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.
[Math] Infinite Field Theory and Category Theory
ct.category-theorygr.group-theoryreference-request
Related Solutions
The class of simple groups isn't elementary. To see this, first note that if it were, then an ultraproduct of simple groups would be simple. But an ultraproduct of the finite alternating groups is clearly not simple. (An $n$-cycle cannot be expressed as a product of less than $n/3$ conjugates of $(1 2 3)$ and so an ultraproduct of $n$-cycyles doesn't lie in the normal closure of the ultraproduct of $(1 2 3)$. )
It turns out that an ultraproduct $\prod_{\mathcal{U}} Alt(n)$ has a unique maximal proper normal subgroup and the corresponding quotient $G$ is an uncountable simple group. This group $G$ has the property that a countable group $H$ is sofic if and only if $H$ embeds into $G$. For this reason, $G$ is said to be a universal sofic group.
As for your third question, Shelah has constructed a group $G$ of cardinality $\omega_{1}$ which has no uncountable proper subgroups. Clearly $Z(G)$ is countable. Consider $H = G/Z(G)$. Then $H$ also has no uncountable proper subgroups.Furthermore, every nontrivial conjugacy class of $H$ is uncountable and it follows that $H$ is simple.
I hesitate to let this out, but there's always this cute little note that I learned from another MO answer (I don't know which one): https://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf. Maybe this will satisfy your curiosity, but I maintain that it takes a warped mind to identify such a categorical formulation of integration as the "right" way to think about integrals.
The advantage of categorical thinking in my view is that it helps to organize computations and arguments involving several different kinds of structures at the same time. For instance, (co)homology is all about capturing useful invariants associated to a complicated structure (e.g. a geometric object) in a much simpler structure (e.g. an abelian group). When we want to determine how the invariants behave under certain operations on the complicated structure (e.g. products, (co)limits) it helps to have a theory already set up to tell us what will happen to the simpler structure. That's where category theory comes into its own, and instances of this paradigm are so ubiquitous in algebra and topology that category theory has taken on a life of its own. It seems that people working in those areas have found it convenient to build categorical constructions into the foundations of their work in order to emphasize generality (one can treat algebraic varieties and solutions to diophantine equations on virtually the same footing), keep track of different notions of equivalence (e.g. homotopy versus homeomorphism), build new kinds of spaces (e.g. groupoids), and to achieve many other aims.
In many kinds of analysis, this kind of abstraction isn't necessary because there's often only one structure to keep track of: $\mathbb{R}$. When you think about it, analysis is only possible because we are willing to seriously overburden $\mathbb{R}$. Take, for example, the expression "$\frac{d}{dt}\int_X f_t(x) d\mu(x)$" and consider all of the different ways real numbers are being used. It is used as a geometric object (odds are X is built out of some construction involving the real numbers or a subspace thereof), a way to give $X$ additional structure (it wouldn't hurt to guess that $\mu$ is a real valued measure), a parameter ($t$), and a reference system ($f$ probably takes values in $\mathbb{R}$ or something related to it). In algebraic geometry, one would probably take each of these roles seriously and understand what kind of structure they are meant to bring to the problem. But part of the power and flexibility of analysis is that we can sweep these considerations under the rug and ultimately reduce most complications to considerations involving the real numbers.
All that being said, the tools of category theory and homological algebra actually have started to make their way into analysis. Because of the fact that analysts generally consider problems tied to certain very specific kinds of structure, they have historically focused on providing the sharpest and most detailed solutions to their problems rather than extracting the crude, qualitative invariants for which cohomological thinking is most appropriate. However, as analysts have become more and more attuned to the deep relationships between functional analysis and geometry, they have turned to ideas from category theory to help keep things organized. K-theory and K-homology have become indispensable tools in operator theory; there is even a bivariant functor $KK(-,-) $ from the category of C-algebras to the category of abelian groups relating the two constructions, and many deep theorems can be subsumed in the assertion that there is a category whose objects are C-algebras and whose morphism spaces are given by $KK(A,B)$. Cyclic homology and cohomology has also become extremely relevant to the interface between analysis and topology.
So ultimately I think it all comes down to what kinds of subtleties are most relevant in a given problem. There is just something fundamentally different about the kind of thinking required to estimate the propagation speed of the solution operator for a nonlinear PDE compared to the kind of thinking required to relate the fixed point theory in characteristic 0 of a linear group acting on a variety to that in characteristic p.
Best Answer
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.