Is the modulo operation a forgetful functor from small infinite to small finite sets?

Question

Pretty self explanatory: My thinking is that the modulo operation is a universal and natural way to map any infinite small set down to a finite set, thus meriting the forgetful functor label.

I'm having a hard time finding reference to modulo operations in this context given the many uses of word modulo and the red herrings of modules and moduli space. Physics also has the concept of moduli fields whose "families of global minima." sound like what a modulo operation would do, yet I could find no reference of it within my search.

I would love any advice on how to approach this or better yet, some form of confirmation in the literature. It may well be that the modulo operation is hidden in every one of those spaces; that insight would be equally valuable to me.

Thanks.

Answer 1

A related concept from category theory is the coequalizer. It is a generalization of taking the quotient of a set by an equivalence relation, and I believe it might encompass what you are looking for in modular operations.

The integers modulo $n$ can be considered a quotient of the set of integers $\mathbb{Z}$ by the equivalence relation $R_n$="$a$ and $b$ are integers with the same remainder modulo $n$". If we consider $R_n$ to be a subset of $\mathbb{Z}\times \mathbb{Z}$, we can define the integers modulo $n$ as a coequalizer: let $f$ and $g$ be the first and second components of $R$, respectively, then the integers is just the coequalizer of $f$ and $g$.

Interestingly, and maybe closer to what you're looking for, the floor/ceiling modulo operators are adjoint functors: consider the set $\mathbb{Z}$ of integers and $\mathbb{R}$ of real numbers as poset categories (where the elements of the set are objects, and there's an arrow between $x\rightarrow y$ if and only if $x \leq y$).

Then there's an inclusion functor $i:\mathbb{N}\hookrightarrow \mathbb{R}$; its left adjoint is the ceiling functor, and its right adjoint is the floor functor. (!)

After all, $$\lceil x \rceil \leq n \iff x \leq i(n)$$ $$ i(n) \leq x \iff n \leq\lfloor x \rfloor$$

If you consider right adjoints to be almost a kind of forgetful functor, then the flooring function is one.

Answer 2

You can indeed make sense of division and remainders (and hence the "$a$ mod $b$" operation) for ordinals; just beware that they won't necessarily follow the rules you're used to.

For $\alpha,\beta$ ordinals, let $[{\alpha\over\beta}]$ denote the supremum of the set of ordinals $\gamma$ such that $\beta\gamma<\alpha$. EXERCISE: if this $\gamma$ is a limit ordinal, then $\beta\gamma=\alpha$.

Now we can let $\alpha$ mod $\beta$ be the unique ordinal $\theta$ such that $[{\alpha\over\beta}]+\theta=\alpha$. It's a good exercise to show that this $\theta$ in fact exists and is unique.

Caution: Remember that ordinal multiplication and addition are not commutative.

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As to a connection with forgetful functors, I think you're misusing terminology. First of all, a functor is a map between categories. So if you want to view "$-$ mod $\beta$" as a functor from $ON$ to $\beta$, we need to first figure out how $ON$ and $\beta$ are categories. Since mod is all about the additive structure, one reasonable guess is to view $ON$ and $\beta$ as monoids - that is, one-element categories, whose arrows correspond to ordinals (or ordinals $<\beta$) and where composition corresponds to addition. This works, but the mod maps aren't really forgetful in any sense that I'm aware of here. Usually "forgetful functor" is used to describe something like the obvious functors from:

• Abelian groups to Groups

that just take in an object, leave it unchanged, and view it as an element of a broader category; or things like

• Rings to Groups

• Groups to Sets

which take in an object, and forget some of the structure on that object (e.g. the ring multiplication or the group operation).

Now, "forgetful functor" is ultimately an informal term, and indeed there's a point of view according to which every functor is forgetful; but I still don't really see "mod" as an example of a forgetful functor in any nice way. I would argue that "quotienting out" (which is what mod does) is different from the kind of structural loss that forgetful functors are named for.