Please correct me, mathematicians.
I see a lot of people saying that a category with a single object is a monoid.
This seems wrong to me.
Let’s take a look at a category with a single object. It’s got an object we’ll call *. Then, it’s got a bunch of morphisms that each map * to *. It’s also got an operation called “composition of morphisms.”
When one morphism from this category, which is a thing that maps * to *, takes part in said operation with another morphism from this category, which is also a thing that maps * to *, you end up with a morphism, which is also a thing that maps * to *.
Now, to me, each of those three things is the same thing. They do the same thing (map * to *), so they’re the same thing.
Just for the sake of argument, call the first morphism partaking in the operation “e” and call the second morphism partaking in the operation “A.” And, because the second and third morphisms taking part in the operation are the same thing to me, you can call the third morphism on the right side of the equation “A” too.
So, you have: e . A = A. Hence, the first thing is acting as an identity. But the second thing is acting as an identity too, by the same reasoning. In fact, I see all of the morphisms in this particular category as identities.
So, how can such a category be a *monoid*, which definitionally has one identity element only?
PS: Why do people say that a monoid is a category with one object? A monoid is a set and a binary operation, while a category is a set (of morphisms), another set (of objects) and a binary operation. So, isn’t a category a definitionally bigger structure than a monoid? Shouldn’t you be saying that a category with one object can be seen as a monoid with an extra set containing a single object? Or are y’all just ignoring the objects because they’re worthless or something?
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> I see a lot of people saying that a category with a single object is a monoid.
Yes, count me in, too.
> This seems wrong to me.
> Now, to me, each of those three things is the same thing. They do the same thing (map * to *), so they’re the same thing.
There are two misconceptions I see here:
(a) You say that they do the same thing and hence are the same thing. It is correct that they “do” the same, in the sense that they map from * to *. But two maps can do vastly different things. Take for example functions ℝ → ℝ, there are many different ones such as x→x , x→1, x→x+1, x→x³ and so on. They all map from ℝ to ℝ but clearly are not the same.
(b) Arrows/morphisms in a category are often not actual _maps_. Instead, they are abstract formal things that just exist and do their own thing (that being: compose). You can decide(!) that the morphisms from * to * are _by definition_ the integers ℤ, and composition is addition; or multiplication; or whatever you want as long as it satisfies two rules (identity and associativity).
However, the (in)famous _Yoneda Lemma_ among other things tells you that morphisms _can_ be _realized_ (this is actually a word with a precise meaning) as certain sets and maps between them. But the single object * would not correspond to the empty set or a set with only one element, but a larger one: namely the monoid it defines, as the underlying set after ignoring the composition. In particular, there are many self-maps on that set.
> A monoid is a set and a binary operation, while a category is a set (of morphisms), another set (of objects) and a binary operation. So, isn’t a category a definitionally bigger structure than a monoid? Shouldn’t you be saying that a category with one object can be seen as a monoid with an extra set containing a single object?
The statement is not that they are exactly the same, but they encode the very same information: a binary operation which has an identity and is associative. One calls it multiplication or an operation, the other composition while including the (useless) extra datum of an object.
The extra object does not add anything, it is really just there on a notational level. It compares to an extra bracket that is unnecessary: “(2·3)+1” and “2·3+1” _mean exactly the same_, but they are _not literally the same_ as strings of symbols. This, but on a more abstract level with monoids.
On a categorical level, one would say they are _isomorphic_ or _equivalent_. Or more precisely and stronger (warning, serious category lingo ahead!), the category of monoids is _equivalent_ to C1, where C1 is the _full_ (no morphisms missing) subcategory of categories with only one object inside the category of all categories.
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