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On left and right model categories and left and right Bousfield localizations
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
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
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
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
(∞, n)-Cat as a closed model category
A characterization of simplicial localization functors and a discussion of DK equivalences
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)
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, andExpand
On the Q construction for exact quasicategories
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 aExpand
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 compatibilityExpand
On Reedy Model Categories
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 determineExpand