Subobject classifier: Difference between revisions

Revision as of 06:47, 13 October 2006

Revision as of 06:56, 13 October 2006

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== Introductory example ==

As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset ''j'':''U'' of→ ''X'' we can assign the function ''gχ_j'' from ''X'' to Ω that maps precisely the elements of ''U'' to 1 (see [[indicator function|characteristic function]]). Every function from ''X'' to Ω arises in this fashion from precisely one subset ''U''.

== Definition ==

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with the following property:

:for each [[monomorphism]] ''j'': ''U'' → ''X'' there is a unique morphism ''gχ_j'': ''X'' -> Ω such that the following [[commutative diagram]]

''U'' -> 1

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<i>

1

|

v

''g&chi;_j'': ''X'' -> &Omega;

</i>

The morphism ''g&chi;_j'' is then called the '''classifying morphism''' for the subobject represented by ''j''.

== Further examples ==