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''in a category''
 
''in a category''
  
A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648001.png" /> of a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648002.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648003.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648004.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648005.png" />) implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648006.png" /> (in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648007.png" /> can be cancelled on the right). An equivalent definition of a monomorphism is: For any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648008.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m0648009.png" /> the mapping of sets induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480010.png" />,
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A morphism $\mu : A \to B$ of a [[Category|category]] $\mathfrak{K}$ for which $\alpha \, \mu = \beta \, \mu$ ($\alpha$, $\beta$ from $\mathfrak{K}$) implies that $\alpha = \beta$ (in other words, $\mu$ can be cancelled on the right). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$,
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$$
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\operatorname{Hom} \left({X, A}\right) \to \operatorname{Hom} \left({X, B}\right),
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480011.png" /></td> </tr></table>
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must be injective. The product of two monomorphisms is a monomorphism. Each left divisor of a monomorphism is a monomorphism. The class of all objects and all monomorphisms of an arbitrary category $\mathfrak{K}$ forms a subcategory of $\mathfrak{K}$ (usually denoted by $\operatorname{Mon} \mathfrak{K}$).
 
 
must be injective. The product of two monomorphisms is a monomorphism. Each left divisor of a monomorphism is a monomorphism. The class of all objects and all monomorphisms of an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480012.png" /> forms a subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480013.png" /> (usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480014.png" />).
 
  
 
In the category of sets (cf. [[Sets, category of|Sets, category of]]) monomorphisms are the injections (cf. [[Injection|Injection]]). Dual to the notion of a monomorphism is that of an [[Epimorphism|epimorphism]].
 
In the category of sets (cf. [[Sets, category of|Sets, category of]]) monomorphisms are the injections (cf. [[Injection|Injection]]). Dual to the notion of a monomorphism is that of an [[Epimorphism|epimorphism]].
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====Comments====
 
====Comments====
In the first definition above, composition of morphisms is written in  "diagram order"  (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064800/m06480015.png" /> means  "a followed by m" ). If, as is frequently done, the opposite convention is employed, then monomorphisms are morphisms which can be cancelled on the left.
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In the first definition above, composition of morphisms is written in  "diagram order"  (that is, $\alpha \, \mu$ means  "a followed by m" ). If, as is frequently done, the opposite convention is employed, then monomorphisms are morphisms which can be cancelled on the left.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>
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Latest revision as of 05:35, 12 January 2017

in a category

A morphism $\mu : A \to B$ of a category $\mathfrak{K}$ for which $\alpha \, \mu = \beta \, \mu$ ($\alpha$, $\beta$ from $\mathfrak{K}$) implies that $\alpha = \beta$ (in other words, $\mu$ can be cancelled on the right). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$, $$ \operatorname{Hom} \left({X, A}\right) \to \operatorname{Hom} \left({X, B}\right), $$

must be injective. The product of two monomorphisms is a monomorphism. Each left divisor of a monomorphism is a monomorphism. The class of all objects and all monomorphisms of an arbitrary category $\mathfrak{K}$ forms a subcategory of $\mathfrak{K}$ (usually denoted by $\operatorname{Mon} \mathfrak{K}$).

In the category of sets (cf. Sets, category of) monomorphisms are the injections (cf. Injection). Dual to the notion of a monomorphism is that of an epimorphism.

References

[1] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
[2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)


Comments

In the first definition above, composition of morphisms is written in "diagram order" (that is, $\alpha \, \mu$ means "a followed by m" ). If, as is frequently done, the opposite convention is employed, then monomorphisms are morphisms which can be cancelled on the left.

References

[a1] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Monomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monomorphism&oldid=15107
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article