Higher Category Theory — Mathematics Knowledge System

Overview

Higher category theory extends ordinary category theory by allowing morphisms between morphisms (2-morphisms), morphisms between those (3-morphisms), and so on. An n-category has morphisms at every level up to n (or all levels, in the case of \infty-categories). These structures arise naturally in homotopy theory, mathematical physics (topological field theories), and the study of algebraic structures with coherence conditions. The central challenge is correctly formulating the coherence conditions that govern the associativity and unit laws at each level.

Intuition

In ordinary category theory, the diagram of morphisms f: A -> B and g: B -> C either commutes (g \circ f = h) or it does not — equality is strict. In a 2-category, there can be 2-morphisms \alpha: f => g (a morphism between morphisms), and the question is not whether two composites are equal but whether there is a coherent 2-morphism between them. Going higher, 3-morphisms mediate between 2-morphisms, and so on. The challenge: at each level, associativity holds only up to the next level's morphisms, creating an infinite tower of coherence conditions.

Formal Definition

Definition

A strict 2-category (or strict \omega-category) has objects, 1-morphisms, and 2-morphisms with strictly associative and unital composition at each level. A bicategory (weak 2-category) has these only up to coherent invertible 2-morphisms (associators and unitors) satisfying the Mac Lane pentagon and triangle identities.

\alpha_{f,g,h}: (h \circ g) \circ f \xRightarrow{\sim} h \circ (g \circ f)

Associator 2-isomorphism in a bicategory

associator

\lambda_f: \mathrm{id}_B \circ f \xRightarrow{\sim} f, \quad \rho_f: f \circ \mathrm{id}_A \xRightarrow{\sim} f

Left and right unitor 2-isomorphisms

unitors

\text{Pentagon: } (\alpha_{f,g,h} \circ \mathrm{id}_k) \cdot \alpha_{f \circ g, h, k} \cdot (\mathrm{id}_f \circ \alpha_{g,h,k}) = \alpha_{f, g \circ h, k} \cdot \alpha_{f \circ g \circ h, k}

Pentagon coherence identity for associators in a bicategory

pentagon

\pi_n(X, x) = \mathrm{Hom}_{\Pi_{\infty}(X)}(x, x)[\text{loop spaces}]

The fundamental \infty-groupoid of a space X encodes all homotopy groups via its morphisms

fundamental-infinity-groupoid

Notation

Notation	Meaning
$\mathbf{2\text{-}Cat}$	Category of (strict) 2-categories
$\mathbf{Bicat}$	Category of bicategories
$\alpha: f \Rightarrow g$	2-morphism (natural transformation between 1-morphisms)
$\alpha \star \beta$	Horizontal composition of 2-morphisms
$\alpha \cdot \beta$	Vertical composition of 2-morphisms
$\Pi_\infty(X)$	Fundamental \infty-groupoid of topological space X

Theorems

Theorem 1: Coherence for Bicategories (Mac Lane-Stasheff type)

Every bicategory is equivalent (as a bicategory) to a strict 2-category. Equivalently, any diagram of isomorphisms in a bicategory built from associators and unitors commutes.

Theorem 2: Baez-Dolan Stabilization Hypothesis

A k-tuply monoidal n-category (an (n+k)-category with only trivial j-morphisms for j < k) is the same as a symmetric monoidal n-category when k \geq n+2. This was proved by Lurie using \infty-categorical methods.

Theorem 3: Homotopy Hypothesis (Grothendieck)

The \infty-groupoids (\infty-categories in which all morphisms at all levels are invertible) are equivalent, as a homotopy theory, to homotopy types of topological spaces. Formally, \infty\text{-}\mathbf{Gpd} \simeq \mathbf{Top}.

Worked Examples

Objects of Span(Set) are sets. A 1-morphism from A to B is a span: a set S with functions s: S -> A and t: S -> B.
$A \xleftarrow{s} S \xrightarrow{t} B$
A 2-morphism between spans (S, s, t) and (S', s', t') is a function f: S -> S' such that s' \circ f = s and t' \circ f = t.
$f: S \to S', \quad s' \circ f = s, \quad t' \circ f = t$
Composition of 1-morphisms is by pullback: (A <- S -> B) composed with (B <- T -> C) gives A <- S x_B T -> C. This is only associative up to canonical isomorphism (the pullback is not strictly associative), making Span(Set) a bicategory, not a strict 2-category.

Answer: Span(Set) is a bicategory where objects are sets, 1-morphisms are spans S: A -> B (a set S with maps to A and B), 2-morphisms are morphisms of spans. Composition is by pullback, associative only up to coherent 2-isomorphism.

Practice Problems

Difficulty 9/10

What is a monoidal category, and how does it arise as a special case of a 2-category?

Difficulty 10/10

Show that the Eckmann-Hilton argument implies that a monoid in the category of monoids is abelian.

Difficulty 10/10

State the Homotopy Hypothesis and explain its significance for the foundations of mathematics.

Common Mistakes

Common Mistake

Assuming strict 2-categories and bicategories are fundamentally different structures.

Every bicategory is equivalent to a strict 2-category (Mac Lane coherence), so for most purposes one can work with strict 2-categories. However, naturally occurring examples (spans, cobordisms, profunctors) are inherently bicategorical.

Common Mistake

Confusing horizontal and vertical composition of 2-morphisms.

Vertical composition (\alpha \cdot \beta) composes 2-morphisms with the same source and target 1-morphism (along the 1-morphism direction). Horizontal composition (\alpha \star \beta) composes 2-morphisms whose underlying 1-morphisms compose. The interchange law (\alpha \star \beta) \cdot (\gamma \star \delta) = (\alpha \cdot \gamma) \star (\beta \cdot \delta) must be satisfied.

Historical Background

2-categories appeared in the work of Bénabou (1967) on bicategories, where composition is only associative and unital up to coherent 2-morphisms. Street and Kelly developed 2-category theory in the 1970s. Higher categories in full generality were championed by Ross Street and, above all, John Baez and James Dolan, who in 1995 proposed the Periodic Table of n-categories and the Baez-Dolan stabilization hypothesis. The field exploded with Jacob Lurie's foundational work on \infty-categories in the 2000s.

1967
Bénabou introduces bicategories (weak 2-categories)
Jean Bénabou
1973
Kelly-Street: coherence for 2-categories
Gregory Maxwell Kelly, Ross Street
1995
Baez-Dolan: Periodic Table and cobordism hypothesis (TFT conjecture)
John Baez, James Dolan
2006
Lurie: HTT — Higher Topos Theory via quasi-categories
Jacob Lurie

Summary

Higher categories have morphisms at every level: n-morphisms for 0 <= k <= n (or all k in the \infty case), with composition associative and unital up to higher morphisms.
Bicategories are weak 2-categories where associativity and unitality hold up to coherent invertible 2-morphisms (associators and unitors) satisfying pentagon and triangle identities.
The Homotopy Hypothesis identifies \infty-groupoids with homotopy types, bridging algebraic topology and category theory.
The cobordism hypothesis (Baez-Dolan-Lurie) classifies topological field theories by the structure of the target (\infty,n)-category, with TFTs determined by fully dualizable objects.
The Periodic Table of n-categories (Baez-Dolan) organizes degenerate higher categories: a k-tuply degenerate n-category is a (n-k)-category with extra commutativity.

References

BookLeinster, T. Higher Operads, Higher Categories. London Mathematical Society, 2004.
PaperBaez, J. & Dolan, J. Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36 (1995).
PaperLurie, J. On the Classification of Topological Field Theories. Current Developments in Mathematics, 2008.