Clark Barwick

On the algebraic K-theory of higher categories

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.

From operator categories to higher operads

In this paper we introduce the notion of an operator category and two different models for homotopy theory of

On exact ∞-categories and the Theorem of the Heart

\infty

Fibrations in ∞-Category Theory

-operads over an operator category -- one of which extends Luries theory of

Relative categories: Another model for the homotopy theory of homotopy theories

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ON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS

-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category

On the Unicity of the Homotopy Theory of Higher Categories

\Lambda(\Phi)

Spectral Mackey functors and equivariant algebraic K-theory (I)

attached to a perfect operator category

A characterization of simplicial localization functors and a discussion of DK equivalences

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On (Enriched) Left Bousfield Localization of Model Categories

that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman--Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads