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| In [[mathematics]], it can be shown that every [[function (mathematics)|function]] can be written as the composite of a [[surjective]] function followed by an [[injective]] function. '''Factorization systems''' are a generalization of this situation in [[category theory]].
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| ==Definition==
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| A '''factorization system''' (''E'', ''M'') for a [[category (category theory)|category]] '''C''' consists of two classes of [[morphisms]] ''E'' and ''M'' of '''C''' such that:
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| #''E'' and ''M'' both contain all [[isomorphisms]] of '''C''' and are closed under composition.
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| #Every morphism ''f'' of '''C''' can be factored as <math>f=m\circ e</math> for some morphisms <math>e\in E</math> and <math>m\in M</math>.
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| #The factorization is ''functorial'': if <math>u</math> and <math>v</math> are two morphisms such that <math>vme=m'e'u</math> for some morphisms <math>e, e'\in E</math> and <math>m, m'\in M</math>, then there exists a unique morphism <math>w</math> making the following diagram commute:
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| [[Image:Factorization_system_functoriality.png|center]]
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| == Orthogonality ==
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| Two morphisms <math>e</math> and <math>m</math> are said to be ''orthogonal'', denoted <math>e\downarrow m</math>, if for every pair of morphisms <math>u</math> and <math>v</math> such that <math>ve=mu</math> there is a unique morphism <math>w</math> such that the diagram
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| [[Image:Factorization_system_orthogonality.png|center]]
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| commutes. This notion can be extended to define the orthogonals of sets of morphisms by
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| :<math>H^\uparrow=\{e\quad|\quad\forall h\in H, e\downarrow h\}</math> and <math>H^\downarrow=\{m\quad|\quad\forall h\in H, h\downarrow m\}.</math>
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| Since in a factorization system <math>E\cap M</math> contains all the isomorphisms, the condition (3) of the definition is equivalent to
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| :(3') <math>E\subset M^\uparrow</math> and <math>M\subset E^\downarrow.</math> | |
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| == Equivalent definition ==
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| The pair <math>(E,M)</math> of classes of morphisms of '''C''' is a factorization system if and only if it satisfies the following conditions:
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| #Every morphism ''f'' of '''C''' can be factored as <math>f=m\circ e</math> with <math>e\in E</math> and <math>m\in M.</math>
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| #<math>E=M^\uparrow</math> and <math>M=E^\downarrow.</math>
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| == Weak factorization systems == | |
| Suppose ''e'' and ''m'' are two morphisms in a category '''C'''. Then ''e'' has the ''left lifting property'' with respect to ''m'' (resp. ''m'' has the ''right lifting property'' with respect to ''e'') when for every pair of morphisms ''u'' and ''v'' such that ''ve''=''mu'' there is a morphism ''w'' such that the following diagram commutes. The difference with orthogonality is that ''w'' is not necessarily unique.
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| [[Image:Factorization_system_orthogonality.png|center]]
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| A '''weak factorization system''' (''E'', ''M'') for a category '''C''' consists of two classes of morphisms ''E'' and ''M'' of '''C''' such that :
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| #The class ''E'' is exactly the class of morphisms having the left lifting property wrt the morphisms of ''M''.
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| #The class ''M'' is exactly the class of morphisms having the right lifting property wrt the morphisms of ''E''.
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| #Every morphism ''f'' of '''C''' can be factored as <math>f=m\circ e</math> for some morphisms <math>e\in E</math> and <math>m\in M</math>.
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| == References ==
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| * {{cite journal
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| | author = [[Peter J. Freyd|Peter Freyd]], [[Max Kelly]]
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| | year = 1972
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| | title = Categories of Continuous Functors I
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| | journal = Journal of Pure and Applied Algebra
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| | volume = 2
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| }}
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| [[Category:Category theory]]
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