Among all of the standard algebraic structures that a student typically encounters in an introduction to abstract algebra (groups, rings, fields, modules), commutative rings are somehow special: the opposite category behaves like a category of spaces, so much so that an entire field of mathematics is dedicated to doing geometry based on it.
In general, suppose you find yourself in some category. What sort of behavior could you look for that might qualify as “behaving like a category of spaces”?
One thing to look for is distributivity. Recall that a distributive category is a category with finite products and finite coproducts such that finite products distribute over finite coproducts; more explicitly, the natural maps
should be isomorphisms, and also the natural maps should be isomorphisms, where denotes the initial object. (Curiously, distributive categories are themselves like categorified versions of commutative rings.)
This is a pretty good test. The following familiar categories are distributive:
- More generally, any bicartesian closed category, and in particular any topos
These are all reasonable candidates for categories of “spaces.” On the other hand, the following familiar categories are not distributive:
- More generally, any nontrivial category with a zero object, and in particular any abelian category
You might object that there is also an entire field of mathematics dedicated to treating groups as geometric objects. I contend that the geometric object a group describes is actually a groupoid, and is distributive!