2-category

A 2-category¹ is motivated from several perspectives:

A 2-category is a category in the next dimension: While categories have a set² of morphisms between objects, a 2-category has a category of morphisms between objects. Because the objects of each “hom-category” need not form a set, the “identities” involved are higher isomorphisms having coherence relations of their own, similar to the associator and unitors of a monoidal category.
A 2-category is the oidification of a monoidal category, so that each object has a monoidal category of endomorphisms.

Thus, in terms of collections, a 2-category $ℭ$ is a mathematical object consisting of: #m/def/cat/2

a collection of objects, $ℭ 0$ ;
for any $𝐴, 𝐵 \in ℭ 0$ , a category $ℭ (𝐴, 𝐵)$ ;

Notation

We call the elements of the collection $ℭ 1 (𝐴, 𝐵) : = ℭ (𝐴, 𝐵) 0$ 1-morphisms, and notate $𝑓 \in ℭ 1 (𝐴, 𝐵)$ with

𝑓 : 𝐴 \to 𝐵 .

For any $𝑓, 𝑔 \in ℭ 1 (𝐴, 𝐵)$ we call the elements of the set $ℭ 2 (𝐴, 𝐵) (𝑓, 𝑔) : = ℭ (𝐴, 𝐵) (𝑓, 𝑔)$ 2-morphisms, and notate $𝛽 \in ℭ 2 (𝐴, 𝐵) (𝑓, 𝑔)$ with

𝛽 : 𝑓 \Rightarrow 𝑔 : 𝐴 \to 𝐵 .

for any $𝐴, 𝐵, 𝐶 \in ℭ 0$ , a 1-composition bifunctor; $(\circ) : ℭ (𝐵, 𝐶) \times ℭ (𝐴, 𝐵) \to ℭ (𝐴, 𝐶);$
for any $𝐴 \in ℭ 0$ , a 1-identity $1 𝐴 \in ℭ 1 (𝐴, 𝐴)$ , sometimes by abuse denoted $𝐴$ ;
for any $𝐴, 𝐵, 𝐶, 𝐷 \in ℭ 0$ , a natural isomorphism called the associator with components $𝛼 ℎ, 𝑔, 𝑓 : (ℎ \circ 𝑔) \circ 𝑓 \Rightarrow ℎ \circ (𝑔 \circ 𝑓) : 𝐴 \to 𝐷$ for indices $ℎ \in ℭ 1 (𝐶, 𝐷)$ , $𝑔 \in ℭ 1 (𝐵, 𝐶)$ , and $𝑓 \in ℭ 1 (𝐴, 𝐵)$ ;
for any $𝐴, 𝐵 \in ℭ 0$ , a natural isomorphism called the left-unitor with components $𝜆 𝑓 : 𝟙 𝐵 \circ 𝑓 \Rightarrow 𝑓 : 𝐴 \to 𝐵$ for index $𝑓 \in ℭ (𝐴, 𝐵)$ ; and
for any $𝐴, 𝐵 \in ℭ 0$ , a natural isomorphism called the right-unitor with components $𝜌 𝑓 : 𝑓 \circ 𝟙 𝐴 \Rightarrow 𝑓 : 𝐴 \to 𝐵$ in index $𝑓 \in ℭ (𝐴, 𝐵)$ ;

satisfying the so-called triangle identity

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and pentagon identity

for any $𝐴 𝑓 \to 𝐵 𝑔 \to 𝐶 ℎ \to 𝐷 𝑖 \to 𝐸$ in $ℭ$ . Together these diagrams ensure that the operation of $(\circ)$ is unital associative up to canonical natural isomorphism, by the Coherence theorem for 2-categories.

Examples

Just as $𝖲 𝖾 𝗍$ is the fundamental example of a category, $ℭ 𝔞 𝔱$ is the fundamental example of a 2-category

2-category

Examples

See also

Footnotes