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Discreteness and completeness for $Θ_n$-models of $(\infty,n)$-categories

Julia E. Bergner

TL;DR

The paper addresses the problem of modeling $(\infty,n)$-categories using $\Theta_n$-diagrams with either discreteness or completeness constraints. It develops cartesian, cofibrantly generated model structures on $\mathcal{SS}ets^{\Theta_n^{op}}$ that impose discreteness at selected levels and proves these variants are equivalent to the standard $\Theta_n$-space model, while also analyzing when discreteness and completeness yield distinct models. A key contribution is the precise characterization of Dwyer–Kan equivalences for $\Theta_n$-spaces, achieved by transporting to complete Segal objects in $\Theta_{n-1}$-spaces via a Quillen equivalence and comparing enriched mapping objects. The results unify multiple approaches to $(\infty,n)$-categories, enable bottom-up discreteness, and pave the way for hybrid diagram models bridging Segal-type constructions across dimensions, with potential applications to monoidal higher categories and beyond.

Abstract

We establish cartesian model structures for variants of $Θ_n$-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 $Θ_n$-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 $Θ_n$-diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the $Θ_n$-space model, generalizing the one given by Rezk for complete Segal spaces.

Discreteness and completeness for $Θ_n$-models of $(\infty,n)$-categories

TL;DR

The paper addresses the problem of modeling -categories using -diagrams with either discreteness or completeness constraints. It develops cartesian, cofibrantly generated model structures on that impose discreteness at selected levels and proves these variants are equivalent to the standard -space model, while also analyzing when discreteness and completeness yield distinct models. A key contribution is the precise characterization of Dwyer–Kan equivalences for -spaces, achieved by transporting to complete Segal objects in -spaces via a Quillen equivalence and comparing enriched mapping objects. The results unify multiple approaches to -categories, enable bottom-up discreteness, and pave the way for hybrid diagram models bridging Segal-type constructions across dimensions, with potential applications to monoidal higher categories and beyond.

Abstract

We establish cartesian model structures for variants of -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 -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 -diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the -space model, generalizing the one given by Rezk for complete Segal spaces.
Paper Structure (10 sections, 32 theorems, 107 equations)

This paper contains 10 sections, 32 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Some background
  3. Complete Segal spaces and Segal categories
  4. $\Theta_n$-spaces
  5. Segal and complete Segal objects in $\Theta_n$-spaces
  6. Dwyer-Kan equivalences for $\Theta_n$-spaces
  7. Results on diagram categories with discreteness assumptions
  8. Model structures for $\Theta_n$-models with discreteness assumptions
  9. Comparison of models
  10. Proof of Proposition \ref{['thesis5.8']}

Key Result

Theorem 2.1

hirsch Let $\mathcal{M}$ be a category that has all small limits and colimits. Suppose that $\mathcal{M}$ has a class of weak equivalences that satisfies the two-out-of-three property and that is closed under retracts. Let $I$ and $J$ be sets of maps in $\mathcal{M}$ that satisfy the following condi Then there is a cofibrantly generated model category structure on $\mathcal{M}$ in which $I$ is a s

Theorems & Definitions (82)

  • Theorem 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Theorem 3.7
  • Definition 3.8
  • ...and 72 more