# Infinite Field Theory and Category Theory

I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were very difficult to define directly had simple and elegant categorical definitions. For example, the direct definition of the free group is rather long and arduous, whereas the categorical definition, i.e. any function $S\to G$, where $G$ is a group factors through a homomorphism from the free group generated by $S$ to $G$, is quite simple. However, for the most part, it seems to me that categorical methods are most easily used on infinite groups, and in particular, infinite abelian groups. Despite this limitation, categorical methods seemed so natural that I couldn't help but wonder if they can be applied to field theory with similar results. So my question is: (1) is it beneficial to study infinite field theory in the generality that category theory necessitates, and (2) are there any good books that use this approach.

-
I think the answer to your question is 'no'. The category of fields is not very nice, especially compared to many other algebraic categories. Both products and coproducts do not exist. There is also no left adjoint to the forgetful functor, meaning the analog of the free group construction mentioned does not exist for fields. Furthermore, it seems to me that most of the definitions and constructions one comes across in field theory have some real content, and cannot simply be placed in an abstract categorical framework. –  Mike Skirvin Sep 17 '10 at 1:46
Ricky, one problem is that your construction doesn't satisfy any kind of universal property. –  Mike Skirvin Sep 17 '10 at 2:39
The class of all groups forms a variety (see en.wikipedia.org/wiki/Variety_%28universal_algebra%29); in general, varieties are very well behaved, and many of the familiar construction have nice categorical descriptions (products, coproducts, free objects, generators-and-relations, etc). The class of all fields (or even the class of all fields of a given characteristic) do not form a variety (fields are not defined equationally), and so in general the category lacks many of the nice properties that varietal categories have. –  Arturo Magidin Sep 17 '10 at 4:22
One thing that might be worth pointing out is that the categorical definition of a free group doesn't allow you to avoid the classical construction of them -- you still have to prove that free groups exists! Moreover, it is almost impossible to prove any substantial theorems about free group directly from the universal property (for instance, I'd be shocked to see a "categorical" proof that subgroups of free groups are free groups). It's important to know the categorical definition since it tells you the "function" of free groups within the ecosystem of groups, but you can't stop there... –  Andy Putman Sep 17 '10 at 4:27
I strongly agree with Andy's main point, but: Subgroups of free groups are free because of (1) the classification of covering spaces of X in terms of subgroups of $\pi_1(X)$, (2) the fact that every free group is $\pi_1$ of a graph, (3) the fact that every covering space of a graph is a graph and (4) the fact that $\pi_1$ of every graph is free. (2) and (4) in turn depend on the van Kampen Theorem, which can be proved without the explicit construction of free groups. –  Tom Goodwillie Sep 17 '10 at 11:46

I think Mike Skirvin's comment above should be expanded into an answer. There are no homomorphisms at all between fields of different characteristic. Hence one has to look at the category of fields of a fixed characteristic $p$.
An elementary fact about fields is that they have no nontrivial ideals.
It follows that all homomorphisms between fields are 1-1. This implies that there are no free fields of any characteristic $p$ (except for the free field of char $p$ over the empty set of generators).

Finally, as Mike Skirvin pointed out in his comment, there are in general no products of fields, even of a fixed characteristic. I think this sufficiently explains why categorial constructions are not very useful in field theory.

-

Category theory can be useful for certain aspects of (infinite) field theory, but really requires that you do not restrict just to fields. I am meaning modern approaches to Galois theory. That is not really field theory as such but intersects very nicely with that area. To see how category theory interacts with field theory look at the book by Borceux and Janelidze: Galois theories , volume 72 of Cambridge Studies in Advanced Mathematics , Cambridge University Press.

That goes from a fairly classical viewpoint to a categorical one, but the classical one is, of course, looked at from a categorical viewpoint.

-

The category of fields is a full subcategory of the category of rings. The latter has nice categorical properties, but the subcategory does not inherit these properties. Of course, you may apply every theorem of category theory to the category of fields, but the problem is, that homomorphisms of fields are rather rigid. If $F$ is a field, it is very rare that there is a nice description for the Hom-functor $Hom(F,-)$. For example, if $F=\mathbb{Q}(t)$ is a function field, then $Hom(F,L)$ is empty if $char(L)>0$, and otherwise can be identified with the elements of $L$, which are transcendental over $\mathbb{Q}$. Anyway, in the theory of algebraic extensions, it is good to know that $Hom_K(K(\alpha),L)$ consists of the roots of $\alpha$ in $L$ of the minimal polynomial of $\alpha$ over $K$. But this already takes place in the category of rings.

Let $K$ be a field. It is a useful fact that the category of field extensions of $K$ is directed in the sense that for every pair of extensions $L/K, L'/K$ there is an extension $M/K$ together with $K$-homomorphisms $L \to M, L' \to M$. Namely, $L \otimes_K L'$ is a nontrivial ring, so we may mod out some maximal ideal to get such a field. The same works for infinitely many extensions and can be used to show the existence of an algebraic closure of $K$. Note that the algebraic closure does not have a honest universal property, but it's construction reminds of other universal objects. The fact that the extensions over $K$ is directed also implies that for schemes $X,Y$ over $S$, the natural map $|X \times_S Y| \to |X| \times_{|S|} |Y|$ is surjective (where $|X|$ is the underlying topological space of $X$).

I suspect that there are applications of category theory to the category of fields, as the others already have pointed out. However, sometimes this category can be useful: Let $X$ be a scheme. Then the colimit of the $X(K)$, where $K$ runs through the category of fields, is just the underlying set of $X$. This is useful when you try to recover a scheme just by a given universal property.

-