2-Categories

Core Idea

A 2-category is a category enriched over Cat: it has objects (0-cells), morphisms between objects (1-cells), and morphisms between morphisms (2-cells or 2-morphisms). The 2-cells can be composed in two ways: vertically (composing 2-cells along shared 1-cells, like composing natural transformations) and horizontally (composing 2-cells along shared 0-cells, like whiskering). These two compositions must satisfy the interchange law. The primary example is Cat itself, where objects are categories, 1-cells are functors, and 2-cells are natural transformations. Strict 2-categories require associativity and unit laws to hold on the nose; the weaker notion of bicategory allows them to hold only up to coherent isomorphism.

How It's Best Learned

Take Cat as the running example. Identify the 0-cells (small categories), 1-cells (functors), and 2-cells (natural transformations). Practice vertical composition (composing two natural transformations α: F ⇒ G and β: G ⇒ H) and horizontal composition (whiskering a natural transformation with a functor). Verify the interchange law on a concrete example. Then consider the bicategory of spans as a non-strict example.

Common Misconceptions

• A 2-category is not the same as a double category; 2-categories have one type of 1-cell, while double categories have two (horizontal and vertical).
• Strict and weak (bi)categories are genuinely different notions; not every bicategory is equivalent to a strict 2-category in the naive sense, though the coherence theorem for bicategories provides a strictification result.
• The interchange law is not a consequence of the other axioms; it is an independent condition that constrains how vertical and horizontal composition interact.

Explainer

An ordinary category has objects and morphisms between objects. A 2-category adds a third level: morphisms between morphisms, called 2-cells or 2-morphisms. You already know the paradigmatic example from your prerequisites: in the functor category [C, D], objects are functors and morphisms are natural transformations. A 2-category makes this structure explicit and formalizes two independent ways of composing 2-cells that coexist in [C, D] and in Cat itself.

Take Cat as the running example throughout. Its 0-cells (objects) are small categories, its 1-cells (morphisms between objects) are functors F: C → D, and its 2-cells (morphisms between 1-cells) are natural transformations α: F ⇒ G between functors with the same source and target. Vertical composition of 2-cells stacks them end-to-end along a shared 1-cell: if α: F ⇒ G and β: G ⇒ H are natural transformations between the same two categories, their vertical composite β ∘ α: F ⇒ H is the natural transformation whose component at each object X is β_X ∘ α_X. This is exactly the composition you know from functor categories. Each hom-category Hom(C, D) is itself a category (with natural transformations as morphisms), and vertical composition is the composition in that category.

Horizontal composition combines 2-cells side by side across different hom-categories. If α: F ⇒ G is a natural transformation between functors C → D, and β: H ⇒ K is a natural transformation between functors D → E, the horizontal composite β ★ α: H∘F ⇒ K∘G is a natural transformation between functors C → E. In Cat, this is whiskering: (β ★ α)_X = β_{GX} ∘ H(α_X) = K(α_X) ∘ β_{FX} (these are equal by naturality of β). The identity 2-cell for horizontal composition on a functor F is the identity natural transformation id_F.

The interchange law (β₂ ∘ β₁) ★ (α₂ ∘ α₁) = (β₂ ★ α₂) ∘ (β₁ ★ α₁) says that composing vertically then horizontally gives the same result as composing horizontally then vertically. Visually: arrange four 2-cells in a 2×2 grid; you can compose the rows first (two vertical composites) then compose the results horizontally — or compose the columns first (two horizontal composites) then compose vertically — and the answer must agree. In Cat, this follows from naturality, but as an axiom in an abstract 2-category it is an independent condition that must be verified.

The strict vs. weak distinction becomes significant when you move beyond Cat. In a strict 2-category, all associativity and unit laws for 1-cell composition hold on the nose as equalities. In a bicategory, they hold only up to specified 2-cell isomorphisms (associators and unitors) satisfying coherence conditions analogous to Mac Lane's pentagon and triangle identities for monoidal categories. The category of spans in a category with pullbacks, profunctors between categories, and cobordisms between manifolds are all naturally bicategories. The coherence theorem guarantees every bicategory is equivalent (as a bicategory) to a strict 2-category, so you can often "strictify" for computational purposes — but the natural presentation of many important examples is inherently weak.