Calculus of Fractions for Quasicategories

We describe a generalization of Gabriel and Zisman's Calculus of Fractions to quasicategories, showing that the two essentially coincide for the nerve of a category. We then prove that the marked Ex-functor can be used to compute the localization of a marked quasicategory satisfying our condition and that the appropriate (co)completeness properties of the quasicategory carry over to its localization. Finally, we present an application of these results to discrete homotopy theory.Comment: 77 pages, comments welcom

Études in Homotopical Thinking: F₁-geometry, Concurrent Computing, and Motivic Measures

This thesis weaves together three papers, each of which provides a use of homotopical intuition in a different field of mathematics. The first applies it to the study of various models of F₁-geometry, focusing mainly on the Bost-Connes algebra. The second endeavors to compare two homotopical models for concurrent computing before introducing a new one as well. Finally, the last paper provides a construction for obtaining derived motivic measures from an abstract six functors formalism and, in particular, applies this idea to obtain a lift of the Gillet-Soulé motivic measure

2-categorical Brown representability and the relation between derivators and infinity-categories

In this thesis, we study the 2-category of infinity-categories, largelywith attention to its relationships with the 2-category of prederivators. We prove that the 2-category of infinity-categories admits a small set of objects detecting equivalences andsatisfies a Brown representability theorem, which we formulate using a new notion of compactly generated 2-category. We show that the canonical2-functor from the 2-category of infinity-categories into the 2-category of prederivators detects equivalences and, under appropriatesize conditions, induces an equivalence on hom-categories. We explain how to extend prederivators defined on the 2-category ofordinary categories to the domain of all infinity-categories using the delocalization theorem. We use theBrown representability theorem to give conditions under which a prederivator is representable by an infinity-category. We also show how to extend derivators defined on categoriesand satisfying a mild size condition to derivators on infinity-categories, using an extensionof Cisinski's theorem on the universality of derivators of spaces. This extension allows us to give conditions under which the small sub-prederivators of quite general derivators are all representable by infinity-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) 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.Comment: 107 pages. Numerous corrections, thanks to an exceptional referee. Final preprint version; accepted at the Journal of Topolog

Enriched ∞\infty-operads

In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.Comment: Accepted version, 59 page

Segalification and the Boardmann-Vogt tensor product

We develop an analog of Dugger and Spivak's necklace formula providing an explicit description of the Segal space generated by an arbitrary simplicial space. We apply this to obtain a formula for the Segalification of $n$-fold simplicial spaces, a new proof of the invariance of right fibrations, and a new construction of the Boardman-Vogt tensor product of $\infty$-operads, for which we also derive an explicit formula.Comment: 21 pages, one figur