Patrick Reany
18 October 2023
Yesterday, I had a sudden realization that the way in which some folk are characterizing how to define
a group using category theory was strange (i.e., confusing) to me. Now, I'm still in the beginning process
of learning category theory myself, so my inability to appreciate the way things are done in category
theory could be because of my own lack of knowledge in the subject. So, to get an unbiased opinion on
the matter, I thought to consult ChatGPT, which has shown itself in the past to be openminded and quick
on the uptake. You can judge the matter for yourself.
I thought to begin the discussion by describing how to define a monoid in category theory. A monoid is a
group minus inverses for every element (so it has an identity element, composition of morphism is the
closed binary 'product', and associativity comes by definition of how categorical morphisms are defined).
Later on, I added to the discussion inverse morphisms, which is added in by merely requiring that every
morphism be an isomorphism (because they have inverses).
Note: After the close of the discussion, I inserted some commments (placed in square brackets) into
the discussion to clarify matters.
Patrick to ChatGPT:
True or false: The morphisms under composition of a category with only one object have the structure of a
monoid (of abstract algebra)?
ChatGPT:
True. In a category with only one object, the morphisms (also called arrows) can be thought of
as endomorphisms of that single object. The composition of morphisms in a category satisfies the
associativity property, which is a requirement for a structure to be considered a monoid.
To be more specific:
There is only one object, let's call it A.
For each morphism f: A -> A (an endomorphism), you can compose it with another endomorphism g: A -> A
to get a new endomorphism (g ◌ f): A -> A.
The identity morphism, denoted as id_A: A -> A, serves as the identity element in this "monoid"
structure. For any endomorphism f: A -> A, we have id_A ◌ f = f and f ◌ id_A = f.
The composition operation is associative, meaning that for endomorphisms f, g, and h: A -> A, we
have (h ◌ g) ◌ f = h ◌ (g ◌ f).
So, in a category with only one object, the set of endomorphisms equipped with composition forms
a monoid.
Patrick:
Thus the morphisms under composition of a category with only one object have the structure of a
group if those morphisms are isomorphism, right?
ChatGPT:
No, the morphisms in a category with only one object, even if they are isomorphisms, do not
necessarily form a group. In a category with a single object, the morphisms are endomorphisms of
that object. The structure of a group requires the existence of inverses for every element, but this
is not necessarily the case for endomorphisms.
In a category with only one object, if all morphisms are isomorphisms, it means that each morphism
has an inverse. However, this does not guarantee the structure of a group, because group axioms also
require associativity.
Associativity is not guaranteed in a category with a single object. In fact, the composition operation
in a category is always assumed to be associative, but in this specific case, the category's only object
has no morphisms going from it to itself other than the identity morphism. Thus, the associativity of the
composition operation is trivial.
So, in summary:
In a category with one object, if all morphisms are isomorphisms, they do have inverses, but it's
not sufficient to conclude that they form a group because the associativity condition for a group is
trivial in this context. They form a monoid.
To have a group structure, you would need to consider a category with multiple objects, and the
morphisms between objects should satisfy the group axioms, including associativity and the existence
of inverses.
[ChatGPT went far off base with that last reply, but was able to correct itself with a little
help from my next prompt.]
Pattrick:
Well, wouldn't all morphisms (arrows) being isomorphisms guarantee inverses for each arrow?
ChatGPT:
Yes, you are correct. If all the morphisms (arrows) in a category are isomorphisms, it does
indeed guarantee the existence of inverses for each arrow. In a category, an isomorphism is
defined as a morphism that has an inverse, so if all morphisms are isomorphisms, then each
arrow has an inverse. In this case, you do have a group structure with respect to composition,
given that the composition operation is also associative.
So, in a category with only one object, where all morphisms are isomorphisms, you do have a
group structure with respect to composition. I apologize for any confusion in my previous
response. Thank you for clarifying this point.
Patrick:
Some people say that a 'group is a category with one object and the morphisms are inverses'
[i.e., isomorphims]. Am I being too nitpicky to say that the groupness of this structure is not
in the category per se, but, rather, lies in the set of its morphisms under composition? Or, are
those claims really the same thing?
ChatGPT:
Your observation is correct, and the distinction you're making is important. While it is
often said that a group can be thought of as a category with one object where the morphisms
are inverses, it's more accurate to say that the group structure arises from the set of morphisms
(group elements) under composition.
The statement that a group is a category with one object and inverses is a way to provide an
intuitive analogy between these two mathematical structures. It highlights the structural
similarities between groups and categories with one object where all morphisms are isomorphisms.
However, it's important to recognize that the group structure resides in the set of elements
(morphisms) and the binary operation (composition), not in the category itself.
In a group, you have a set of elements (group elements) and an associative binary operation
(group composition) that satisfies the group axioms, which include the existence of inverses.
In a category with one object where all morphisms are isomorphisms, you do indeed have a collection
of morphisms, each with an inverse, and a composition operation. This collection, along with the
composition operation, forms a monoid (a mathematical structure similar to a group but without the
requirement of inverses). The group structure is realized within this collection when you consider
the inverses of the isomorphisms, and this structure is a subset of the category's morphisms
under composition.
So, your observation is not nitpicky but rather a precise way to distinguish between the category and
the group. The category provides the framework, and the group structure arises from the properties
of the morphisms and their composition.
I think it's fascinating that ChatGPT seems to understand how to correctly characterize human
behavior as intuitive, when it itself presumably is incapable of using intuition --- at least in the
way that humans do.
I think that ChatGPT says it well that the category provides the "framework" on which the group
structure resides. That's an elegant way of expressing it.
It seems to me that the central concept under investigation in category theory is the associative
composable morphism, together with all that that implies. For example, the reason natural
transformation are 'natural' is because they must be morphisms in the image category. As such,
they are ready-made to compose with the morphisms already in the category, and this is the reason
the diagram of image objects under a functor-pair commute --- they have to by the nature of
composable morphisms.