Category theory; homological algebra Definition

2-category

Builds on Enriched category Category of small categories Category Functor Natural transformation

Definition

A 2-category (or strict 2-category) is a category enriched over the category of small categories $\mathbf{Cat}$ with the Cartesian product as tensor product. Explicitly, a 2-category $\mathcal{K}$ consists of:

a class of objects;
for each pair of objects $a, b$ , a category $\mathcal{K}(a, b)$ of 1-morphisms $f \colon a \to b$ and 2-morphisms $\theta \colon f \Rightarrow g$ , whose composition is called vertical;
for each triple $a, b, c$ , a functor $\circ \colon \mathcal{K}(b, c) \times \mathcal{K}(a, b) \to \mathcal{K}(a, c)$ , called horizontal composition;
for each object $a$ , an identity 1-morphism $1_a$ ;

such that horizontal composition is strictly associative and $1_a$ and its identity 2-morphism are strict units for it.

Remarks

That horizontal composition is a functor is the interchange law: $(\psi' \cdot \psi) \circ (\theta' \cdot \theta) = (\psi' \circ \theta') \cdot (\psi \circ \theta)$ , where $\cdot$ is vertical composition. Example: $\mathbf{Cat}$ , with small categories as objects, functors as 1-morphisms and natural transformations as 2-morphisms, is a 2-category; composition of functors is strictly associative. Non-example: rings, bimodules and bimodule maps, with composition of bimodules given by the tensor product, do not form a 2-category. Take an $R$ - $S$ -bimodule $M$ , an $S$ - $T$ -bimodule $N$ and a $T$ - $U$ -bimodule $P$ . The bimodules $(M \otimes_S N) \otimes_T P$ and $M \otimes_S (N \otimes_T P)$ are isomorphic but not equal in general, so strict associativity fails; they form a bicategory instead.

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Tag	`8TPR96GV7F`
URI	`m2://8TPR96GV7F`
Kind	definition
Writing	W0 stub
MSC	18N10
Defines	2-category
Uses	Enriched category, Category of small categories, Category, Functor, Natural transformation
Changed	2 November 2026

2-category

Definition

Remarks

Reading path

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