Lagmental Vicfred

A Pullback Is a Fibre Product of Morphisms by Vicfred

Last updated: Sun 17 May 2026

tags: mathematics, category theory, functors, adjunctions

The pullback consists of pairs with the same image and is universal among all such compatible pairs. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Definitions first

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.

$$ X\times_ZY=\{(x,y):f(x)=g(y)\} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \operatorname{Hom}(T,X\times_ZY)\cong\operatorname{Hom}(T,X)\times_{\operatorname{Hom}(T,Z)}\operatorname{Hom}(T,Y) $$

A small case in full

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \begin{array}{ccc}X\times_ZY&\longrightarrow&Y\\\downarrow&&\downarrow{\scriptstyle g}\\X&\xrightarrow{\ f\ }&Z\end{array} $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad X\times_ZY=\{(x,y):f(x)=g(y)\},\\[5pt] \mathsf{C}\;&:\quad \operatorname{Hom}(T,X\times_ZY)\cong\operatorname{Hom}(T,X)\times_{\operatorname{Hom}(T,Z)}\operatorname{Hom}(T,Y). \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Hom}(T,X\times_ZY)\cong\operatorname{Hom}(T,X)\times_{\operatorname{Hom}(T,Z)}\operatorname{Hom}(T,Y) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Sat 23 March 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.