nLab homotopy factorization system

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.

References

• 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