K-theory
• Algebraic K-theory of finitely projective modules on ${\mathbb{E}}_{\infty }$-rings
• From Six Functors Formalisms to Derived Motivic Measures