Units and Counits

Summary

An adjunction $F⊣G$ is equivalent to natural transformations $\eta :1_{\mathcal{A}}\Rightarrow GF$ (unit) and $\epsilon :FG\Rightarrow 1_{\mathcal{B}}$ (counit) satisfying the triangle identities. The unit ${\eta }_A$ is the universal map from $A$ into $G$-objects; the counit ${\epsilon }_B$ is the universal map from $F$-objects onto $B$.

Overview

The hom-set bijection $\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$ of an adjunction can be “encoded” in just two natural transformations: the unit $\eta :1\Rightarrow GF$ and counit $\epsilon :FG\Rightarrow 1$. These satisfy identities that allow full reconstruction of the bijection, and they are often the most convenient way to specify or work with an adjunction.

Main Content

Definition: Unit and Counit of an Adjunction

For an adjunction $F⊣G$ with bijection $\overline{(-)}$:

• The unit $\eta :1_{\mathcal{A}}\Rightarrow GF$ has components ${\eta }_A=\overline{1_{FA}}:A\to GFA$. It is the transpose of the identity on $FA$.
• The counit $\epsilon :FG\Rightarrow 1_{\mathcal{B}}$ has components ${\epsilon }_B={\overline{1_{GB}}}^{-1}:FGB\to B$. It is the inverse-transpose of the identity on $GB$.

Intuition:

• ${\eta }_A:A\to GFA$ is the “insertion of generators” — the canonical map from $A$ into the object $GFA$ obtained by freely constructing and then forgetting.
• ${\epsilon }_B:FGB\to B$ is the “evaluation” — the canonical map from the free object on the underlying set of $B$ back to $B$ itself.

Triangle Identities

Theorem: Triangle Identities (BCT, Ch. 2.2)

The unit $\eta$ and counit $\epsilon$ of an adjunction $F⊣G$ satisfy:

1. $(G\epsilon )\circ (\eta G)=1_G$, i.e., for each $B$: $G{\epsilon }_B\circ {\eta }_{GB}=1_{GB}$
2. $(\epsilon F)\circ (F\eta )=1_F$, i.e., for each $A$: ${\epsilon }_{FA}\circ F{\eta }_A=1_{FA}$

Conversely, any pair of natural transformations $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$ satisfying these identities determines an adjunction.

Mnemonic: The triangle identities say that going “around the triangle” composed with the unit/counit is the identity. Diagrammatically:

$$G\stackrel{\eta G}{\to }GFG\stackrel{G\epsilon }{\to }G=G\stackrel{1_G}{\to }G$$

Recovering the Bijection from Unit/Counit

Given $\eta$ and $\epsilon$ satisfying the triangle identities, the bijection $\overline{(-)}$ and its inverse are:

$$\bar{f}=Gf\circ {\eta }_A\text{for }f:FA\to B$$ $${\tilde{g}}^{-1}={\epsilon }_B\circ Fg\text{for }g:A\to GB$$

One checks that these are mutual inverses using the triangle identities.

Examples

Example: Unit/Counit for Free/Forgetful (BCT, Ch. 2.2)

For the adjunction $F⊣U$ between the free group functor and the forgetful functor:

• Unit ${\eta }_X:X\to UFX$: the inclusion of a set $X$ into the underlying set of the free group $FX$ — “generators into their free group.”
• Counit ${\epsilon }_G:FUG\to G$: the homomorphism from the free group on the underlying set of $G$ to $G$ itself, sending each generator $g$ to itself — “evaluate the free group.”

Triangle identity 1: $U{\epsilon }_G\circ {\eta }_{UG}=1_{UG}$ — inserting generators and then evaluating is the identity on the underlying set.

Example: Unit/Counit for Product/Hom (BCT, Ch. 2.2)

For the cartesian closed adjunction $(-)\times B⊣(-{)}^B$ in a cartesian closed category:

• Unit ${\eta }_A:A\to (A\times B{)}^B$: the curried identity, ${\eta }_A(a)(b)=(a,b)$.
• Counit ${\epsilon }_C:C^B\times B\to C$: evaluation, ${\epsilon }_C(f,b)=f(b)$.

Connections

• Unit/counit gives an equivalent definition of adjunctions; see Adjoint Functors for the hom-set definition.
• Monadicity: from an adjunction $F⊣G$, the composite $T=GF:\mathcal{A}\to \mathcal{A}$ with multiplication $\mu =G\epsilon F:GF\circ GF\Rightarrow GF$ and unit $\eta :1\Rightarrow GF$ forms a monad (not covered in Leinster but a direct extension).
• The adjoint functor theorems (Adjoint Functor Theorems) produce adjoints by constructing universal maps, which are exactly the unit components.

See Also

• Adjoint Functors — The hom-set definition
• Adjunctions via Initial Objects — Universal maps and the unit’s universal property
• Adjoints and Limits — How units/counits are used in preservation proofs
• Cartesian Closed Categories — the Product/Hom adjunction example (unit = currying, counit = evaluation) is the canonical CCC instance
• Adjunctions Induce Monads — the T=GF monad induced by the adjunction this note defines