Bicategories and 2-Categories — Mathematics Knowledge System

Overview

A bicategory (or weak 2-category) is a higher-categorical structure with objects (0-cells), morphisms between objects (1-cells), and morphisms between morphisms (2-cells). Composition of 1-cells is associative and unital only up to coherent 2-cell isomorphisms (the associator and unitors). A (strict) 2-category requires composition to be strictly associative. Key examples: Cat (categories, functors, natural transformations), Rel (sets, relations, inclusions), Span(C) (spans in a category), and the bicategory of algebras, bimodules, and bimodule maps. The Homotopy Hypothesis: n-groupoids model homotopy n-types.

Intuition

A 2-category has: objects (like categories), 1-morphisms between objects (like functors), and 2-morphisms between 1-morphisms (like natural transformations). The prototypical example is Cat: categories are objects, functors are 1-morphisms, and natural transformations are 2-morphisms. Composition of functors is strictly associative in Cat. In a bicategory, the composition of 1-morphisms is only 'associative up to isomorphism' -- there is a coherent 2-cell alpha: (h o g) o f ~= h o (g o f) (an associator), similar to how monoidal categories have associators for tensor products.

Formal Definition

Definition

A bicategory B consists of: a collection of objects (0-cells); for each pair of objects a, b, a category B(a,b) (whose objects are 1-cells and morphisms are 2-cells); for each triple a,b,c, a composition functor o: B(b,c) x B(a,b) -> B(a,c); for each a, an identity 1-cell id_a in B(a,a); natural isomorphisms: associator alpha_{f,g,h}: (h o g) o f ~= h o (g o f) and unitors lambda_f: id_b o f ~= f, rho_f: f o id_a ~= f; satisfying the pentagon axiom (for alpha) and triangle axiom (for alpha, lambda, rho).

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

Associator 2-cell

\lambda_f: \mathrm{id}_b \circ f \xrightarrow{\sim} f,\quad \rho_f: f \circ \mathrm{id}_a \xrightarrow{\sim} f

Unitor 2-cells

\mathbf{Cat}: \text{categories, functors, natural transformations -- a strict 2-category}

Key example

\text{Bicategory} \simeq \text{Strict 2-category (up to biequivalence)}

Coherence theorem

Notation

Notation	Meaning
$a, b, c$	0-cells (objects) of bicategory
$f: a \to b$	1-cell (morphism of objects)
$\alpha: f \Rightarrow g$	2-cell (morphism of morphisms)
$\bullet \text{ (vertical), } \circ \text{ (horizontal)}$	Two kinds of composition of 2-cells

Theorems

Theorem 1: Coherence Theorem for Bicategories

Every bicategory is biequivalent to a strict 2-category. Consequently, any diagram built from the associator, unitors, and their inverses (between the same 1-cell functors) commutes. This is the bicategorical analogue of Mac Lane's monoidal coherence theorem. Proof: embed the bicategory into its strictification via the Yoneda embedding, replacing the bicategory with the 2-category of pseudofunctors from B^op to Cat.

Theorem 2: Interchange Law

In a bicategory, there are two ways to compose 2-cells: vertically (alpha followed by beta between same 1-cells: B(a,b)(f,g) x B(a,b)(g,h) -> B(a,b)(f,h)) and horizontally (natural transformation of composites). The interchange law: (beta_2 * alpha_2) o (beta_1 * alpha_1) = (beta_2 o beta_1) * (alpha_2 o alpha_1), where o is vertical and * is horizontal composition. This is equivalent to composition being a functor.

Theorem 3: Bicategory of Rings and Bimodules

The bicategory Bim has: 0-cells = rings; 1-cells from R to S = (R,S)-bimodules; 2-cells = bimodule maps; composition of 1-cells = tensor product: M: R->S and N: S->T compose to N tensor_S M: R->T. The identity 1-cell at R is R itself (as an (R,R)-bimodule). This is a non-strict bicategory (associativity of tensor product is only up to canonical iso).

Worked Examples

1
Objects (0-cells): sets A, B, C, ...
2
1-cells from A to B: spans, i.e., diagrams A <- M -> B (a set M with maps to A and B).
3
2-cells between spans (A <- M -> B) and (A <- N -> B): a function f: M -> N making both triangles commute (morphism of spans).
4
Composition of spans: (A <- M -> B) and (B <- N -> C) compose to (A <- M x_B N -> C) via the pullback M x_B N.
$A \leftarrow M \times_B N \rightarrow C$
5
Composition of spans is only associative up to the canonical pullback isomorphism (M x_B N) x_C P ~= M x_B (N x_C P). So Span(Set) is a bicategory, not a strict 2-category.

✓ Answer

Span(Set): objects = sets, 1-cells = spans, 2-cells = span morphisms, composition = pullback. This is a bicategory (associativity holds up to canonical isomorphism).

Practice Problems

Hardfree response

What is a monoidal category as a bicategory? Explain the delooping construction B(C) for a monoidal category C.

Common Mistakes

Common Mistake

Thinking that a 2-category is just a category where objects are categories.

A 2-category has three levels of structure: 0-cells (objects), 1-cells (morphisms between objects), and 2-cells (morphisms between morphisms). You cannot recover the full 2-categorical structure just from knowing that the objects are categories. The crucial additional data is: what the 2-cells are (in Cat, they are natural transformations), and how the two kinds of composition (vertical and horizontal) interact (via the interchange law). A 2-category with trivial 2-cells (all identity) is just an ordinary category.

Quiz

In the bicategory Cat, the 2-cells are:

Functors between categoriesNatural transformations between functorsObjects of categoriesAdjoint pairs of functors

Historical Background

Benabou introduced bicategories in 1967 to capture structures where composition is only associative up to isomorphism. The strict version (2-categories) had been studied by Ehresmann and Kelly earlier. The coherence theorem for bicategories (any bicategory is biequivalent to a strict 2-category) was proved by Mac Lane and Pare in the 1980s. Grothendieck's homotopy hypothesis (n-groupoids model homotopy n-types) stimulated the development of higher category theory. Street's work in the 1970s-80s developed the theory systematically. Today, (oo,1)-categories (quasi-categories, complete Segal spaces) and (oo,n)-categories are active research areas.

1967
Benabou introduces bicategories
Jean Benabou
1970s
Street and Kelly develop 2-categorical algebra systematically
Ross Street, G. Max Kelly
1983
Grothendieck proposes homotopy hypothesis in 'Pursuing Stacks'
Alexander Grothendieck
2006
Lurie's work on (oo,1)-categories and higher algebra
Jacob Lurie

Summary

Bicategory: objects (0-cells), morphisms (1-cells), morphisms of morphisms (2-cells). Associativity up to 2-iso.
Coherence: every bicategory is biequivalent to a strict 2-category.
Examples: Cat (strict 2-cat), Span(C), Bim (rings and bimodules), B(C) for monoidal C.
Monoidal category = bicategory with one object (delooping correspondence).

References

BookBenabou, J. Introduction to Bicategories. In Reports of the Midwest Category Seminar, 1967.
BookLeinster, T. Basic Bicategories. ArXiv: math/9810017, 1998.
WebsitenLab -- Bicategory