factorization system

A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:

1. E and M both contain all isomorphisms of C and are closed under composition.
2. Every morphism f of C can be factored as $f=m\backslash circ\; e$ for some morphisms $e\backslash in\; E$ and $m\backslash in\; M$.
3. The factorization is functorial: if $u$ and $v$ are two morphisms such that $vme=m\text{'}e\text{'}u$ for some morphisms $e,\; e\text{'}\backslash in\; E$ and $m,\; m\text{'}\backslash in\; M$, then there exists a unique morphism $w$ making the following diagram commute:

center

Remark: $(u,v)$ is a morphism from $me$ to $m\text{'}e\text{'}$ in the arrow category.

Two morphisms $e$ and $m$ are said to be orthogonal, denoted $e\backslash downarrow\; m$, if for every pair of morphisms $u$ and $v$ such that $ve=mu$ there is a unique morphism $w$ such that the diagram

center

commutes. This notion can be extended to define the orthogonals of sets of morphisms by

:$H^\backslash uparrow=\backslash \{e\backslash quad|\backslash quad\backslash forall\; h\backslash in\; H,\; e\backslash downarrow\; h\backslash \}$ and $H^\backslash downarrow=\backslash \{m\backslash quad|\backslash quad\backslash forall\; h\backslash in\; H,\; h\backslash downarrow\; m\backslash \}.$

Since in a factorization system $E\backslash cap\; M$ contains all the isomorphisms, the condition (3) of the definition is equivalent to

:(3') $E\backslash subseteq\; M^\backslash uparrow$ and $M\backslash subseteq\; E^\backslash downarrow.$

Proof: In the previous diagram (3), take $m:=\; id\; ,\backslash \; e\text{'}\; :=\; id$ (identity on the appropriate object) and $m\text{'}\; :=\; m$.

The pair $(E,M)$ of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:

1. Every morphism f of C can be factored as $f=m\backslash circ\; e$ with $e\backslash in\; E$ and $m\backslash in\; M.$
2. $E=M^\backslash uparrow$ and $M=E^\backslash downarrow.$

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.

center

A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:{{harvtxt|Riehl|2014|loc=§11.2}}

1. The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M.
2. The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
3. Every morphism f of C can be factored as $f=m\backslash circ\; e$ for some morphisms $e\backslash in\; E$ and $m\backslash in\; M$.

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,

• $(C\; \backslash cap\; W,\; F)$ is a weak factorization system,

• $(C,\; F\; \backslash cap\; W)$ is a weak factorization system, and

• $W$ satisfies the two-out-of-three property: if $f$ and $g$ are composable morphisms and two of $f,g,g\backslash circ\; f$ are in $W$, then so is the third.{{harvtxt|Riehl|2014|loc=§11.3}}

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 $F\backslash cap\; W,$ and it is called a trivial cofibration if it belongs to $C\backslash cap\; W.$ An object $X$ is called fibrant if the morphism $X\backslash rightarrow\; 1$ to the terminal object is a fibration, and it is called cofibrant if the morphism $0\backslash rightarrow\; X$ from the initial object is a cofibration.Valery Isaev - On fibrant objects in model categories.

{{refbegin}}

• {{cite journal

| author = Peter Freyd, Max Kelly

| year = 1972

| title = Categories of Continuous Functors I

| journal = Journal of Pure and Applied Algebra

| volume = 2

}}

• {{Citation|

author = Riehl|first=Emily|authorlink = Emily Riehl|

title = Categorical homotopy theory |

publisher = Cambridge University Press|

year = 2014|

isbn = 978-1-107-04845-4|

mr = 3221774|

doi = 10.1017/CBO9781107261457}}

{{refend}}

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

Category:Category theory