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$(\infty,n)$-Limits I: Definition and first consistency results

Lyne Moser, Nima Rasekh, Martina Rovelli

TL;DR

This work defines an $(\infty,n)$-limit for diagrams in $(\infty,n)$-categories using double $(\infty,n-1)$-categories of cones within the complete Segal object model. It proves that this notion is self-consistent across dimensions and compatible with existing theories: homotopy 2-limits in 2-categories and ordinary $(\infty,1)$-limits, via dimension-raising functors and nerve constructions. The paper emphasizes a fibrational, cone-centered perspective with accessible cone/slice calculus in double categories, enabling practical calculations and coherent comparisons across models. It lays groundwork for weighted $(\infty,n)$-limits, Kan extensions, and broader applications in presentable $(\infty,n)$-categories and higher topoi.

Abstract

We give a model-independent definition of limits for diagrams valued in an $(\infty,n)$-category. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of $(\infty,1)$-limits for $(\infty,1)$-categories, and with itself across different values of $n$.

$(\infty,n)$-Limits I: Definition and first consistency results

TL;DR

This work defines an -limit for diagrams in -categories using double -categories of cones within the complete Segal object model. It proves that this notion is self-consistent across dimensions and compatible with existing theories: homotopy 2-limits in 2-categories and ordinary -limits, via dimension-raising functors and nerve constructions. The paper emphasizes a fibrational, cone-centered perspective with accessible cone/slice calculus in double categories, enabling practical calculations and coherent comparisons across models. It lays groundwork for weighted -limits, Kan extensions, and broader applications in presentable -categories and higher topoi.

Abstract

We give a model-independent definition of limits for diagrams valued in an -category. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of -limits for -categories, and with itself across different values of .
Paper Structure (10 sections, 43 theorems, 98 equations)

This paper contains 10 sections, 43 theorems, 98 equations.

Table of Contents

  1. Limits in an (infinity,n)-category
  2. Limits in an (infinity,n)-category
  3. Self-consistency with respect to varying n
  4. Homotopy limits in a 2-category
  5. Homotopy limits in a 2-category
  6. Consistency with respect to strict context
  7. Auxiliary results
  8. Model structure for weakly horizontally invariant double categories
  9. Characterization of homotopy 2-limits
  10. Properties of double (infinity,1)-nerve of double categories

Key Result

Proposition 1

Given a diagram $K\colon\cJ\to\cC$ between categories, an object $\ell$ of $\cC$ is a limit object of $K$ if and only if there is an object $(\ell,\lambda)$ in $\cC\mathrm{one}_{\cJ}K$ such that the canonical projection induces an isomorphism of categories

Theorems & Definitions (97)

  • Proposition : folklore
  • Theorem : Grandis--Paré, Moser--Sarazola--Verdugo
  • Theorem : clingman--Moser
  • Definition
  • Theorem A
  • Theorem B
  • Definition 1.1.1
  • Definition 1.1.2
  • Remark 1.1.1
  • Definition 1.1.3
  • ...and 87 more