Initial object: Difference between revisions

From Cattheory
No edit summary
 
Line 18: Line 18:


===Examples of categories without initial objects===
===Examples of categories without initial objects===
* The [[category of fields]] has no initial object. Every field has a unique field embedding of its prime subfield; however, there are multiple isomorphism classes of prime fields.
* The [[category of nontrivial finite groups]] has no initial object.


==Related notions==
==Related notions==

Latest revision as of 12:30, 25 December 2008

This article is about a basic definition in category theory. The article text may, however, contain more material. Rate its utility as a basic definition article on the talk page
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in category theory

Definition

Suppose is a category. An initial object in is an object such that for any object , there is a unique morphism from to .

If an initial object exists, then any two initial objects are isomorphic, and there is a unique isomorphism between them. In particular, the automorphism group of any initial object is trivial.

Examples

Examples of initial objects

Examples of categories without initial objects

Related notions