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
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.
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:
where denotes indexing from some index set, , using the index term, and represents some family of sets indexed by and is some element of the respective set. Then, based on readings from Category Theory texts, I have gathered that the following is equivalent notation to the formal set theory definition: . 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 given the following:
Where I am confused is the following:
What is the difference between and and 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?
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!
Zelko, Jacob. Clarifying Understanding of Coproducts. https://jacobzelko.com/10022022214249-understanding-coproducts. October 2 2022.