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:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
GFG \arrow[rr, "G\varepsilon"] && G \\
& G \arrow[ul, "\eta G"] \arrow[ur, "id"']
\end{tikzcd}
\end{document}

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
F \arrow[rr, "F\eta"] \arrow[dr, "id"'] && FGF \arrow[dl, "\varepsilon F"] \\
& F
\end{tikzcd}
\end{document}

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

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} && \mathcal{C} && \mathcal{C} && \mathcal{C} \\ \\ \mathcal{D} && \mathcal{D} && \mathcal{D} && \mathcal{D} 
\arrow[""{name=0, anchor=center, inner sep=0}, equals, from=1-1, to=1-3] 
\arrow["F", from=1-1, to=3-1]
\arrow["F", from=1-3, to=3-3]
\arrow[""{name=1, anchor=center, inner sep=0}, equals, from=1-5, to=1-7] 
\arrow[""{name=2, anchor=center, inner sep=0}, "F"'{pos=0.3}, from=1-7, to=3-5] 
\arrow[""{name=3, anchor=center, inner sep=0}, "G"{pos=0.8}, from=3-1, to=1-3] 
\arrow[""{name=4, anchor=center, inner sep=0}, equals, from=3-1, to=3-3] 
\arrow["G"', from=3-5, to=1-5] 
\arrow[""{name=5, anchor=center, inner sep=0}, equals, from=3-5, to=3-7] 
\arrow["G"', from=3-7, to=1-7]
\arrow["\eta"', shift right=4, Rightarrow, from=0, to=3] 
\arrow["\eta"', shift right=4, Rightarrow, from=1, to=2] 
\arrow["\varepsilon", shift left=4, Rightarrow, from=2, to=5] 
\arrow["\varepsilon", shift left=4, Rightarrow, from=3, to=4]
\end{tikzcd}
\end{document}

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 in general such a statement can’t be made.

Examples of Adjunctions

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

References

maclane1978-categories riehl2016-category-theory