Adjoint Functors - Adjunctions in Full

Adjunctions in Full

There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

An adjunction between categories C and D consists of

  • A functor F : C ← D called the left adjoint
  • A functor G : C → D called the right adjoint
  • A natural isomorphism Φ : homC(F–,–) → homD(–,G–)
  • A natural transformation ε : FG → 1C called the counit
  • A natural transformation η : 1D → GF called the unit

An equivalent formulation, where X denotes any object of C and Y denotes any object of D:

For every C-morphism there is a unique D-morphism such that the diagrams below commute, and for every D-morphism there is a unique C-morphism in C such that the diagrams below commute:

From this assertion, one can recover that:

  • The transformations ε, η, and Φ are related by the equations
\begin{align} f = \Phi_{Y,X}^{-1}(g) &= \varepsilon_X\circ F(g) & \in & \, \, \mathrm{hom}_C(F(Y),X)\\ g = \Phi_{Y,X}(f) &= G(f)\circ \eta_Y & \in & \, \, \mathrm{hom}_D(Y,G(X))\\ \Phi_{GX,X}^{-1}(1_{GX}) &= \varepsilon_X & \in & \, \, \mathrm{hom}_C(FG(X),X)\\ \Phi_{Y,FY}(1_{FY}) &= \eta_Y & \in & \, \, \mathrm{hom}_D(Y,GF(Y))\\ \end{align}
  • The transformations ε, η satisfy the counit-unit equations
\begin{align} 1_F &= \varepsilon F\circ F\eta\\ 1_G &= G\varepsilon \circ \eta G \end{align}
  • Each pair is a terminal morphism from F to X in C
  • Each pair is an initial morphism from Y to G in D

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. We will demonstrate the equivalence of these situations below.

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