the cedar ledge

# Fiber Products (Pullbacks)

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

# Bibliography

Not Available

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

### Basics of a Fiber Product

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:

$\pi_{1}: X {x}_{Z}Y \rightarrow X$
$\pi_{2}: X {x}_{Z}Y \rightarrow Y$

How I would understand that, is by saying that

### Basics of the Pullback

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.

## How To Cite

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

## Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: January 17, 2023. Website built with Franklin.jl and the Julia programming language.