# nLab homotopy factorization system

Homotopy factorization systems

# Homotopy factorization systems

## Idea

A homotopy factorization system in a model category is a presentation of an orthogonal factorization system in its underlying (∞,1)-category.

## The enriched case

### Definition

Let $V$ be a monoidal model category (with cofibrant unit object) and $M$ a $V$-enriched model category. In the case $V =$ SSet, the following definition is found in Bousfield, section 6.

###### Definition

A homotopy factorization system in $M$ is a pair $(L,R)$ of classes of maps such that:

1. Every map in $L$ is a cofibration, and every map in $R$ is a fibration.
2. If $i:A\to B$ is in $L$ and $p:X\to Y$ is in $R$, then the induced pullback power
$[i,p] : [B,X] \to [A,X] \times_{[A,Y]} [B,Y]$

is an acyclic fibration in $V$.

3. Every morphism in $M$ factors as a map in $L$ followed by a map in $R$.
4. $L$ and $R$ are closed under retracts.

### Remarks

It follows that $(L,R)$ is in fact a weak factorization system. For on the one hand; the underlying-set functor $V(I,-) : V\to Set$ takes acyclic fibrations to surjections since $I$ is cofibrant; thus $(L,R)$ have the lifting property. And on the other hand, if $i$ has the left lifting property against $R$, then factoring it and applying the retract argument implies $i\in L$, and dually.

Note that $[i,p]$ is automatically a fibration, since $i$ is a cofibration and $p$ a fibration; thus the content of assertion (2) is that this map is a weak equivalence. If $A$ (hence also $B$) is cofibrant and $Y$ (hence also $X$) is fibrant, then the pullback $[A,X] \times_{[A,Y]} [B,Y]$ is pullback of two fibrations between fibrant objects and thus a homotopy pullback; thus in this case the condition is equivalent to asking that the square

is a homotopy pullback square. If $V$ is right proper, then the condition for the pullback to be a homotopy pullback can be weakened to “$A$ (hence also $B$) is cofibrant OR $Y$ (hence also $X$) is fibrant”, and thus becomes automatic if either all objects of $M$ are fibrant or all objects of $M$ are cofibrant.

## The unenriched case

### Definition

A hierarchy of notions of “homotopy factorization system” for unenriched model categories can be found in Joyal, Appendix F. Let $M$ be a model category and $(L,R)$ a pair of classes of maps. Write $C$ for the class of cofibrations, $F$ for the class of fibrations, $M_{c}$ for the subcategory of cofibrant objects, $M_f$ for the subcategory of fibrant objects, $M_{c f}$ for the subcategory of fibrant and cofibrant objects, $C_{c f}$ for the class of cofibrations between fibrant and cofibrant objects, etc.

###### Definition

• $(L,R)$ is a weak homotopy factorization system if
1. $L$ and $R$ are closed under weak equivalence in the arrow category $M^\to$, and
2. $(L\cap C_{c f}, R\cap F_{c f})$ is a weak factorization system in $M_{c f}$.
• $(L,R)$ is a homotopy factorization system if it is a weak homotopy factorization system and in addition
1. If $f\in L$ and $g f\in L$, then $g\in L$.
2. If $g\in R$ and $g f\in R$, then $f\in R$.
• $(L,R)$ is a strong homotopy factorization system if it is a homotopy factorization system and in addition
1. $(L\cap C, R\cap F)$ is a weak factorization system in $M$.

## Relation between definitions

The relation between the enriched and unenriched notions is unclear to the author of this page, but here are some things that can be said.

###### Proposition

Suppose either every object of $M$ is fibrant and cofibrant, or $V$ is right proper and either every object of $M$ is fibrant or every object of $M$ is cofibrant. Then given an enriched hfs, by closing $L$ and $R$ under weak equivalence in $M^\to$ we obtain an unenriched weak hfs $(L',R')$.

###### Proof

Since $(L,R)$ is a wfs, to show that $(L',R')$ is an unenriched weak hfs, it suffices to show that $L'\cap C_{c f} = L \cap C_{c f}$ and dually. Note that any morphism in $C_{c f}$ is both cofibrant and fibrant in the Reedy model structure on $M^\to$; hence if two such morphisms are weakly equivalent in $M^\to$, there is a single weak equivalence relating them. But the property of being a homotopy pullback square is preserved under weak equivalence; and under the given hypotheses, as remarked above, the homotopy lifting property can be expressed in terms of such a square, and is thus preserved by weak equivalences between cofibrations. The proof for $R$ is dual (using the other Reedy model structure).

It is unclear whether or under what conditions this weak hfs is a hfs or a strong hfs.

In the converse direction, the following are proven by Joyal:

• An unenriched weak hfs $(L,R)$ is determined by $(L\cap C_{c f}, R\cap F_{c f})$ (called its center).
• If $(L,R)$ is an unenriched weak hfs, then $L\cap C_C$ has the left lifting property against $R\cap F_f$.
• If $(L,R)$ is an unenriched weak hfs, then every morphism from a cofibrant object to a fibrant one factors as a map in $L\cap C_c$ followed by one in $R\cap F_f$.

###### Proposition

If $(L,R))$ is an unenriched weak hfs, the following are equivalent:

1. If $f\in L$ and $g f\in L$, then $g\in L$.
2. If $g\in R$ and $g f\in R$, then $f\in R$.
3. The codiagonal of any map in $L\cap C_c$ belongs to $L$.
4. The diagonal of any map in $R\cap F_f$ belongs to $R$.
5. if $f\in L\cap C_{c f}$ and $g f \in L\cap C_{c f}$ and $g\in C$, then $g\in L$.
6. if $g\in R\cap F_{c f}$ and $g f \in R\cap F_{c f}$ and $f\in F$, then $f\in R$.

The closure under diagonals and codiagonals suggests that some kind of homotopy orthogonality should exist, using simplicial resolutions rather than enrichment.

• A. K. Bousfield, Constructions of factorization systems in categories, Journal of Pure and Applied Algebra 9 (1977) 207-220, pdf

• Andre Joyal, Notes on quasi-categories, pdf