Lagmental Vicfred

An Adjunction Is a Natural Hom-Set Bijection by Vicfred

tags: mathematics, category theory, functors, adjunctions

A left adjoint and right adjoint translate maps in opposite categories without choosing coordinates. I want the notation, the mechanism, and the failure mode visible at the same time.

Statement

Category theory records objects through their morphisms. A functor \(F:\mathcal C\to\mathcal D\) preserves identities and composition, while a natural transformation \(\eta:F\Rightarrow G\) compares functors uniformly.

$$ F:\mathcal C\rightleftarrows\mathcal D:G $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \operatorname{Hom}_{\mathcal D}(F X,Y)\cong\operatorname{Hom}_{\mathcal C}(X,GY) $$

Worked algebra

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \mathbf{Set}(X\times A,Y)\cong\mathbf{Set}\!\left(X,Y^A\right),\qquad f(x,a)\longleftrightarrow\bigl(x\mapsto(a\mapsto f(x,a))\bigr) $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad F:\mathcal C\rightleftarrows\mathcal D:G,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Hom}_{\mathcal D}(F X,Y)\cong\operatorname{Hom}_{\mathcal C}(X,GY). \end{aligned} $$

Caveat

An isomorphism of objects is stronger than a natural bijection of underlying sets unless that bijection respects all morphisms in the relevant category.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Hom}_{\mathcal D}(F X,Y)\cong\operatorname{Hom}_{\mathcal C}(X,GY) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Wed 28 February 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.