Lagmental Vicfred

Yoneda Identifies an Object through All Maps into It by Vicfred

Last updated: Thu 31 July 2025

tags: mathematics, category theory, functors, adjunctions

Natural transformations from a representable functor to F are determined by one element of F at the representing object. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

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.

$$ h^A=\operatorname{Hom}_{\mathcal C}(A,-) $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \operatorname{Nat}(h^A,F)\cong F(A) $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{aligned}\eta&\longmapsto\eta_A(1_A),\\x\in F(A)&\longmapsto\bigl(\eta_X(f)=F(f)(x)\bigr).\end{aligned} $$

Conceptual compression

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad h^A=\operatorname{Hom}_{\mathcal C}(A,-),\\[5pt] \mathsf{C}\;&:\quad \operatorname{Nat}(h^A,F)\cong F(A). \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Nat}(h^A,F)\cong F(A) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Wed 19 July 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.