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Clarifying Understanding of Coproducts

Date: October 2 2022

Summary: A line of questioning to ensure I understand what coproducts and disjoint unions are

Keywords: #question #zulip #category #theory #coproducts #disjoint #union #set #archive


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Table of Contents

    1. Reading Motivation
    2. Question
  1. How To Cite
  2. References:
  3. Discussion:

Reading Motivation

This is a question to clarify my understand of coproducts as I have largely spent some time in understanding what they are and how to interpret them.


Hi everyone,

I was hoping I could get some clarification on my understanding of coproducts as I believe I have finally wrapped my head around them, but I want to be clear that I do. To understand coproducts, I started with the set theory definition of a coproduct:

iI{(x,i):xAi} \bigcup_{i \in I} \{(x, i): x \in A_{i}\}

where iIi \in I denotes indexing from some index set, II, using the index term, ii and AiA_{i} represents some family of sets indexed by ii and xx is some element of the respective AiA_{i} set. Then, based on readings from Category Theory texts, I have gathered that the following is equivalent notation to the formal set theory definition: iIAi\bigsqcup_{i \in I} A_{i}. As an example to make sure I am on the right track, here is a small simple problem and its solution according to my understanding:

Find iIAi\bigsqcup_{i \in I} A_{i} given the following:

>I={1,2,3}> > I = \{1, 2, 3\} >
>A={{1,2,3},{2,4,6}}> > A = \{\{1, 2, 3\}, \{2, 4, 6\}\} >

Solution: iIAi={(1,1),(2,1),(3,1),(2,2),(4,2),(6,2)}\bigsqcup_{i \in I} A_{i} = \{(1, 1), (2, 1), (3, 1), (2, 2), (4, 2), (6, 2)\}

Where I am confused is the following:

  1. What is the difference between iIAi\bigsqcup_{i \in I} A_{i} and iIAi\coprod_{i \in I} A_{i} and X1X2X_{1} \coprod X_{2} syntax notation (the latter notation concerns two specific sets)?

I have seen all notations when referring to coproducts when viewed from both set theory and category theory perspectives. Is it a matter of aesthetic or is there a functional/semantic difference occurring?

  1. I have read that coproducts are also known as disjoint unions.

I read about disjoint unions from a set theory perspective and also a category theory perspective but cannot really discern a functional (i.e. practical) difference from coproducts. Are coproducts and disjoint unions really effectively the same idea?

Thanks all and let me know if I can clarify anywhere!

How To Cite

Zelko, Jacob. Clarifying Understanding of Coproducts. October 2 2022.



CC BY-SA 4.0 Jacob Zelko. Last modified: May 19, 2024. Website built with Franklin.jl and the Julia programming language.