202506271206

Tags : Category Theory

Adjunctions

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Definition

An adjunction is a pair of Functors that have a special relation to one another: A pair of functors $F:C\leftrightarrows D:G$ form an adjunction if there is, for every $c:C$ and $d:D$ there is a natural isomorphism

$$D(Fc,d)\cong C(c,Gd)$$

which is natural in both variables. Here way say that $F$ is the left-adjoint of $G$ and $G$ is the right-adjoint of $F$.

Also deriving from what seems to be music notation, we match up the functions across the natural isomorphism as follows:

$$Fc\stackrel{f^{♯}}{\to }d\leftrightsquigarrow c\stackrel{f^{♭}}{\to }Gd$$

When categories $C$ and $D$ are locally isomorphic, $D(Fc,d)$ and $C(c,Gd)$ are both hom-sets and hence the natural bijection can be written as the natural isomorphism between the following functors:

which also makes it easier to make sense of naturality in the 2 variables, naturality is $D$ can be depicted by the commutative diagram:

and commutativity in $C$ can be depicted by the diagram:

Note

An Equivalent definition is given in Unit and Counit as Universal Arrows.

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References

• Functors
• Natural Transformation
• Examples of Adjunctions
• Unit and Counit as Universal Arrows
• Equivalence between definitions of Adjunction