Categorical bridge

In category theory, a discipline in mathematics, a bridge between categories 𝔸 and 𝔹 is a category ℍ such that 𝔸 and 𝔹 are disjoint full subcategories of ℍ and Obℍ = Ob𝔸 ∪ Ob𝔹. Morphisms of 𝔸 and 𝔹 are called homomorphisms and the rest (passing between 𝔸 and 𝔹) are called heteromorphisms.

In notation: ℍ : 𝔸 ⇋ 𝔹.

As an example, the empty bridge between two categories is just their disjoint union.

A directed bridge from 𝔸 to 𝔹 is a bridge without arrows of the form B → A (where B ∈ Ob𝔹 and A ∈ Ob𝔸). We can easily see that directed bridges and profunctors (i.e. functors F : 𝔸^op × 𝔹 → Set) are eventually the same [by identifying F(A,B) with the set of heteromorphisms A → B].

Bridge morphism

A morphism between bridges ℍ, 𝕂 : 𝔸 ⇋ 𝔹 is just a functor φ : ℍ → 𝕂 which is identical on both 𝔸 and 𝔹, i.e. φ∣_𝔸 = id_𝔸 and φ∣_𝔹 = id_𝔹.

Profunctors (directed bridges)

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