PlanetMath: inverse image of a morphism

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inverse image of a morphism

(Definition)

Let $f:A\to B$ be a morphism in a category $\mathcal{C}$ . Let $\operatorname{im}(f)$ be the image of and $i:\operatorname{im}(f)\to B$ be a representing monomorphism. The inverse image of is the pullback of $f:A\to B$ and $i: \operatorname{im}(f) \to B$ :

$\displaystyle \xymatrix@+=3pc{ {C}\ar[r] \ar[d] &{A}\ar[d]^{f} \ {\operatorname{im}(f)}\ar[r]^i &{B} }$

is sometimes denoted by $f^{-1}(B)$ . Since the diagram is a pullback and

is monomoprhic, the inverse image $f^{-1}(B)$ is a subobject of

(see this entry for more detail.)

For example, in $% latex2html id marker 318 $ \textbf{Set}$$ , the category of sets, the inverse image, in the sense above, of a morphism $f:A\to B$ is just the inverse image of as a function: clearly,

$\displaystyle f^{-1}(B)=\lbrace a\in A\mid f(a)\in B\rbrace$

is a set (a subset of

). Let $j:f^{-1}(B)\to A$ be the canonical inclusion, and $\overline{f}: f^{-1}(B)\to \operatorname{im}(f)$ be the induced function by restricting the domain of

to $f^{-1}(B)$ and the range to $\operatorname{im}(f)$ . The diagram above is clearly commutative. Suppose there is a set

and two functions $g:S\to A$ and $h:S\to \operatorname{im}(f)$ such that $f\circ g= i\circ h$ . Define $k:S\to f^{-1}(B)$ by

. This is a well-defined function, since $f(g(s))=i(h(s))=h(s)\in B$ , or $g(s)\in f^{-1}(B)$ . Furthermore,

, and $\overline(f)(k(s))=f(k(s))=f(g(s))=i(h(s))=h(s)$ . Finally, it is easy to see that

is unique.

Remark. The inverse coimage of a morphism is dually defined.

Bibliography

1: C. Faith Algebra: Rings, Modules, and Categories I, Springer-Verlag, New York (1973)

"inverse image of a morphism" is owned by CWoo.

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Also defines:

inverse image, inverse coimage

This object's parent.

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Cross-references: easy to see, well-defined, commutative, range, domain, induced, inclusion, canonical, subset, function, category of sets, subobject, diagram, pullback, monomorphism, image, category, morphism
There are 9 references to this entry.

This is version 6 of inverse image of a morphism, born on 2008-09-02, modified 2008-09-22.
Object id is 10977, canonical name is InverseImageOfAMorphism.
Accessed 299 times total.

Classification:

AMS MSC:

18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

Pending Errata and Addenda

None.

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