Zero object: Difference between revisions
(New page: ==Definition== An object in a category is termed a '''zero object''' if it is both an defining ingredient::initial object and a defining ingredient::final object.) |
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An object in a category is termed a '''zero object''' if it is both an [[defining ingredient::initial object]] and a [[defining ingredient::final object]]. | An object in a category is termed a '''zero object''' if it is both an [[defining ingredient::initial object]] and a [[defining ingredient::final object]]. | ||
==Examples== | |||
===Examples of categories with a zero object=== | |||
* The [[category of groups]] has a zero object: the trivial group. This is both an initial and a final object. | |||
* The [[category of pointed sets]] has a zero object: a one-point set. This is both an initial and a final object. | |||
===Examples of categories without a zero object=== | |||
* The [[category of sets]] has no zero object. This is because the initial object (the empty set) is not isomorphic to the final object (the one-point set). | |||
* The [[category of topological spaces]] has no zero object. This is because the initial object (the empty topological space) is not isomorphic to the final object (the one-point topological space). | |||
* The [[category of fields]] has neither an initial object nor a final object. Hence, it has no zero object. | |||
==Related notions== | |||
* [[Initial object]] | |||
* [[Final object]] | |||
Latest revision as of 12:28, 25 December 2008
Definition
An object in a category is termed a zero object if it is both an initial object and a final object.
Examples
Examples of categories with a zero object
- The category of groups has a zero object: the trivial group. This is both an initial and a final object.
- The category of pointed sets has a zero object: a one-point set. This is both an initial and a final object.
Examples of categories without a zero object
- The category of sets has no zero object. This is because the initial object (the empty set) is not isomorphic to the final object (the one-point set).
- The category of topological spaces has no zero object. This is because the initial object (the empty topological space) is not isomorphic to the final object (the one-point topological space).
- The category of fields has neither an initial object nor a final object. Hence, it has no zero object.