A homotopy factorization system in a model category is a presentation of an orthogonal factorization system in its underlying (∞,1)-category.
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.
A homotopy factorization system in $M$ is a pair $(L,R)$ of classes of maps such that:
is an acyclic fibration in $V$.
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.
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.
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.
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 $(L',R')$.
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:
If $(L,R))$ is an unenriched weak hfs, the following are equivalent:
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
Last revised on February 12, 2021 at 23:47:02. See the history of this page for a list of all contributions to it.