Journal ArticleDOI
Reedy categories and the \varTheta -construction
Julia E. Bergner,Charles Rezk +1 more
TLDR
In this paper, the authors used the notion of multi-Reedy categories to prove that, if a C is a Reedy category, then C is also a reedy category.Abstract:
We use the notion of multi-Reedy category to prove that, if $$\mathcal C $$
is a Reedy category, then $$\varTheta \mathcal C $$
is also a Reedy category. This result gives a new proof that the categories $$\varTheta _n$$
are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.read more
Citations
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Higher Categories and Homotopical Algebra
TL;DR: In this paper, the authors provide an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century.
Journal ArticleDOI
Univalence for inverse diagrams and homotopy canonicity
TL;DR: In this paper, a homotopical version of the relational and gluing models of type theory is described, and generalization to inverse diagrams and oplax limits is proposed.
Journal ArticleDOI
Univalence for inverse diagrams and homotopy canonicity
TL;DR: A homotopical version of the relational and gluing models of type theory that uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory.
A type theory for synthetic ∞-categories
Emily Riehl,Michael Shulman +1 more
TL;DR: In this paper, a synthetic theory of $( ∞, 1)$-categories within homotopy type theory is proposed, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities.
Journal ArticleDOI
Higher quasi-categories vs higher Rezk spaces
TL;DR: The notion of n-quasi-categories was introduced in this article as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn.
References
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Book
Model categories and their localizations
TL;DR: Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localisation of model categories Existence of right Bousfield localizations Fiberwise localization Homotopy theory in model categories as discussed by the authors.
Journal ArticleDOI
A Cellular Nerve for Higher Categories
TL;DR: In this paper, the authors realize Joyal' cell category Θ as a dense subcategory of the category of ω-categories, and show that the resulting homotopy category of A -algebras (i.e., weak ωcategories) is equivalent to the homotonicity of compactly generated spaces.
Journal ArticleDOI
On an extension of the notion of Reedy category
Clemens Berger,Ieke Moerdijk +1 more
TL;DR: In this paper, the authors extend the classical notion of Reedy categories to allow non-trivial automorphisms, such as Segal's category Γ and Connes' cyclic category Λ.