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 underling (∞,1)-category.

The enriched case

Definition

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

Definition

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

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

    is an acyclic fibration in VV.

  3. Every morphism in MM factors as a map in LL followed by a map in RR.
  4. LL and RR are closed under retracts.

Remarks

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

Note that [i,p][i,p] is automatically a fibration, since ii is a cofibration and pp a fibration; thus the content of assertion (2) is that this map is a weak equivalence. If AA (hence also BB) is cofibrant and YY (hence also XX) is fibrant, then the pullback [A,X]× [A,Y][B,Y][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 VV is right proper, then the condition for the pullback to be a homotopy pullback can be weakened to “AA (hence also BB) is cofibrant OR YY (hence also XX) is fibrant”, and thus becomes automatic if either all objects of MM are fibrant or all objects of MM 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 MM be a model category and (L,R)(L,R) a pair of classes of maps. Write CC for the class of cofibrations, FF for the class of fibrations, M cM_{c} for the subcategory of cofibrant objects, M fM_f for the subcategory of fibrant objects, M cfM_{c f} for the subcategory of fibrant and cofibrant objects, C cfC_{c f} for the class of cofibrations between fibrant and cofibrant objects, etc.

Definition

  • (L,R)(L,R) is a weak homotopy factorization system if
    1. LL and RR are closed under weak equivalence in the arrow category M M^\to, and
    2. (LC cf,RF cf)(L\cap C_{c f}, R\cap F_{c f}) is a weak factorization system in M cfM_{c f}.
  • (L,R)(L,R) is a homotopy factorization system if it is a weak homotopy factorization system and in addition
    1. If fLf\in L and gfLg f\in L, then gLg\in L.
    2. If gRg\in R and gfRg f\in R, then fRf\in R.
  • (L,R)(L,R) is a strong homotopy factorization system if it is a homotopy factorization system and in addition
    1. (LC,RF)(L\cap C, R\cap F) is a weak factorization system in MM.

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

Proof

Since (L,R)(L,R) is a wfs, to show that (L,R)(L',R') is an unenriched weak hfs, it suffices to show that LC cf=LC cfL'\cap C_{c f} = L \cap C_{c f} and dually. Note that any morphism in C cfC_{c f} is both cofibrant and fibrant in the Reedy model structure on M M^\to; hence if two such morphisms are weakly equivalent in M 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 RR 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)(L,R) is determined by (LC cf,RF cf)(L\cap C_{c f}, R\cap F_{c f}) (called its center).
  • If (L,R)(L,R) is an unenriched weak hfs, then LC CL\cap C_C has the left lifting property against RF fR\cap F_f.
  • If (L,R)(L,R) is an unenriched weak hfs, then every morphism from a cofibrant object to a fibrant one factors as a map in LC cL\cap C_c followed by one in RF fR\cap F_f.

Proposition

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

  1. If fLf\in L and gfLg f\in L, then gLg\in L.
  2. If gRg\in R and gfRg f\in R, then fRf\in R.
  3. The codiagonal of any map in LC cL\cap C_c belongs to LL.
  4. The diagonal of any map in RF fR\cap F_f belongs to RR.
  5. if fLC cff\in L\cap C_{c f} and gfLC cfg f \in L\cap C_{c f} and gCg\in C, then gLg\in L.
  6. if gRF cfg\in R\cap F_{c f} and gfRF cfg f \in R\cap F_{c f} and fFf\in F, then fRf\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

Created on January 28, 2019 at 01:51:43. See the history of this page for a list of all contributions to it.