Higher categories

Builds on

Required prerequisites, all the way down: 21

You will learn to

• Define strict 2-categories and bicategories, and show that a bicategory with one object is a monoidal category.
• Work in the 2-category Cat with vertical and horizontal composition and the interchange law.
• Describe the difficulties of defining weak n-categories and the main approaches, separating established theory from open questions.

Definition

• A strict 2-category has objects, 1-morphisms and 2-morphisms with vertical and horizontal compositions satisfying associativity, unit and interchange laws.
• A bicategory weakens the associativity and unit laws for 1-morphisms to coherent invertible 2-morphisms.
• A pseudofunctor preserves composition and identities up to coherent invertible 2-morphisms.

Theorem

• Categories, functors and natural transformations form a strict 2-category Cat.
• A bicategory with one object is the same thing as a monoidal category.
• Every bicategory is biequivalent to a strict 2-category.
• Not every tricategory is triequivalent to a strict 3-category; Gordon, Power and Street showed that Gray-categories are needed.

Common misconceptions

• Assuming strictification works in every dimension.
• Thinking there is one agreed definition of weak n-category for all n; several definitions exist, and comparing them is ongoing research.

Leads to

Optional discoveries

• Homotopy type theory Higher-dimensional structure also appears in type theory: the iterated identity types of a type form a weak ∞-groupoid (Lumsdaine; van den Berg and Garner).
• ← Categories
• ← Enriched categories
• ← Large cardinals

Further reading

• Tom Leinster, Higher Operads, Higher Categories (2004)
• Niles Johnson and Donald Yau, 2-Dimensional Categories (2021)