Patrick Reany
4 November 2023
I am currently spending a great deal of time to learn category theory. I find it very difficult, for many reasons, being often confused or bewildered. On this occasion, my confusion was about how there is a function (morphism) from the empty set to other sets. In set theory, a function is defined as the mapping of the elements of one set to another set. If the original set is empty, it has no elements to be mapped, so no function can be defined in this case. According to ChatGPT, this is correct in set theory, but the category of Set extends that definition for its own purposes. The point is that by use of the following chat I had with the chatbot, I was able to better understand how there can be a "function" from the empty set. (It's like defining 0! = 1 as a convenient extension of regular factorials.)
On 2 November 2023, I had this discussion with ChatGPT 3.5
Please explain to me how that, in the category of Set, it is possible to have a morphism from the empty set to some other set?
ChatGPT:
In category theory, a category consists of objects and morphisms (also called arrows) that connect these objects. The category of Set is a category in which objects are sets, and morphisms are functions between sets.
One key feature of category theory is that it allows for morphisms between any two objects, including the empty set. This means that there can be morphisms from the empty set (denoted as ∅) to other sets.
When you consider the category of Set, you have to define what these morphisms mean. In this context, a morphism from the empty set to another set can be thought of as a function, even though the empty set itself does not have any elements. Let's delve into this a bit more:
Empty Set (∅): The empty set is a set that contains no elements. In terms of category theory, it is an object in the category of Set.
Other Sets: Consider any other set, which we'll call "A," for instance.
Morphism (Arrow): Now, you can define a morphism (arrow) from the empty set (∅) to set A as a function, which takes elements from the empty set and maps them to elements in set A. However, since the empty set has no elements, there are no elements to map. So, this function is essentially empty, and it's called the unique empty function or the initial morphism.
In this way, there exists a morphism from the empty set to any other set in the category of Set. However, this morphism doesn't carry any information or elements, as there are no elements to map from the empty set.
This concept is a result of the way category theory is defined and allows for a broad and flexible framework to study various mathematical structures, including the category of Set, where the empty set is considered an object like any other set, and morphisms can be defined between any pair of objects.
Patrick:
Are you saying that in the category of Set, one simply constructs Set from the beginning so that the empty set has one function to every other set (and one to itself)? (That is, it's a design feature.) This procedure would, by fiat, construct the category of Set so that the empty set is an initial object of Set. Is that the motivation behind it? If so, what's so important about Set having an initial object?
ChatGPT:
Yes, in the category of Set, it is a design feature to consider the empty set (∅) as an initial object. This is a specific choice made in the construction of the category of Set and is not something that arises naturally from the definition of a category. It's an important design choice in category theory to ensure that Set forms a well-behaved category, and it has its advantages.
The motivation behind considering the empty set as an initial object in the category of Set is primarily to ensure that the category satisfies certain properties and to facilitate various aspects of category theory:
Completeness and Universal Mapping Properties: An initial object in a category serves as a kind of "starting point" or a universal object. It allows you to define and discuss concepts such as universal mapping properties and the existence of unique morphisms from the initial object to any other object in the category. For Set, this means there is a unique empty function to every other set, as you mentioned.
Structure and Generality: Designing Set with an initial object ensures that the category has a uniform structure, where all objects, including the empty set, have a consistent behavior with respect to morphisms.
Compatibility with Other Categories: Defining Set with an initial object allows it to be compatible with other categories in category theory, where initial objects have important roles in the study of limits, colimits, and various universal constructions.
Conceptual Clarity: Having the empty set as an initial object simplifies the definition and discussion of concepts like products, coproducts, and terminal objects in the category of Set.
So, while the choice of the empty set as an initial object in the category of Set is a design decision, it is motivated by the desire to ensure that the category behaves well and can be studied in a way that is consistent with other categories in category theory. It simplifies and generalizes the study of sets and functions, making it a useful choice for a category that plays a central role in many areas of mathematics and computer science.
Patrick:
I see your justification for it. It just occurs to me that if we are to include the empty set in Set (which we must!) then we are obliged by the rules of category construction to have a morphism from the empty set to itself, and thus there's no reason to stop there. Might as well add a morphism to every other set at the same time.
ChatGPT:
Your point is valid, and it's a matter of how categories are constructed. In the category of Set, you could indeed choose to include morphisms from the empty set to itself and to every other set in a consistent manner. This would be a legitimate construction, and it would certainly satisfy the rules of category theory.
Including such morphisms for the empty set is a way to maintain a consistent structure within the category and make it clear that the empty set is truly an object in the category with morphisms defined to and from it, even if those morphisms don't carry any meaningful information due to the empty set's lack of elements.
It's worth noting that different authors and mathematicians may make slightly different choices in how they formally define the category of Set or other categories, and these choices can be influenced by pedagogical preferences, the specific needs of the mathematical context, or the desire for simplicity and elegance in the formalism. The important thing is that the chosen construction should be internally consistent and allow for the study of sets and functions in a meaningful and coherent way.
Patrick:
Are you saying that some mathematicians do not include the empty set in the category of Set?
ChatGPT:
No, I'm not suggesting that mathematicians exclude the empty set from the category of Set. The empty set is universally included in the category of Set as an object, and it's a fundamental part of set theory. All mathematicians include the empty set in the category of Set; it's a standard practice in mathematics.
What I was discussing in my previous responses was the choice of how to define the morphisms involving the empty set within the category of Set. In most presentations of the category of Set, the empty set is included, and morphisms are defined from the empty set to other sets, even though these morphisms may not carry meaningful information due to the empty set's lack of elements.
The inclusion of the empty set as an object in Set is a foundational aspect of set theory, and it's a widely accepted and fundamental concept in mathematics. Different presentations may make slight variations in how morphisms involving the empty set are defined, but the empty set itself is always considered part of the category of Set.
Conclusion:
I don't always start off by inquiring with an LLM when I run into these frequent difficulties understanding technical concepts. But quite often, once I've exhausted the search engines and YouTube without success, I can get the answers I need from the chatbots (at the very least, reasonable answers). ChatGPT first and BingChat second.
When mathematicians introduce the category of Set and demonstrate that the empty set is the initial object of the set, they treat as unremarkable that one can have a "function" from the empty set to some other set. This is one of the frustrating aspects of the teaching of category theory, at least to me, which is the high number of things that should be explained but aren't, presumably in the interest of brevity.