module Cat.Bi.Diagram.Adjunction where

Adjunctions in a bicategoryπŸ”—

Let be a bicategory, be objects, and and be 1-cells. Generalising the situation where and are functors, we say they are adjoints if there exist 2-cells and (the unit and counit respectively), satisfying the equations


called the triangle identities (because of their shape) or zigzag identities (because it sounds cool).

  record _⊣_ {a b : B.Ob} (f : a B.↦ b) (g : b B.↦ a) : Type β„“' where
      Ξ· : B.β‡’ (g B.βŠ— f)
      Ξ΅ : (f B.βŠ— g) B.β‡’

      zig : ≑ B.λ← f B.∘ (Ξ΅ B.β—€ f) B.∘ B.α← f g f B.∘ (f B.β–Ά Ξ·) B.∘ B.ρ→ f
      zag : ≑ B.ρ← g B.∘ (g B.β–Ά Ξ΅) B.∘ B.Ξ±β†’ g f g B.∘ (Ξ· B.β—€ g) B.∘ B.Ξ»β†’ g

Working in a fully weak bicategory means the triangle identities, rather than simply expressing a compatibility relation between and as is the case for adjoint functors, instead exhibit a complicated compatibility relation between and the structural isomorphisms (the unitors and associator) of the ambient bicategory.

We have taken pains to draw the triangle identities as triangles, but counting the morphisms involved you’ll find that they really want to be commutative pentagons instead (which we draw in this website as commutative altars):