Algebraic Geometry Series 1: An introduction to Category Theory
This is intended to be a series of articles, that fleshes out the bare skeleton of Algebraic Geometry. I will be closely following Ravi Vakil’s treatment of the topic, supplemented with other treatments.
A category consists of a “collection” of objects, and for each pair of objects, a set of morphisms (or arrows) between them.
Note that the objects being considered need not be distinct. In the category of sets, for instance, the identity morphism is also a legal morphism. The set of morphisms between objects and is referred to as Mor. Morphisms compose as expected: if Mor and Mor, then Mor. Composition is associative: . Also, for each object Obj, there exists an identity morphism id. If Mor, then id. Also, if Mor, then id.
Fibered products: These can be best studied with a diagram. Suppose we have and , then the fibered product is an object such that it is universal for any object mapping to both and . For example, let us consider the category comprising of all ideals of , and a morphism exists if . Then, if and , we can see both lie inside . Hence, we can choose . Now every object that maps to both and needs to be generated by a multiple of . Hence, will be generated by , or .
Functors: A covariant functor is like a homomorphism between categories and . A contravariant functor is slightly different. It maps Mor to Mor. For example, the identity functor is covariant, while the functor mapping topological open sets ( implies ) to differentiable functions on them is a contravariant functor.
I would like to discuss the contravariant functor mentioned above. Let us consider the category of open sets in , such that morphisms exist between two objects only when the domain is a subset (not necessarily proper) of the range. For example, there exists a morphism between and . Now consider a differentiable function on , like . Clearly, it is also differentiable on every subset of , including . Hence, there is a mapping from the set of differentiable functions on to those on (functions map to themselves). However, this mapping may not be surjective. For example, is differentiable on but not on .