Adjoint functor

An adjunction of functors is an adjunction in $ℭ 𝔞 𝔱$ . #m/def/cat Let $𝖣$ , $𝖢$ be categories. A pair of functors $𝐹 : 𝖣 ⇆ 𝖢 : 𝑈$ form an adjunction, written

or compactly $𝐹 ⊣ 𝑈 : 𝖣 \to 𝖢$ , iff there is a natural isomorphism in $𝖲 𝖾 𝗍 (𝖢 𝐨 𝐩 \times 𝖣)$ of hom-sets¹

𝜑 : 𝖣 (𝐹 \times 1 𝖣) ≅ 𝖢 (1 𝖢 \times 𝑈) : 𝜑 - 1 𝜑 𝐶, 𝐷 : 𝖣 (𝐹 𝐶, 𝐷) ≅ 𝖢 (𝐶, 𝑈 𝐷) : 𝜑 - 1 𝐶, 𝐷 .

When adjoints exist they are unique up to natural isomorphism, hence we call $𝐹$ the left adjoint of $𝑈$ , and $𝑈$ the right adjoint of $𝐹$ .

Proof of uniqueness

By duality, it suffices to prove right adjoints are unique up to natural isomorphism. Suppose $𝐹 ⊣ 𝑈, 𝑉 : 𝖣 ⇆ 𝖢$ . Then by adjunction

𝖢 (1 \times 𝑈) ≅ 𝖣 (𝐹 \times 1) ≅ 𝖢 (1 \times 𝑉)

hence for any object $𝐷 \in 𝖣$ we have

よ (𝑈 𝐷) ≅ よ (𝑉 𝐷)

naturally and thus

𝑈 𝐷 ≅ 𝑈 𝑉

naturally (see Yoneda embedding).

The name comes from an analogy to the Adjoint operator. In the archetypal examples, we think of $𝑈$ as forgetful and $𝐹$ as free — See Free-forgetful adjunction.

Unit and coünit

We can equivalently rephrase the condition for an adjunction in terms of a unit or coünit, so named since they form the corresponding data for a monad or comonad induced by the adjunction respectively.

There exists a natural transformation $𝜂 : 1 \Rightarrow 𝑈 𝐹 : 𝖢 \to 𝖢$ called the unit of adjunction such that for any objects $𝐶 \in 𝖢$ , $𝐷 \in 𝖣$ , and morphism $𝑓 \in 𝖢 (𝐶, 𝑈 𝐷)$ , there exists a unique adjunct $𝑓 ♯ \in 𝖣 (𝐹 𝐶, 𝐷)$ such that $𝑓 = (𝑈 𝑓 ♯) 𝜂 𝐶$ .
There exists a natural transformation $𝜖 : 𝐹 𝑈 \Rightarrow 1 : 𝖢 \to 𝖢$ called the coünit of adjunction such that for any objects $𝐶 \in 𝖢$ , $𝐷 \in 𝖣$ , and morphism $𝑔 \in 𝖣 (𝐹 𝐶, 𝐷)$ , there exists a unique adjunct $𝑔 ♯ \in 𝖢 (𝐶, 𝑈 𝐷)$ such that $𝑔 = 𝜖 𝐷 (𝐹 𝑔 ♯)$ .

To see that either of these are necessary and sufficient, note²

𝜂 𝐶 = 𝜑 𝐶, 𝐹 𝐶 (1 𝐹 𝐶) 𝜑 - 1 (𝑓) = 𝑓 ♯ 𝜖 𝐷 = 𝜑 - 1 𝑈 𝐷, 𝐷 (1 𝑈 𝐷) 𝜑 (𝑔) = 𝑓 ♭

This gives us another perspective on adjunctions: They are a weakening of Equivalence of categories.

Properties

#state/tidy | #lang/en | #SemBr

2010. Category theory, §9 ↩
2020. From categories to homotopy theory, p. 40 ↩

Adjoint functor

Unit and coünit

Properties

Footnotes