Abstract algebra/Category theory

Category theory is the study of categories, which are collections of objects and morphisms (or arrows), or from one object to another.

Definitions & Notations

A category consists of a class G of objects and for every pair of objects A, B in G a class Hom(A, B) of things called morphisms or arrows from A to B. This may be thought of as a directed graph where G are the points and Hom(A, B) are the directed lines between them.

For every three objects A, B and C in G there is a map o called composition:

o : Hom(A, B) × Hom(B, C) → Hom(A, C)

We write f o g for the composition of f and g

Composition is associative, i.e. for and A, B, C, D in G, for any f in Hom(A, B), g in Hom(B, C), h in Hom(C, D),

(f o g) o h = f o (g o h)

For every object A in the category there is a special map i_A in Hom(A, A) which we call the identity of A. This has the properties:

for any object B in G, any g in Hom(A, B), i_A o g = g

for any object B in G, any h in Hom(B, A), h o i_A= h

[Note if j_A is another identity for A, the axioms imply that

j_A = j_A o i_A= i_A,

so the identity for each object is unique.]

Some examples of categories

• $𝐒𝐞𝐭$, the category whose objects are sets, and whose morphisms are maps between the sets.

• The category whose objects are open subsets of ${ℝ}^n$ and whose morphisms are continuous (differentiable, smooth) maps between them.

• The category whose objects are smooth (differentiable,topological) manifolds, and morphisms are smooth (differentiable,continuous) maps.

• $k-𝐕𝐞𝐜𝐭$, the category whose objects are vector spaces over a field $k$ (for example, the real or complex numbers), with morphisms linear maps.

• $𝐆𝐫𝐨𝐮𝐩$, the category of groups, and homomorphisms between them.

In all the examples I have given thus far, the objects have been sets with the morphisms given by set maps between them. This is not always the case. There are some categories where this is not possible, and others where the category doesn't naturally appear in this way. For example:

• Let $𝒢$ be any category. Then its opposite category ${𝒢}^{op}$ is a category with the same objects, and all the arrows reversed. More formally, a morphism in ${𝒢}^{op}$ from an object $X$ to $Y$ is a morphism from $Y$ to $X$ in $𝒢$.

• Let $G$ be any group. Then we can define a category with a single object, with morphisms from that object to itself given by elements of $G$ with composition given by multiplication in $G$.