## Waldhausen additivity: classical and quasicategorical.(English)Zbl 1423.19002

Generalizing various algebraic $$K$$-theory constructions from the framework of categories to the context of quasi-categories, or categories up to homotopy, has been a topic of much recent interest. Some of these modern treatments have addressed this question from the point of view of characterizing algebraic $$K$$-theory as some kind of universal property; for example, Blumberg, Gepner, and Tabuada develop algebraic $$K$$-theory for stable quasi-categories in this way. For the still more general framework of Waldhausen quasi-categories, which are natural generalizations of Waldhausen categories (or categories with cofibrations), Barwick takes a similar approach. In this paper under review, the authors address the question of how to set up and prove essential theorems for the algebraic $$K$$-theory of Waldhausen quasi-categories, but in a way that more closely resembles the classical treatment. The main result is a version of the Additivity Theorem for Waldhausen quasi-categories.
The authors begin by re-proving the Additivity Theorem for (ordinary) Waldhausen categories, but in a way amenable to generalization to quasi-categories. This approach makes the passage to quasi-categories more transparent, and thus it is likely to be accessible to readers more familiar with Waldhausen’s original constructions but perhaps less familiar with the setting of quasi-categories. Further facilitating this goal, the paper includes quite a bit of background material about quasi-categories that are used in the proof. The paper also includes some results on split exact sequences in a Waldhausen quasi-category.

### MSC:

 19D10 Algebraic $$K$$-theory of spaces 55U10 Simplicial sets and complexes in algebraic topology 55U40 Topological categories, foundations of homotopy theory 18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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