Adjunctions

A functor \(\mathcal{C}\xrightarrow{L}\mathcal{D}\) is left adjoint to a functor \(\mathcal{D}\xrightarrow{R}\mathcal{C}\)

• For any \(c \in C\) and \(d \in D\), there is an isomorphism of hom-sets: \(\alpha_{c,d}: \mathcal{C}(c,R(d)) \xrightarrow{\cong} \mathcal{D}(L(c),d)\) that is natural in c and d.

• Given a morphism \(c \rightarrow{f} R(d)\) in \(\mathcal{C}\), its image \(g:=\alpha_{c,d}(f)\) is called its mate (and vice-versa)

• To denote the adjunction we write \(L \dashv R\) or

  

Linked by

• Galois connections are adjoint: referenced
• Currying: referenced

Galois connections between preorders are the same as ?adjunctions between the corresponding categories.

The adjunction called ?currying says \(\mathbf{Set}(A \times B,C)\cong \mathbf{Set}(A,C^B)\)

Linked by

• Exercise 3-73: referenced

Examples of adjunctions

1. Free constructions: given a set you get a free groupmonoidring/vector space/etc. - each of these is a ?left adjoint. The corresponding ?right adjoint throws away the algebraic structure.

2. Given a graph you get a free preorder or a free category, the corresponding right adjoint is the underlying graph of a preorder/category.

3. Discrete things: given any set you get a discrete preordergraphmetric spacecategorytopological space/etc.; each of these is a ?left adjoint and the corresponding ?right adjoint again recovers the set.

4. Codiscrete things: given any set you get a codiscrete preorder, complete graph, codiscrete category, category, etc.; each of these is a ?right adjoint and the ?left adjoint gives the underlying set.

5. Given a group, you can quotient by its commutator subgroup to get an abelian group; this is a ?left adjoint. The ?right adjoint is the inclusion of abelian groups into Grp

Currying was an adjunction between functors in Set, but the example only discussed what the functors did to objects.

1. Given a morphism \(X \xrightarrow{f}Y\), what morphism should \(X \times B \xrightarrow{-\times B}Y\times B\) return?

2. Given a morphism \(X \xrightarrow{f}Y\), what morphism should \(X^ B \xrightarrow{(-)^B}Y^B\) return?

3. Consider \(\mathbb{N}\times \mathbb{N}\xrightarrow{+}\mathbb{N}\). Currying \(+\), we get a certain function \(\mathbb{N}\xrightarrow{p}\mathbb{N}^\mathbb{N}\). What is \(p(3)\)?

Solution(1)

1. This morphism maps \((x,b)\mapsto (f(x),b)\)

2. This morphism takes in a function \(B \xrightarrow{bx} X\) and composes with f to give \(B \xrightarrow{bx;f} Y\)

3. It takes a number and returns a function which adds three to that number.