Category theory MOC

Adjoint functor

An adjunction of functors is an adjunction in Category of small categories. cat Let 𝖣, 𝖒 be categories. A pair of functors 𝐹 :𝖣 ⇆𝖒 :π‘ˆ form an adjunction, written

A quiver diagram.

or compactly 𝐹 βŠ£π‘ˆ :𝖣 →𝖒, iff there is a natural isomorphism in [[Functor category|{(𝖒𝐨𝐩×𝖣)}]] of hom-sets1

πœ‘:𝖣(𝐹×1𝖣)≅𝖒(1π–’Γ—π‘ˆ):πœ‘βˆ’1πœ‘πΆ,𝐷:𝖣(𝐹𝐢,𝐷)≅𝖒(𝐢,π‘ˆπ·):πœ‘βˆ’1𝐢,𝐷.

When adjoints exist they are unique up to natural isomorphism, hence we call 𝐹 the left adjoint of π‘ˆ, and π‘ˆ the right adjoint of 𝐹.

The name comes from an analogy to the Adjoint operator. In the archetypal examples, we think of π‘ˆ as forgetful and 𝐹 as free β€” See Free-forgetful adjunction.

Unit and coΓΌnit

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

  • There exists a natural transformation πœ‚ :1 β‡’π‘ˆπΉ :𝖒 →𝖒 called the unit of adjunction such that for any objects 𝐢 βˆˆπ–’, 𝐷 βˆˆπ–£, and morphism 𝑓 βˆˆπ–’(𝐢,π‘ˆπ·), there exists a unique adjunct 𝑓♯ βˆˆπ–£(𝐹𝐢,𝐷) such that 𝑓 =(π‘ˆπ‘“β™―)πœ‚πΆ.

    A quiver diagram.

  • There exists a natural transformation πœ– :πΉπ‘ˆ β‡’1 :𝖒 →𝖒 called the coΓΌnit of adjunction such that for any objects 𝐢 βˆˆπ–’, 𝐷 βˆˆπ–£, and morphism 𝑔 βˆˆπ–£(𝐹𝐢,𝐷), there exists a unique adjunct 𝑔♯ βˆˆπ–’(𝐢,π‘ˆπ·) such that 𝑔 =πœ–π·(𝐹𝑔♯).

    A quiver diagram.

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

πœ‚πΆ=πœ‘πΆ,𝐹𝐢(1𝐹𝐢)πœ‘βˆ’1(𝑓)=π‘“β™―πœ–π·=πœ‘βˆ’1π‘ˆπ·,𝐷(1π‘ˆπ·)πœ‘(𝑔)=𝑓♭

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

Properties

tidy | en | SemBr

Footnotes

  1. 2010. Category theory, Β§9 ↩

  2. 2020. From categories to homotopy theory, p. 40 ↩

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