Discrete 2-Segal Spaces

Complete Segal spaces arising from simplicial categories

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-08-04616-3 ◽

2008 ◽

Vol 361 (01) ◽

pp. 525-546 ◽

Cited By ~ 5

Author(s):

Julia E. Bergner

Keyword(s):

Segal Spaces

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Univalent categories and the Rezk completion

Mathematical Structures in Computer Science ◽

10.1017/s0960129514000486 ◽

2015 ◽

Vol 25 (5) ◽

pp. 1010-1039 ◽

Cited By ~ 12

Author(s):

BENEDIKT AHRENS ◽

KRZYSZTOF KAPULKIN ◽

MICHAEL SHULMAN

Keyword(s):

Category Theory ◽

Type Theory ◽

Dependent Type Theory ◽

Equivalence Of Categories ◽

Segal Spaces ◽

Definition Of ◽

Categorical Semantics ◽

Univalent Foundations ◽

Dependent Type

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.

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Higher Quasi-Categories vs Higher Rezk Spaces

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/s1865243315000021 ◽

2014 ◽

Vol 14 (3) ◽

pp. 701-749 ◽

Cited By ~ 8

Author(s):

Dimitri Ara

Keyword(s):

Category Structure ◽

Model Category ◽

Segal Spaces

AbstractWe introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk Θn-spaces, showing that n-quasi-categories are a model for (∞, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.

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