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On left and right model categories and left and right Bousfield localizations

- C. Barwick
- Mathematics
- 1 November 2010

We verify the existence of left Bouseld localizations and of enriched left Bouseld localizations, and we prove a collection of useful technical results characterizing certain brations of (enriched)… Expand

On the Unicity of the Homotopy Theory of Higher Categories

- C. Barwick, Christopher J. Schommer-Pries
- Mathematics
- 30 November 2011

We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces,… Expand

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

- C. Barwick, D. M. Kan
- Mathematics
- 8 November 2010

We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

On the algebraic K-theory of higher categories

- C. Barwick
- Mathematics
- 16 April 2012

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)… Expand

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

- C. Barwick, D. M. Kan
- Mathematics
- 1 March 2012

Abstract In a previous paper, we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of “relative categories”. Here,… Expand

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

- C. Barwick
- Mathematics
- 1 April 2014

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable infinity-category, and… Expand

On the Q construction for exact quasicategories

- C. Barwick
- Mathematics
- 21 January 2013

We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction. This construction yields a quasicategory whose weak homotopy type is a… Expand

On exact $\infty$-categories and the Theorem of the Heart

- C. Barwick
- Mathematics
- Compositio Mathematica
- 20 December 2012

The new homotopy theory of exact$\infty$-categories is introduced and employed to prove a Theorem of the Heart for algebraic $K$-theory (in the sense of Waldhausen). This implies a new compatibility… Expand

On Reedy Model Categories

- C. Barwick
- Mathematics
- 21 August 2007

The sole purpose of this note is to introduce some elementary results on the structure and functoriality of Reedy model categories. In particular, I give a very useful little criterion to determine… Expand

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