Factorization system

In mathematics, it can be shown that every 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.

Definition

A factorization system (''E'', ''M'') for a category C consists of two classes of morphisms ''E'' and ''M'' of C such that:
#''E'' and ''M'' both contain all isomorphisms 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:

''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'' 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 limits and colimits,

* $(C \cap W, F)$ is a weak factorization system,

* $(C, F \cap W)$ is a weak factorization system, and

* $W$ satisfies the two-out-of-three property: if $f$ and $g$ are composable morphisms and two of $f,g,g\circ f$ are in $W$, then so is the third.

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

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External links

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

Category theory