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Given a [[Category (mathematics)|category]] ''C'' and a [[morphism]] | |||
<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]]: | |||
#There exists a morphism <math>g\colon X\to I</math> such that ''f'' = ''hg''. | |||
#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''. | |||
Note the following: | |||
# ''g'' is unique. | |||
# ''m'' is monic. | |||
# ''h''=''lm'' already implies that ''m'' is unique. | |||
# ''k''=''mg'' | |||
[[Image:Image diagram category theory.svg]] | |||
The image of ''f'' is often denoted by im ''f'' or Im(''f''). | |||
One can show that a morphism ''f'' is [[Monomorphism|monic]] if and only if ''f'' = im ''f''. | |||
==Examples== | |||
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. | |||
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: | |||
:im ''f'' = ker coker ''f'' | |||
This holds especially in [[Abelian category|abelian categories]]. | |||
==See also== | |||
*[[Subobject]] | |||
*[[Coimage]] | |||
*[[Image (mathematics)]] | |||
==References== | |||
*Section I.10 of {{Mitchell TOC}} | |||
{{DEFAULTSORT:Image (Category Theory)}} | |||
[[Category:Category theory]] | |||
Revision as of 03:29, 30 January 2014
Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property:
- There exists a morphism such that f = hg.
- For any object Z with a morphism and a monomorphism such that f = lk, there exists a unique morphism such that h = lm.
Note the following:
- g is unique.
- m is monic.
- h=lm already implies that m is unique.
- k=mg
The image of f is often denoted by im f or Im(f).
One can show that a morphism f is monic if and only if f = im f.
Examples
In the category of sets the image of a morphism is the inclusion from the ordinary image to . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.
In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism can be expressed as follows:
- im f = ker coker f
This holds especially in abelian categories.
See also
References
- Section I.10 of Template:Mitchell TOC