We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, "functors between two homotopy theories form a homotopy theory", or more precisely that the category of such models has a well-behaved internal hom-object.

Abstract. 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 ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory.More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors.Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. We also study fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks.

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 simple universal property. Using this, we give new, higher categorical proofs of the approximation, additivity, and fibration theorems of Waldhausen in this article. As applications of this technology, we study the algebraic K‐theory of associative rings in a wide range of homotopical contexts and of spectral Deligne–Mumford stacks.

We axiomatise the theory of ( ∞ , n ) (\infty ,n) -categories. We prove that the space of theories of ( ∞ , n ) (\infty ,n) -categories is a B ( Z / 2 ) n B(\mathbb {Z}/2)^n . We prove that Rezk’s complete Segal Θ n \Theta _n spaces, Simpson and Tamsamani’s Segal n n -categories, the first author’s n n -fold complete Segal spaces, Kan and the first author’s n n -relative categories, and complete Segal space objects in any model of ( ∞ , n − 1 ) (\infty , n-1) -categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of ( Z / 2 ) n (\mathbb {Z}/2)^n .