2-category theory MOC

Bicategory

A bicategory is motivated from several perspectives:

1. A bicategory is a category in the next dimension: While categories have a set¹ of morphisms between objects, a bicategory 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.
2. A bicategory is the oidification of a monoidal category, so that each object has a monoidal category of endomorphisms.

Thus, in terms of collections, a bicategory ℭ is a mathematical object consisting of: #m/def/cat/bi

1. a collection of objects, ℭ0;
2. for any 𝐴,𝐵 ∈ℭ0, a category ℭ(𝐴,𝐵);

3. for any 𝐴,𝐵,𝐶 ∈ℭ0, a composition bifunctor; (∘):ℭ(𝐵,𝐶)×ℭ(𝐴,𝐵)→ℭ(𝐴,𝐶);
4. for any 𝐴 ∈ℭ0, a distinguished 𝟙𝐴 ∈ℭ1(𝐴,𝐴), sometimes by abuse denoted 𝐴;
5. for any 𝐴,𝐵,𝐶,𝐷 ∈ℭ0, a natural isomorphism called the associator with components 𝛼ℎ,𝑔,𝑓 :(ℎ ∘𝑔) ∘𝑓 ⇒ℎ ∘(𝑔 ∘𝑓) :𝐴 →𝐷 for indices ℎ ∈ℭ1(𝐶,𝐷), 𝑔 ∈ℭ1(𝐵,𝐶), and 𝑓 ∈ℭ1(𝐴,𝐵);
6. for any 𝐴,𝐵 ∈ℭ0, a natural isomorphism called the left-unitor with components 𝜆𝑓 :𝟙𝐵 ∘𝑓 ⇒𝑓 :𝐴 →𝐵 for index 𝑓 ∈ℭ(𝐴,𝐵); and
7. for any 𝐴,𝐵 ∈ℭ0, a natural isomorphism called the right-unitor with components 𝜌𝑓 :𝑓 ∘𝟙𝐴 ⇒𝑓 :𝐴 →𝐵 in index 𝑓 ∈ℭ(𝐴,𝐵);

satisfying the so-called triangle identity

and pentagon identity

for any 𝐴𝑓→𝐵𝑔→𝐶ℎ→𝐷𝑖→𝐸 in ℭ. Together these diagrams ensure that the operation of ( ∘) is unital associative up to canonical natural isomorphism, by the Coherence theorem for bicategories.

Examples

• Just as 𝖲𝖾𝗍 is the fundamental example of a category, ℭ𝔞𝔱 is the fundamental example of a bicategory

See also

• Notions of 2-functor

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