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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

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Table of Contents

    1. Reading Motivation
    2. Basics of a Fiber Product
    3. Basics of the Pullback
  1. How To Cite
  2. References:
  3. Discussion:

Reading Motivation

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:

XxZY:={(x,w,y)f(x)=w=g(y)} X{x}_{Z}Y := \{(x, w, y) | f(x) = w = g(y)\}

Which has two projection functions:

π1:XxZYX \pi_{1}: X {x}_{Z}Y \rightarrow X
π2:XxZYY \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 XX and YY over ZZ is any set WW for which we have an isomorphism W -approx-> X {x}_{Z} Y. In this case, WW is the pullback.

How To Cite

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

References:

Discussion:

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