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My question is somewhat similar to this previous question, but from a slightly different perspective. Is there any textbook on elementary number theory that develops the properties of $\mathbb{Z}$ as, say, the initial object in the category of commutative rings with identity? I am looking for something that presupposes a knowledge of category theory at the level of Categories for the Working Mathematician.

Edit: I had no idea that this question would provoke the storm of criticism that is has. My intention was not to imply that number theory is best learned from a categorical perspective, or that number theory should be subsumed by category theory. I was simply wondering what sort of interesting things one could say about $\mathbb{Z}$ from a category-theoretic perspective. So, I'll narrow the question: "Are there any good sources for learning about the properties of a natural numbers object in an arbitrary topos (possibly well-pointed and satisfying the axiom of choice)?"

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To be honest I am not sure this kind of perspective is really necessary for any elementary number theory. What sort of application of the perspective did you have in mind? –  Qiaochu Yuan Jun 19 '11 at 13:27
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Ugh. What is this world coming to? –  Marty Jun 19 '11 at 15:25
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@Marty: a less argumentative response would be great. –  Todd Trimble Jun 19 '11 at 17:13
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Oh, for goodness sake. If you want to learn elementary number theory, forget about categories completely and pick up something like Ireland/Rosen. This question is only slightly less absurd than asking for a category-theoretical introduction to convex optimization or bayesian statistics. –  Dan Petersen Jun 19 '11 at 17:58
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I notice Marty's comment is far and away the most popular one here, judging by votes. This puzzles me, because I find it very difficult to extract any message from the comment except an unexplained "You disgust me". It's the lack of explanation I object to. There are perspectives in and approaches to mathematics which I as well would want to express my strong distaste for, even vitriolically so. If Marty had deigned to clarify his comments in any way, that would be potentially useful. But as it stands, I can't make out quite what it is that he is objecting to or why; that deserves 19 upvotes? –  Sridhar Ramesh Jun 20 '11 at 2:04

3 Answers 3

up vote 22 down vote accepted

I have to admit that this is not really an answer, but rather some sort of meta-answer with some very general remarks which I hope do not bore everyone reading this; it just seems to me that this is necessary to indicate that it is rather misguided, as Yemon already says in the comments and I strongly agree with, to ask such a question if some book introduces elementary number by means of category theory.

Mathematics is all about the nontrivial, unexpected relationships. Category Theory is not really about finding such relationships, but rather about the correct setting, language and color some theory is developed. This point of view does not really contradict the hitherto development of category theory into a huge area of mathematics in its own right, full of nontrivial deep theorems; namely because often there is some geometric or whatever background which is our real motiviation. There are ubiquitous examples (model categories, topoi, stacks, $\infty$-categories, ...) which I don't want to elaborate here.

Anyway, as I said, mathematics really starts when something unexpected happens, which does not follow from general category theory. For example, the covariant functor $\hom(X,-)$ is always continuous, but when is it also cocontinuous, or respects at least filtered colimits? It turns out that this leads to a natural finiteness condition on $X$, namely we call $X$ then finitely presented. But finally to arrive at the question, $\mathbb{Z}$ is easily seen to be a inital object in the category of rings, but what theorems from category theory are known about initial objects? Well there is nothing to say, expect that every two initial objects are canonical isomorphic, which is just a trivial consequence of the definition. So $\hom(\mathbb{Z},-)$ is easy to describe, but what about the contravariant functor $\hom(-,\mathbb{Z})$? What happens when you plug in $\mathbb{Z}[x,y,z]/(x^n+y^n=z^n)$ for some fixed $n>2$? Does category theory help you to understand this? This example also shows that although the Yoneda-Lemma says that an object $X$ of a category is determined by its functor $\hom(X,-)$, it does not say you anything about the relationship of $X$ with other objects, for example when we just reverse the arrows. Instead, we have to use a specific incarnation of the category and its objects in order derive something which was not there just by abstract nonsense.

Perhaps related questions are more interesting: Which investigations in elementary number theory have led to some category theory (for example, via categorification), which was then applied to other categories as well, thus establishing nontrivial analogies? Or for the other direction, which general concepts become interesting in elementary number theory after some process of decategorification? But in any case, it should be understood that you have to digest elementary number theory before that ...

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My friend Owen once explained to me that you can view the integers as the initial object in the category of groups. I'm not sure if initial object is the right term, but I just mean that an element in a group gives rise to a map $k \mapsto g^k$. By playing with this property and forgetful functors, you can see that this object is naturally a ring. Most properties are immediate. But try to start with this asymmetric construction and prove that the multiplication is commutative. Is there a nice proof? –  Phil Isett Aug 10 '11 at 1:59
    
The trivial group is the initial object in the category of groups. The group of integers $(\mathbb{Z},+)$ has a different universal property, namely it represents the forgetful functor from (abelian) groups to sets. Or in other words, it is the free (abelian) group on one generator. This proof works with your map $k \mapsto g^k$. Now it seems to me that you want to endow the free group $\langle g \rangle$ on one generator $g$ with a ring structure and show that it is commutative, without using any explicit construction. –  Martin Brandenburg Aug 10 '11 at 8:17
    
First one has to prove that $G=\langle g \rangle$ is abelian. For this, remark that the centralizer of $g$ is a subgroup which contains $g$, thus it is $G$ (by the universal property applied to the inclusion). Thus for every $x$, the centralizer of $x$ is a subgroup which contains $g$, thus it is $G$ (see above). This means that $G$ is abelian. –  Martin Brandenburg Aug 10 '11 at 8:21
    
Now we would like to define a bilinear map $G \times G \to G$, or equivalently a homomorphism $G \to \hom(G,G)$. By the universal property this corresponds to one specific homomorphism $G \to G$, and again by the universal property this corresponds to a specific element of $G$, which we choose to be $g$. Thus the bilinear map is characterized by $(g,g)=g$. In order to show $(x,y)=(y,x)$ for all $x,y \in G$, we use the same trick as above: –  Martin Brandenburg Aug 10 '11 at 8:30
    
First observe that $\{x : (x,g)=(g,x)\}$ is a subgroup of $G$ which obviously contains $g$, thus it is $G$. Then for all $x$, we have that $\{y : (x,y)=(y,x)\}$ is a subgroup which contains $g$, thus it is $G$. This proves $(x,y)=(y,x)$. It remains to show associativity $(x,(y,z))=((x,y),z)$, this works exactly with the same trick, it may be formally reduced to the trivial equation $(g,(g,g))=((g,g),g)$. Also, it may be shown that $(g,x)=x$ for all $x$, i.e. $g$ is a unit. Thus we get a commutative ring. –  Martin Brandenburg Aug 10 '11 at 8:32

Here is one answer, although I am not sure it is along the lines of what Daniel Miller was after. For decades now, Lawvere and Schanuel have spearheaded a project on what they call "objective number theory", which you could consider a kind of structural approach to number theory, in that some aspects of the theory of numbers are re-interpreted by decategorifying analogous results for things like you say, the initial commutative ring satisfying such and such an identity seen as a decategorification of some category with appropriate properties, like an extensive category equipped with some isomorphism that expresses the identity.

Unfortunately, I don't know of a lot in print about this; I've seen Schanuel give talks on it once or twice. Here's one paper by Schanuel on this (which I haven't read yet myself).

A flavor of the type of thing this is about can be gleaned by looking at Andreas Blass's nice paper, Seven Trees in One, which is about a structural solution to the equation $x^7 = x$ in terms of binary trees. The starting observation is that the linear species of binary trees satisfies the data type equation $x = x^2 + 1$, and $x^7 = x$ can be derived as a formal consequence in the theory of commutative rigs; the operations used here can be interpreted structurally in terms of extensive categories with an object satisfying the appropriate identity.

Edit 1: Here is a youtube video to go along with "seven trees in one". There are a number of illustrations out there with more detail; here is one by Dan Piponi, who also contributes to MO under a pseudonym.

Edit 2: And I should definitely mention this very nice paper on Objective Number Theory by Tom Leinster and Marcel Fiore.

Edit 3: And finally (I hope this is my last edit), there is a beautiful article at the ncatlab by John Baez and James Dolan, on zeta functions from a "categorified" or species-theoretic point of view.

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I searched Math Reviews for books with Anywhere = categor* AND MSC Primary = 11 and then looked at the 25 matches and only one of them was an elementary Number Theory text, and the categories involved are not the kind wanted here ("Each section of the text concludes with about 30--40 problems which are divided up into three categories...."). I submit this as evidence that what OP asks for doesn't exist.

EDIT: In response to OP's edit, I searched MR for Anywhere = natural numbers object AND Anywhere = topos and got 92 matches. I don't know if any of them do what OP wants. I looked at the reviews of a few, the ones that actually referred to "natural numbers" in the title, and only saw one that had anything I wuld recognize as Number Theory.

Carol Szasz, Das Objekt "ganze Zahlen'' in einem elementaren Topos, Proceedings of the national conference on algebra (Iaşi, 1984), An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 31 (1985), suppl., 88–89, MR0858194 (88a:18006).

After defining an integral number object in terms of morphisms, the reviewer (Roswitha Harting) writes, "For an integral number object of $(Z,0,s)$ the author then defines in the obvious way morphisms $a$ (addition) and $m$ (multiplication), and obtains that $(Z,a,0)$ is an abelian group."

But wait, I hear you say, that's integral number object, what about natural number object?

"Furthermore the author insinuates that there may exist topoi having an integral number object but no natural number object. For this it should be mentioned, that in the case of the integral number object being decidable, the existence of a natural number object follows."

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+1 for addressing the question itself (something I admit I was too lazy to attempt) –  Yemon Choi Jun 20 '11 at 0:29
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Well, the newest question of Daniel isn't very number-theoretic, not really. I think there could be a great deal to say about the newest question, but it would be better addressed under another question (preferably more sharply focused, after the OP has thought about it more). To Daniel Miller: I would start off with something like Mac Lane and Moerdijk's book on topos theory, and see what they say about natural number objects in a topos. I am skeptical about the value of people who are not category theorists saying too much more about this particular topic, here in this thread. –  Todd Trimble Jun 21 '11 at 0:22
    
Given the number of proofs of say, quadratic reciprocity, surely there should be a category theory proof of it, if the original question has a workable answer. The list rzuser.uni-heidelberg.de/~hb3/rchrono.html gives some possibles. A more intrepid challenge would be to prove something like Fermat's 2-square theorem on primes 1 mod 4 via only methods that "develop[s] the properties of $Z$ as, say, the initial object in the category of commutative rings with identity" –  Junkie Jun 21 '11 at 1:05
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@Junkie: At this stage of the discussion, I'm not sure it's a productive to fall back on the original formulations of the OP, as given, e.g., in your quotation. The applications of category theory to number theory tend to be much, much subtler, as perhaps you already know. (I tend to think of the OP's formulations as fairly groping in character to begin with, as I hope he won't mind my saying.) Meanwhile, there are of course proofs of quadratic reciprocity which have a kind of structural character (I am thinking of Zolotarev's) which might be amenable to "categorification" of some sort. –  Todd Trimble Jun 21 '11 at 1:40
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Quite incidentally, I'll also mention that I happen to be having these days quite a few conversations with a certain category theorist about class field theory, but again, the particular way in which category theory is entering these discussions is most unlikely to be guessed by the casual onlooker. So straw-man arguments against the possibility of category theory being applied to number theory are only useful perhaps as a warning to the young that their first ideas might not be the most fruitful. Otherwise, it's a mighty big world out there. –  Todd Trimble Jun 21 '11 at 1:49

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