CATEGORIES-Definition of a category
المؤلف:
STEVE AWODEY
المصدر:
Category Theory
الجزء والصفحة:
4-5
4-2-2021
819
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.
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