Initial object: Difference between revisions
(New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C}</math> is a category. An '''initial object''' in <math>\mathcal{C}</math> is an object <math>A \in \operatorname{Ob}\mathcal{C}</mat...) |
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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. | 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=== | |||
* The empty set is an initial object in the [[category of sets]]. | |||
* A one-point set is an initial object in the [[category of pointed sets]]. | |||
* The trivial group is an initial object in the [[category of groups]]. | |||
* The ring of integers is an initial object in the [[category of unital rings]]. | |||
* The empty topological space is an initial object in the [[category of topological spaces]]. | |||
===Examples of categories without initial objects=== | |||
==Related notions== | ==Related notions== | ||
Revision as of 12:24, 25 December 2008
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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
- The empty set is an initial object in the category of sets.
- A one-point set is an initial object in the category of pointed sets.
- The trivial group is an initial object in the category of groups.
- The ring of integers is an initial object in the category of unital rings.
- The empty topological space is an initial object in the category of topological spaces.