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.