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$(\infty,n)$-categories in context

Viktoriya Ozornova, Martina Rovelli

TL;DR

This note surveys the landscape of $(\infty,n)$-categories and $(\infty,n)$-functors, outlining intuitive definitions, several concrete models, and examples relevant to mathematical physics. It explains how higher coherence replaces strict equalities, summarizes prominent models (such as $n$-complicial sets, $n$-fold complete Segal spaces, and categories enriched over $(\infty,n-1)$-categories), and discusses the homotopy theory and symmetric monoidal extensions of these structures. The note highlights the role of $(\infty,n)$-categories in formulating extended TQFTs, factorization algebras, and $\mathbb{E}_n$-algebras, including the cobordism hypothesis and related classification theorems. It emphasizes model equivalences, the construction of the $(\infty,1)$-category of $(\infty,n)$-categories, and practical implications for applying higher-categorical language in mathematical physics.

Abstract

This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an $(\infty,n)$-category and $(\infty,n)$-functor, discussing some of the different models that implement them. We also discuss the notions of a symmetric monoidal $(\infty,n)$-category and symmetric monoidal $(\infty,n)$-functor, recalling some important results whose statements employ the language of $(\infty,n)$-categories.

$(\infty,n)$-categories in context

TL;DR

This note surveys the landscape of -categories and -functors, outlining intuitive definitions, several concrete models, and examples relevant to mathematical physics. It explains how higher coherence replaces strict equalities, summarizes prominent models (such as -complicial sets, -fold complete Segal spaces, and categories enriched over -categories), and discusses the homotopy theory and symmetric monoidal extensions of these structures. The note highlights the role of -categories in formulating extended TQFTs, factorization algebras, and -algebras, including the cobordism hypothesis and related classification theorems. It emphasizes model equivalences, the construction of the -category of -categories, and practical implications for applying higher-categorical language in mathematical physics.

Abstract

This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an -category and -functor, discussing some of the different models that implement them. We also discuss the notions of a symmetric monoidal -category and symmetric monoidal -functor, recalling some important results whose statements employ the language of -categories.
Paper Structure (24 sections, 15 theorems, 66 equations)

This paper contains 24 sections, 15 theorems, 66 equations.

Table of Contents

  1. $(\infty,n)$-Categories
  2. Schematic definition
  3. Examples
  4. $(\infty,n)$-Functors
  5. Schematic definition
  6. Examples
  7. The homotopy theory of $(\infty,n)$-categories
  8. Equivalences between $(\infty,n)$-functors
  9. The $(\infty,1)$-category of $(\infty,n)$-categories
  10. Equivalences of $(\infty,n)$-categories
  11. Two examples
  12. Mathematical implementations of $(\infty,n)$-categories
  13. Complicial sets
  14. $n$-Fold complete Segal spaces
  15. Categories enriched over $(\infty,n-1)$-categories
  16. ...and 9 more sections

Key Result

Theorem 3.2.1

There is an $(\infty,1)$-category $(\infty,n)\mathscr{C}at$ that satisfies the axioms from BarwickSchommerPries. In the $(\infty,1)$-category $(\infty,n)\mathscr{C}at$, the objects are the $(\infty,n)$-categories, the morphisms are the $(\infty,n)$-functors, the $2$-morphisms are the equivalences $(

Theorems & Definitions (76)

  • Example 1.2.1
  • Example 1.2.2
  • Example 1.2.3
  • Example 1.2.4
  • Example 1.2.5: Vector spaces & Hilbert spaces
  • Example 1.2.6: Spaces
  • Example 1.2.7: Chain complexes
  • Example 1.2.8: Linear categories
  • Example 1.2.9: Points in a space
  • Example 1.2.10: Spaces and Spans
  • ...and 66 more