Why is the category of fields seemingly so poorly behaved?

Question

Compared to the categories of other “common” algebraic objects like groups and rings, it seems that fields as a whole are missing some important properties:

• There are no initial or terminal objects
• There are no free fields
• No products or coproducts
• Every arrow is a mono (maybe not a bad thing, but still indicates how restrictive the category is)

A logician once told me in passing that part of the reason is that the properties for fields contain a decidedly “weird” property, namely that every element in a field except zero has a multiplicative inverse. If I understood him correctly, this property is sufficiently different from the others that the category of all such objects loses some features. But I have no idea if this was a heuristic or a proven theorem.

Answer 1

There is a precise sense in which the concept of field is not algebraic like, say, the concept of ring or group or vector space etc.: it is a theorem that any kind of mathematical structure that is defined as having a set of elements and some fixed list of total operations of constant finite arity obeying some fixed list of unconditional equations gives rise to a category with certain nice properties (which I omit for the moment). The usual definition of field has a partially defined operation – inversion – as well as an inequality ($0 \ne 1$), which means the theorem is not applicable; the fact that the category of fields does not have the nice properties of algebraic categories tells us there is actually no way of defining fields so that the theorem applies.

So what does being algebraic buy us, and how do we recognise an algebraic category without thinking about the logical form of the definition? Well, a category is equivalent to a category of algebraic structures if and only if it has all of the following properties:

• It has limits for all small diagrams and colimits for small filtered diagrams.
• There is an object $A$ such that the functor $\mathrm{Hom} (A, -)$ has a left adjoint, is monadic, and preserves colimits for small filtered diagrams.

In fact, it follows that such a category has colimits for small diagrams in general, but this fact is not needed in the theorem. Note that the object $A$ is not unique up to isomorphism; this is essentially the phenomenon of Morita equivalence.