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- Title
Hammocks and fractions in relative ∞-categories.
- Authors
Mazel-Gee, Aaron
- Abstract
We study the homotopy theory of ∞<inline-graphic></inline-graphic>-categories enriched in the ∞<inline-graphic></inline-graphic>-category <inline-graphic></inline-graphic> of simplicial spaces. That is, we consider <inline-graphic></inline-graphic>-enriched ∞<inline-graphic></inline-graphic>-categories as presentations of ordinary ∞<inline-graphic></inline-graphic>-categories by means of a “local” geometric realization functor <inline-graphic></inline-graphic>, and we prove that their homotopy theory presents the ∞<inline-graphic></inline-graphic>-category of ∞<inline-graphic></inline-graphic>-categories, i.e. that this functor induces an equivalence <inline-graphic></inline-graphic> from a localization of the ∞<inline-graphic></inline-graphic>-category of <inline-graphic></inline-graphic>-enriched ∞<inline-graphic></inline-graphic>-categories. Following Dwyer-Kan, we define a hammock localization functor from relative ∞<inline-graphic></inline-graphic>-categories to <inline-graphic></inline-graphic>-enriched ∞<inline-graphic></inline-graphic>-categories, thus providing a rich source of examples of <inline-graphic></inline-graphic>-enriched ∞<inline-graphic></inline-graphic>-categories. Simultaneously unpacking and generalizing one of their key results, we prove that given a relative ∞<inline-graphic></inline-graphic>-category admitting a homotopical three-arrow calculus, one can explicitly describe the hom-spaces in the ∞<inline-graphic></inline-graphic>-category presented by its hammock localization in a much more explicit and accessible way. As an application of this framework, we give sufficient conditions for the Rezk nerve of a relative ∞<inline-graphic></inline-graphic>-category to be a (complete) Segal space, generalizing joint work with Low.
- Subjects
HOMOTOPY theory; MATHEMATICAL equivalence; SEGAL algebras; CALCULUS; MATHEMATICAL analysis
- Publication
Journal of Homotopy & Related Structures, 2018, Vol 13, Issue 2, p321
- ISSN
2193-8407
- Publication type
Article
- DOI
10.1007/s40062-017-0184-0