CATEGORIES-Definition of a category

Definition 1.1. A category consists of the following data:

• Objects: A, B, C, . . .

• Arrows: f, g, h, . . .

• For each arrow f, there are given objects

dom(f), cod(f)

called the domain and codomain of f. We write

f : A → B

to indicate that A = dom(f) and B = cod(f). •

Given arrows f : A → B and g : B → C, that is, with

cod(f) = dom(g)

there is given an arrow

g ◦ f : A → C

called the composite of f and g.

• For each object A, there is given an arrow

1A : A → A

called the identity arrow of A.

These data are required to satisfy the following laws:

• Associativity:

h ◦ (g ◦ f)=(h ◦ g) ◦ f

for all f : A → B, g : B → C, h : C → D.

• Unit:

f ◦ 1A = f = 1B ◦ f for all f : A → B.

A category is anything that satisfies this definition—and we will have plenty of examples very soon. For now I want to emphasize that, unlike in Section 1.2, the objects do not have to be sets and the arrows need not be functions. In this sense, a category is an abstract algebra of functions, or “arrows” (sometimes also called “morphisms”), with the composition operation “◦” as primitive. If you are familiar with groups, you may think of a category as a sort of generalized group.