Adjunctions

Builds on

Required prerequisites, all the way down: 19

You will learn to

• Verify adjunctions such as free monoid ⊣ forgetful, − × B ⊣ (−)^B in Set, and the tensor–hom adjunction.
• Pass between the hom-set and the unit–counit definitions of an adjunction using the triangle identities.
• Prove that right adjoints preserve limits and left adjoints preserve colimits, and use this to show that some functors have no adjoint.

Definition

• F : C → D is left adjoint to G : D → C, written F ⊣ G, if there are bijections Hom_D(FA, B) ≅ Hom_C(A, GB) natural in A and B.
• The unit η : id_C ⇒ GF and counit ε : FG ⇒ id_D of an adjunction satisfy the triangle identities εF ∘ Fη = id_F and Gε ∘ ηG = id_G.
• A Galois connection is an adjunction between posets regarded as categories.

Theorem

• A left (or right) adjoint of a functor is unique up to unique natural isomorphism.
• Right adjoints preserve all limits that exist, and left adjoints preserve all colimits that exist.
• Every adjunction F ⊣ G induces a monad GF.
• In Set, − × B ⊣ (−)^B: functions A × B → C correspond to functions A → C^B (currying).

Common misconceptions

• Thinking every functor has an adjoint.
• Mixing up left and right; in a free–forgetful adjunction the free functor is the left adjoint.
• Assuming an adjunction is an equivalence; its unit and counit need not be isomorphisms.

Leads to

Optional discoveries

• The Yoneda lemma A functor G : D → C has a left adjoint if and only if, for each object A of C, the functor Hom_C(A, G(−)) : D → Set is representable; the Yoneda lemma then assembles the representing objects into a functor.Wonder Trail →
• The λ-calculus Currying, the adjunction − × B ⊣ (−)^B, is the basis of function types in the typed λ-calculus.
• ← Fields and Galois theory
• ← Limits and colimits
• ← Modules

Learning pathways

• Graduate preparation

Further reading

• Saunders Mac Lane, Categories for the Working Mathematician (1971)
• Emily Riehl, Category Theory in Context (2016)
• Tom Leinster, Basic Category Theory (2014)