**Date:** October 30 2022

**Summary:** An overview on fiber products (aka pullbacks) and their features within category theory

**Keywords:** ##summary #fiber #product #pullback #category #theory #archive

Not Available

Pullbacks are a central part to category theory so naturally, I would like to know more about them!

Suppose we have the diagram of sets and functions:

X -f-> Z <-g- Y

Its fiber product is defined as:

$X{x}_{Z}Y := \{(x, w, y) | f(x) = w = g(y)\}$Which has two projection functions:

How I would understand that, is by saying that

Suppose we have the diagram of sets and functions:

W -pi*{1}-> X -f-> Z <-g- Y <-pi*{2}- W

The pullback of $X$ and $Y$ over $Z$ is any set $W$ for which we have an isomorphism W -approx-> X {x}_{Z} Y. In this case, $W$ is the pullback.

Zelko, Jacob. *Fiber Products (Pullbacks)*. https://jacobzelko.com/10312022005339-fiber-product-pullback. October 30 2022.

CC BY-SA 4.0 Jacob Zelko. Last modified: January 17, 2023.
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