Adjunction

Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. An adjunction between them consists of:

• Two functors:
  • $F:\mathcal{C}\to \mathcal{D}$ - left adjoint
  • $G:\mathcal{D}\to \mathcal{C}$ - right adjoint
• Two natural transformations:
  • $\eta :1_{\mathcal{C}}\Rightarrow G\circ F$ - unit
  • $\epsilon :F\circ G\Rightarrow 1_{\mathcal{D}}$ - counit

The following triangle identities must commute:

These are equivalent to the zig-zag laws, where the composition of the unit and counit (via whiskering) yields the identity:

Hom-set Definition

An adjunction $F⊣G$ may be defined via a natural isomorphism of hom-sets:

$\mathcal{D}[Fc,d]\cong \mathcal{C}[c,Gd]$ This bijection is natural in both $c\in \mathcal{C}$ and $d\in \mathcal{D}$. For any morphism $f:Fc\to d$, there exists a unique transpose $f^{♯}:c\to Gd$ given by $G(f)\circ {\eta }_c$. Conversely, for $g:c\to Gd$, the transpose $g^{♭}:Fc\to d$ is ${\epsilon }_d\circ F(g)$.

Remarks

With set-based categories, a left adjoint maps to the category with more structure, while a right adjoint preserves or reflects structure, however for an arbitrary adjunction, such a statement can’t be made, since in the Opposite Category adjunction the dual behaviour could be observed.

Examples of Adjunctions

• Free-Forgetful Adjunction
• Product-Slice Adjunction
• Coproduct-Diagonal Adjunction
• Diagonal-Product Adjunction

References

maclane1978-categories riehl2016-category-theory