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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, it can be shown that every
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
can be written as the composite of a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
function followed by an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
function. Factorization systems are a generalization of this situation in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.


Definition

A factorization system (''E'', ''M'') for a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
C consists of two classes of
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
''E'' and ''M'' of C such that: #''E'' and ''M'' both contain all
isomorphisms In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of C and are closed under composition. #Every morphism ''f'' of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. #The factorization is ''functorial'': if u and v are two morphisms such that vme=m'e'u for some morphisms e, e'\in E and m, m'\in M, then there exists a unique morphism w making the following diagram
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
: ''Remark:'' (u,v) is a morphism from me to m'e' in the arrow category.


Orthogonality

Two morphisms e and m are said to be ''orthogonal'', denoted e\downarrow m, if for every pair of morphisms u and v such that ve=mu there is a unique morphism w such that the diagram commutes. This notion can be extended to define the orthogonals of sets of morphisms by :H^\uparrow=\ and H^\downarrow=\. Since in a factorization system E\cap M contains all the isomorphisms, the condition (3) of the definition is equivalent to :(3') E\subseteq M^\uparrow and M\subseteq E^\downarrow. ''Proof:'' In the previous diagram (3), take m:= id ,\ e' := id (identity on the appropriate object) and m' := m .


Equivalent definition

The pair (E,M) of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions: #Every morphism ''f'' of C can be factored as f=m\circ e with e\in E and m\in M. #E=M^\uparrow and M=E^\downarrow.


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'' (respectively ''m'' has the ''
right lifting property In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given c ...
'' 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. A weak factorization system (''E'', ''M'') for a category C consists of two classes of morphisms ''E'' and ''M'' of C such that: #The class ''E'' is exactly the class of morphisms having the left lifting property with respect to each morphism in ''M''. #The class ''M'' is exactly the class of morphisms having the right lifting property with respect to each morphism in ''E''. #Every morphism ''f'' of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences ''W'', fibrations ''F'' and cofibrations ''C'' so that * C has all
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
s and colimits, * (C \cap W, F) is a weak factorization system, and * (C, F \cap W) is a weak factorization system. A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to F\cap W, and it is called a trivial cofibration if it belongs to C\cap W. An object X is called fibrant if the morphism X\rightarrow 1 to the terminal object is a fibration, and it is called cofibrant if the morphism 0\rightarrow X from the initial object is a cofibration.Valery Isaev - On fibrant objects in model categories.


References

* *


External links

* {{Citation, author=Riehl, first=Emily, year=2008, url=http://www.math.jhu.edu/~eriehl/factorization.pdf, title= Factorization Systems Category theory