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strong monomorphism
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(Definition)
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Let
 be a category. A monomorphism  in
 is said to be a strong monomorphism if, whenever we are given the following commutative diagram
with  an epimorphism, then there is a morphism  such that the following is another commutative diagram:
Note that the “diagonal” morphism  is necessarily unique. In other words, a monomorphism is strong iff every epimorphism is orthogonal to it.
Dually, a strong epimorphism is an epimorphism which is orthogonal to every monomorphism in the category.
Remark. Every regular monomorphism is strong (see proof here), and every strong monomorphism is extremal.
Proof. Suppose  is a strong monomorphism and that
 with  epimorphic. Then we have the following commutative diagram
Since  is strong, there is a morphism  such that the diagram below is commutative
This shows that  is a split monomorphism, as
 . But  is epimorphic, we conclude that  is an isomorphism (this fact is proved here). 
- 1
- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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"strong monomorphism" is owned by CWoo.
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(view preamble)
Cross-references: isomorphism, split monomorphism, commutative, diagram, proof, regular monomorphism, orthogonal, iff, morphism, epimorphism, commutative diagram, monomorphism, category
There are 56 references to this entry.
This is version 11 of strong monomorphism, born on 2008-09-15, modified 2008-10-15.
Object id is 11038, canonical name is StrongMonomorphism.
Accessed 489 times total.
Classification:
| AMS MSC: | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\ {A}\ar[r]_{f}&{B} } $](http://images.planetmath.org:8080/cache/objects/11038/l2h/img4.png "$\displaystyle \xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\ {A}\ar[r]_{f}&{B} } $")
![$\displaystyle \xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]\vert{h} \ {A}\ar[r]_{f}&{B} } $](http://images.planetmath.org:8080/cache/objects/11038/l2h/img7.png "$\displaystyle \xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]\vert{h} \ {A}\ar[r]_{f}&{B} } $")
![$\displaystyle \xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\ {A}\ar[r]_{f}&{B} } $](http://images.planetmath.org:8080/cache/objects/11038/l2h/img13.png "$\displaystyle \xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\ {A}\ar[r]_{f}&{B} } $")
![$\displaystyle \xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]\vert{e} \ {A}\ar[r]_{f}&{B} } $](http://images.planetmath.org:8080/cache/objects/11038/l2h/img16.png "$\displaystyle \xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]\vert{e} \ {A}\ar[r]_{f}&{B} } $")