Geometry & Topology

Comparison of models for $(\infty, n)$–categories, I

Julia E Bergner and Charles Rezk

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While many different models for (,1)–categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for (,n)–categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as (,n)–categories. Furthermore, we establish Quillen equivalences between them.

Article information

Geom. Topol., Volume 17, Number 4 (2013), 2163-2202.

Received: 17 April 2012
Revised: 18 March 2013
Accepted: 17 April 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U40: Topological categories, foundations of homotopy theory
Secondary: 55U35: Abstract and axiomatic homotopy theory 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.) 18D20: Enriched categories (over closed or monoidal categories) 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18C10: Theories (e.g. algebraic theories), structure, and semantics [See also 03G30]

$(\infty, n)$–categories $\Theta_n$–spaces enriched categories


Bergner, Julia E; Rezk, Charles. Comparison of models for $(\infty, n)$–categories, I. Geom. Topol. 17 (2013), no. 4, 2163--2202. doi:10.2140/gt.2013.17.2163.

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