# Waldhausen Additivity: classical and quasicategorical

## Abstract

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for the Waldhausen *K*-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen *K*-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.

## Keywords

Waldhausen*K*-theory Additivity Quasicategory Joyal model structure Natural pushout functor

## Notes

### Acknowledgements

Scientific Acknowledgements: We thank David Gepner and Andrew Blumberg for helpful discussions explaining their work with Tabuada [3], some suggestions that lead to Propositions 2.12 and 6.11, and discussions relating to some results in Sect. 5. We also thank Clark Barwick for patiently answering our questions about his work [1] and for some comments about the present paper. We thank Emily Riehl and Dominic Verity for explaining to us their results [37, Corollary 5.2.20] and [38, Theorem 1.1]; their explanations positively impacted Sect. 3.6. We also thank Wolfgang Lück, Niko Naumann, Ulrich Bunke, Georgios Raptis, John Lind, Justin Noel, Mike Shulman, Marcy Robertson, and Parker Lowrey for suggestions, interesting discussions, or the occasional email. We thank the referee for helpful suggestions on the improvement of the manuscript. Thomas Fiore thanks the Regensburg *Sonderforschungsbereich 1085: Higher Invariants* and the Regensburg Mathematics Department for a very stimulating working environment during his September 2015–July 2016 sabbatical visit. Finally, Thomas Fiore also thanks the organizers of the 2007–2008 thematic programme on Homotopy Theory and Higher Categories at the Centre de Recerca Mathemàtica in Bellaterra: André Joyal, Carles Casacuberta, Joachim Kock, Amnon Neeman, and Frank Neumann. André Joyal’s *Advanced Course on Simplicial Methods in Higher Categories* and his accompanying foundational text [24] have been a tremendous influence on the present paper. At the Universitat Autònoma de Barcelona in 2007–2008, Thomas Fiore was supported on Grant SB2006-0085 of the Spanish Ministerio de Educación y Ciencia.

Financial Acknowledgements: Thomas Fiore gratefully acknowledges support from several sources during the genesis of this paper. A Humboldt Research Fellowship for Experienced Researchers supported Thomas Fiore during his September 2015–July 2016 sabbatical at Universität Regensburg. The Max-Planck-Institut für Mathematik supported his research stays in Bonn in May–June 2011 and July 2013. A Small Grant for Faculty Research from the University of Michigan-Dearborn supported some travel costs to Bonn, and a Rackham Faculty Research Grant of the University of Michigan made possible his weekend trip to Regensburg in July, 2011 to discuss with David Gepner. Malte Pieper was supported by the ERC Advanced Grant “KL2MG-interactions” (no. 662400) granted by the European Research Council to Wolfgang Lück.

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