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| Given a [[Category (mathematics)|category]] ''C'' and a [[morphism]]
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| <math>f\colon X\to Y</math> in ''C'', the '''image''' of ''f'' is a [[monomorphism]] <math>h\colon I\to Y</math> satisfying the following [[universal property]]:
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| #There exists a morphism <math>g\colon X\to I</math> such that ''f'' = ''hg''.
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| #For any object Z with a morphism <math>k\colon X\to Z</math> and a monomorphism <math>l\colon Z\to Y</math> such that ''f'' = ''lk'', there exists a unique morphism <math>m\colon I\to Z</math> such that ''h'' = ''lm''.
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| Note the following:
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| # ''g'' is unique.
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| # ''m'' is monic.
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| # ''h''=''lm'' already implies that ''m'' is unique.
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| # ''k''=''mg''
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| [[Image:Image diagram category theory.svg]]
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| The image of ''f'' is often denoted by im ''f'' or Im(''f'').
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| One can show that a morphism ''f'' is [[Monomorphism|monic]] if and only if ''f'' = im ''f''.
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| ==Examples==
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| In the [[category of sets]] the image of a morphism <math>f\colon X \to Y</math> is the inclusion from the ordinary [[image (mathematics)|image]] <math>\{f(x) ~|~ x \in X\}</math> to <math>Y</math>. In many [[Concrete category|concrete categories]] such as [[Category of groups|groups]], [[Category of abelian groups|abelian groups]] and (left- or right) [[Module (mathematics)|modules]], the image of a morphism is the image of the correspondent morphism in the category of sets.
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| In any [[normal category]] with a [[zero object]] and [[Kernel (category theory)|kernels]] and [[Cokernel (category theory)|cokernels]] for every morphism, the image of a morphism <math>f</math> can be expressed as follows:
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| :im ''f'' = ker coker ''f'' | |
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| This holds especially in [[Abelian category|abelian categories]].
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| ==See also==
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| *[[Subobject]]
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| *[[Coimage]]
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| *[[Image (mathematics)]]
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| ==References==
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| *Section I.10 of {{Mitchell TOC}}
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| {{DEFAULTSORT:Image (Category Theory)}}
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| [[Category:Category theory]]
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Revision as of 01:45, 25 February 2014
They call me Ned when i think promoting it . quite good when you say this. For years I've been working as a production and planning expert. New Mexico wherever he's lived for as well as his parents live area. As a girl what I enjoy is playing dominoes although i don't keep time lately. Go to my how do people find out more: http://www.43things.com/entries/view/6513722
Look at my web blog: resinas