Definition:Object (Category Theory)

Definition

Let $\mathbf C$ be a metacategory.

An object of $\mathbf C$ is an object which is considered to be atomic from a category theoretic perspective.

Notation

Objects in a general metacategory are usually denoted with capital letters like $A,B,C,X,Y,Z$.

The collection of objects of $\mathbf C$ is denoted $\mathbf C_0$.

Motivation

An object is a conceptual device introduced mainly to make the discussion of morphisms more convenient.

That objects do not play an important role in category theory is apparent from the fact that the notion of a metacategory can be described while avoiding to mention objects altogether.

Nonetheless the notion of object is one of the two basic concepts of metacategories and as such of category theory.

Also see

• Definition:Morphism (Category Theory)
• Definition:Metacategory

• Results about objects in the context of category theory can be found here.

Sources

• 1971: Saunders Mac Lane: Categories for the Working Mathematician: Chapter I: Categories, Functors, and Natural Transformations
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): object
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): category: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): object
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): category: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): object