A survey of models for (∞, n)-categories

Bergner, Julia E.

doi:10.1201/9781351251624-7

Cited by 26 publications

(4 citation statements)

References 24 publications

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“…We have chosen to follow Rezk's original terminology and refer to these objects as "Θ n -spaces". It is arguably more accurate to call them "complete Segal Θ n -spaces", a convention we adopt in [7]. For the purposes of this paper, however, this specification leads to unwieldy terminology; to avoid having to refer repeatedly to atrocities such as " complete Segal objects in complete Segal Θ n -spaces" in what follows, we have chosen to revert back to the more concise name.…”

Section: θ N -Spacesmentioning

confidence: 99%

Models for $(\infty,n)$-categories with discreteness conditions, I

Bergner¹

2022

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We establish cartesian model structures for variants of Θn-spaces in which we replace some or all of the completeness conditions by discreteness conditions. We prove that they are all equivalent to each other and to the Θn-space model, and we give a criterion for which combinations of discreteness and completeness give non-overlapping models. These models can be thought of as generalizations of Segal categories in the framework of Θn-diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the Θn-space model, generalizing the one given by Rezk for complete Segal spaces.

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Section: θ N -Spacesmentioning

confidence: 99%

Models for $(\infty,n)$-categories with discreteness conditions, I

Bergner¹

2022

Preprint

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“…Studying Fibrations of (∞, n)-Categories: The way we generalized 1-categories to (∞, 1)-categories, we can generalize strict n-categories to (∞, n)-categories [8]. Moreover, similar to the (∞, 1)-categorical case, the study of functors…”

Section: Why the Complete Segal Object Approach?mentioning

confidence: 99%

Quasi-categories vs. Segal spaces: Cartesian edition

Rasekh

2021

J. Homotopy Relat. Struct.

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We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial spaces (due to deBrito [12]), On bisimplicial sets, On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.

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“…Studying Fibrations of (∞, n)-Categories: The way we generalized 1-categories to (∞, 1)-categories, we can generalize strict n-categories to (∞, n)-categories [Ber20]. Moreover, similar to the (∞, 1)-categorical case, the study of functors…”

Section: Why the Complete Segal Object Approach?mentioning

confidence: 99%

Quasi-Categories vs. Segal Spaces: Cartesian Edition

Rasekh

2021

Preprint

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We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent:(1) On marked simplicial sets (due to Lurie [Lur09a]),(2) On bisimplicial spaces (due to deBrito [BdB18]),(3) On bisimplicial sets, (4) On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie.

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A survey of models for (∞, n)-categories

Abstract: We give describe several models for (∞, n)-categories, with an emphasis on models given by diagrams of sets and simplicial sets. We look most closely at the cases when n ≤ 2, then summarize methods of generalizing for all n.

Cited by 26 publications

References 24 publications

Models for $(\infty,n)$-categories with discreteness conditions, I

Models for $(\infty,n)$-categories with discreteness conditions, I

Quasi-categories vs. Segal spaces: Cartesian edition

Quasi-Categories vs. Segal Spaces: Cartesian Edition

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