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General Definition
Given a category
Contents
- There exists a morphism
e : X → I such thatf = m e . - For any object
I ′ e ′ : X → I ′ m ′ : I ′ → Y such thatf = m ′ e ′ v : I → I ′ m = m ′ v .
Remarks:
- such a factorization does not necessarily exist
-
e is unique by definition ofm monic -
v is monic. -
m = m ′ v already implies that m is unique.
The image of
Proposition: If
Second definition
In a category
Remarks:
- Finite bicompleteness of the category ensures that pushouts and equalizers exist.
-
( I m , m ) can be called regular image asm is a regular monomorphism, i.e. the equalizer of a pair of morphism. (Recall also that an equalizer is automatically a monomorphism). - In an abelian category, the cokernel pair property can be written
i 1 f = i 2 f ⇔ ( i 1 − i 2 ) f = 0 = 0 f and the equalizer conditioni 1 m = i 2 m ⇔ ( i 1 − i 2 ) m = 0 m . Moreover, all monomorphisms are regular.
Examples
In the category of sets the image of a morphism
In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism
In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.