Algebraic Geometry Series 1: An introduction to Category Theory

by ayushkhaitan3437

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 \mathcal{C} 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 \iota:\{1\}\to\{1\} is also a legal morphism. The set of morphisms between objects A and B is referred to as Mor(A,B). Morphisms compose as expected: if f\in Mor(A,B) and g\in Mor(B,C), then g\circ f\in Mor(A,C). Composition is associative: f\circ (g\circ h)=(f\circ g)\circ h. Also, for each object A\in Obj(\mathcal{C}), there exists an identity morphism id_A:A\to A. If f\in Mor(A,B), then f\circ id_A=f. Also, if g\in Mor(B,A), then id_A\circ g=g.

Fibered products: These can be best studied with a diagram. Suppose we have \alpha: X\to Z and \beta:Y\to Z, then the fibered product is an object X\times_Z Y such that it is universal for any object A mapping to both X and Y. For example, let us consider the category comprising of all ideals of \Bbb{Z}, and a morphism f:A\to B exists if A\subseteq B. Then, if X=(4) and Y=(6), we can see both lie inside (2). Hence, we can choose Z=(2). Now every object that maps to both (4) and (6) needs to be generated by a multiple of lcm(4,6). Hence, X\times_Z Y will be generated by lcm(4,6), or 12.

Functors: A covariant functor F is like a homomorphism between categories A and B. A contravariant functor G is slightly different. It maps f\in Mor(A,B) to G(f)\in Mor(G(B),G(A)). For example, the identity functor is covariant, while the functor mapping topological open sets (\exists f:A\to B implies A\subseteq B) 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 \Bbb{R}, 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 (0,1) and (0,2). Now consider a differentiable function on (0,2), like x^2. Clearly, it is also differentiable on every subset of (0,2), including (0,1). Hence, there is a mapping from the set of differentiable functions on (0,2) to those on (0,1) (functions map to themselves). However, this mapping may not be surjective. For example, \frac{1}{x-1} is differentiable on (0,1) but not on (0,2).