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Stream: learning: questions

Topic: clarifying understanding of coproducts and disjoint unions

Jacob Zelko (Oct 03 2022 at 01:39):

[Beginner Question]

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:

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

where $i \in I$ denotes indexing from some index set, $I$, using the index term, $i$ and $A_{i}$ represents some family of sets indexed by $i$ and $x$ is some element of the respective $A_{i}$ set. Then, based on readings from Category Theory texts, I have gathered that the following is equivalent notation to the formal set theory definition: $\bigsqcup_{i \in I} A_{i}$. 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 $\bigsqcup_{i \in I} A_{i}$ given the following:

$I = \{1, 2, 3\}$

$A = \{\{1, 2, 3\}, \{2, 4, 6\}\}$

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

Where I am confused is the following:

1. What is the difference between $\bigsqcup_{i \in I} A_{i}$ and $\coprod_{i \in I} A_{i}$ and $X_{1} \coprod X_{2}$ 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?

2. 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?

3. Anything from my current explanation that I sound unclear on or am wrong on?

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

Ralph Sarkis (Oct 03 2022 at 02:47):

In short, the disjoint union of a family of sets is exactly the same thing as its coproducts (up to isomorphism) and the difference in notation is irrelevant (you might have also seen the notations $A+B$ or $\sum_{i \in I} A_i$).

I am failing to see exactly what you are missing (if there is anything), so let me give some details and motivation.

The name 'disjoint union' probably comes from the fact when the $A_i$'s are pairwise disjoint ($A_i \cap A_j = \emptyset$ for all $i\neq j$), the disjoint union and union of the family are isomorphic: $\sqcup_{i \in I} A_i \cong \cup_{i \in I}A_i$. When the $A_i$'s are not disjoint, the disjoint union is taking their union as if they were disjoint using the trick of identifying $x \in A_i$ with $(x,i)$ (so that $(x,i) \neq (x,j)$ if $i\neq j$). This is one of the possible tricks that allows this construction, but there are other ways to "force" the disjointness and then take the union.

In category theory, we prefer not to refer to elements of sets, and we love to extract the essence of a definition to get rid of all unnecessary information $\ast$ . Thus, we came up $\ast\ast$ with the definition of coproducts, which informally says that the coproduct of a family of sets $\{A_i\}_{i \in I}$ is a set that contains copies of each set (via the coprojections $\kappa_i : A_i \to \amalg_{i \in I} A_i$) that are completely independent of each other (via the universal property: any way you want to map the $A_i$'s into $X$, you can combine the functions to map the coproduct into $X$).

This is an appropriate definition because the disjoint union of sets (along with the coprojections $A_i \to \sqcup_{i \in I}A_i = x \mapsto (x,i)$) is the coproduct of the family $\{A_i\}_{i \in I}$ (up to isomorphism). However, it is possible that you have a different opinion about what the essence of disjoint unions is. Then, you might want to try formalizing another categorical definition that captures that essence and verify it coincides with disjoint union.

$\ast$ An example of unnecessary information in this case is that the elements of the disjoint union are pairs of elements of $A = \cup_{i \in I}A_i$ and elements of $I$, that is, $\sqcup_{i \in I} A_i \subseteq A \times I$. This fact has nothing to do with our intuition of disjoint union (indeed, we usually don't bother to refer to elements of the disjoint union with the pairs except when doing very rigorous technical things), and the other "tricks" mentioned above might get rid of it, but inevitably bringing another piece of irrelevant information about the disjoint union.

$\ast\ast$ That is not really accurate, I am only trying to motivate things from this point of view.

John Baez (Oct 03 2022 at 08:28):

Jacob Zelko said:

1. What is the difference between $\bigsqcup_{i \in I} A_{i}$ and $\coprod_{i \in I} A_{i}$ and $X_{1} \coprod X_{2}$ 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?
   

There's no difference, people often use different notations for the same thing in math.

The notation $\bigsqcup$ probably comes from people who write the disjoint union of two sets as $X_1 \sqcup X_2$ so that it looks similar to the union $X_1 \cup X_2$; these people tend to write $\bigcup_{i \in I} X_{i}$ for the union of a bunch of sets and $\bigsqcup_{i \in I} X_{i}$ for their disjoint union. The notation $\coprod$ probably comes from people who noticed that the coproduct is dual to the product: they turned the product symbol $\prod$ upside down.

2. 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?

Probably the best thing to say is that coproducts in the category of sets, and various related categories like the category of topological spaces, are the same as disjoint unions.

Nobody I know calls coproducts 'disjoint unions' in an arbitrary category. For example coproducts in the category of vector spaces are called 'direct sums'. Everyone will understand you if you say 'direct sum of vector spaces', and category theorists will know this is their coproduct, but people would think you don't understand math if you say 'disjoint union of vector spaces'.

Mike Shulman (Oct 03 2022 at 15:37):

Amusingly, however, the coproduct of vector spaces is formally a [[disjoint coproduct]].

Jacob Zelko (Oct 18 2022 at 03:28):

Hi @Ralph Sarkis and @John Baez!

Apologies for taking so long to respond to your great responses! I actually have read the response about 3 or 4 times but was digesting the information.

Ralph, I appreciate your further explanation into the disjoint union and getting more into the intuition of the mechanism. Furthermore, I am glad that according to your understanding, I am not far off with my understandings of the disjoint union -- really breaking down the exact difference between disjoint unions and the coproduct definition was super clarifying.

John, thanks for the clarification on the notation and disentangling my confusion there! This is going directly into my notes! The semantics you introduced about "coproducts" in the category of sets and related categories wasn't something I have really thought much about. As I am primarily working within the category of Set, you made me realize I was, naively, taking this machinery for granted and it is something I need to be more cognizant of.

Jacob Zelko (Oct 18 2022 at 03:29):

Thank you and have a wonderful day!

John Baez (Oct 18 2022 at 17:32):

Great, I'm glad it helped!