Adjunction

Let $\mathcal{C}$ be a 2-category. For a prototypical example, one can consider $\mathcal{C}$ to be the category of categories.

Then an adjunction consists of:

• Two $\mathcal{C}$ objects $A$, $B$.
• Two 1-morphisms:
  • $f:A\to B$ - left adjoint
  • $g:B\to A$ - right adjoint
• Two 2-morphisms:
  • $\eta :1_A\Rightarrow g\circ f$ - adjunction unit
  • $\epsilon :f\circ g\Rightarrow 1_B$ - adjunction counit Such that:
• The following triangles identities commute in their hom-categories:¹

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
g \arrow[r, "\eta\cdot g"] \arrow[dr, "id"] & g\circ f\circ g \arrow[d, "g\cdot\epsilon"] \\
& g 
\end{tikzcd}
\end{document}

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
f \arrow[r, "f\cdot\eta"] \arrow[dr, "id"] & f\circ g\circ f \arrow[d, "\epsilon\cdot f"] \\
& f
\end{tikzcd}
\end{document}

• The zig-zag laws should both hold, composing $\eta$ and $\epsilon$ to get the identity 2-morphism:

\usepackage{tikz-cd}
\begin{document}
% https://q.uiver.app/#q=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
\begin{tikzcd}
A && A && A && A \\ \\ B && B && B && B 
\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["\epsilon", shift left=4, Rightarrow, from=2, to=5] 
\arrow["\epsilon", shift left=4, Rightarrow, from=3, to=4]
\end{tikzcd}
\end{document}

Remarks

Loosely speaking, a left adjoint (eg. free) gets you back to where you started, while a right adjoint (eg. forgetful) preserves structure.

Examples of Adjunctions

Free-Forgetful Adjunction

Product-Slice Adjunction

Coproduct-Diagonal Adjunction

Diagonal-Product Adjunction

Footnotes

1. The first in $\mathcal{C}[B,A]$, and the second in $\mathcal{C}[A,B]$. Here $\cdot$ denotes whiskering. ↩