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Algebraic Geometry Lecture 17 – Categories and stuff
Joe Grant^1
Motivating example: Sets and functions.
Def n. A category C is a collection of objects, ob(C), and morphisms/maps/arrows, mor(C), which each have a source and a target in ob(C). If f ∈ mor(C) and the source of f is s(f ) = X, and its target is t(f ) = Y , then we write f : X → Y.
- For each c ∈ ob(C), there is a unique distinguished morphism idc.
- We have composition: if f : X → Y and g : Y → Z then there is a composite g ◦ f : X → Z, subject to: - Associativity: for each f, g, h ∈ mor(C), we have (f ◦ g) ◦ h = f ◦ (g ◦ h) whenever this is defined. - Identity: f ◦ idX = f and idY ◦f = f.
Examples.
Category Objects Morphisms Set Sets Functions Grp Groups Group homomorphisms Ab Abelian groups Group homomorphisms Top Topological spaces Continuous maps Toph Topological spaces Continuous maps up to homotopy
The above examples all have sets as their objects. This is “nice” because their objects have elements, e.g. x ∈ X ∈ ob(Set). Such categories are called concrete.
Other examples.
A poset (S, 6 ) is a set S with a partial ordering. We can describe this as a category C. Let ob(C) = S and non-identity morphisms be:
for x, y ∈ S there is a unique morphism f : x → y if and only if x 6 y.
Composition follows from the transitive law for posets: if x 6 y and y 6 z then x 6 z.
In sets we like knowing when a function is injective or surjective. We say a morphism m : X → Y is monic (like an injection) if for every f : W → X and f ′^ : W → X we have m ◦ f = m ◦ f ′^ ⇒ f = f ′.
We call a morphism e : X → Y epi (surjective) if for every f : Y → Z and f ′^ : Y → Z we have f ◦ e = f ′^ ◦ e ⇒ f = f ′.
If a morphism f is both monic and epi then we call f and isomorphism. (^1) Notes typed by Lee Butler based on a lecture given by Joe Grant. Any errors are the respon- sibility of the typist. Or the US sub-prime mortgage crisis. 1
2
Remarks.
Let G be a group. Define a category C such that ob(C) = {∗} and mor(C) = G. So groups are one-object categories with invertible morphisms.
In the good old days ( 6 1950) we would write a function with domain X and codomain Y as f (X) ⊂ Y. The notation f : X → Y comes from category theory.
Functors.
Def n. A functor F : C → D for categories C and D is a function that takes ob(C) to ob(D) and mor(C) to mor(D), such that if f : X → Y then F (f ) : F (X) → F (Y ), and:
- F (idc) = idF (c) for all c ∈ ob(C).
- If g, f ∈ mor(C) then F (g ◦ f ) = F (g) ◦ F (f ) in D.
There are a lot of functors.
E.g. 1. The identity functor F : C → C.
E.g. 2. P : Set → Set, the power set functor. Let X ∈ ob(Set), then
P(X) = {Y | Y ⊆ X}.
Let f : X → Y be a function. To define P we need a function F such that F (f ) : P(X) → P(Y ). Let X′^ ∈ P(X), i.e. X′^ ⊆ X. Then define
F (f )(X′) = f (X′) ⊆ Y.
So F (f )(X′) ∈ P(Y ).
E.g. 3. Let G, H be two groups and CG, CH be the categories associated to them. Then a functor F : CG → CH is a homomorphism. There is only one object in each of ob(CG) and ob(CH ) so there are no worries there. On morphisms,
F (fg ◦ fg′^ ) = F (fg ) ◦ F (fg′^ )
is the same as
ϕ(gg′) = ϕ(g)ϕ(g′).
Natural transformations.
Let F : C → D and G : C → D be two functors. We want to define a natural transformation “ F ⇒ G ” or “ F →˙G ”.
We define the natural transformation, η. We want a map F (c) 7 → G(c) for each c ∈ ob(C). This map must be “nice”. Define the map F (c) → G(c) by ηc. So a natural transformation is, for each c ∈ ob(C), an assignment F (c) → G(c) denoted by ηc such that for all f : c → d in C the following diagram commutes:
