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cofibrantly generated model category

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Model category theory

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Contents

Idea

A model category CC is cofibrantly generated if there is a set (meaning: small set, not a proper class) of cofibrations and one of trivial cofibrations, such that all other (trivial) cofibrations are generated from these.

Definition

We need the following general terminology

Definition (cells and injectives)

Let CC be a category with all colimits and let SMor(C)S \subset Mor(C) a class of morphisms. We write

  • rlp(S)rlp(S) for the collection of morphisms with the right lifting property with respect to SS;

  • llp(S)llp(S) for the collection of morphisms with the left lifting property with respect to SS.

Moreover, we also write, now for IMor(C)I \subset Mor(C):

Definition (cofibrantly generated model category)

A model category with all colimits is cofibrantly generated if there is a small set II and a small set JJ such that

  • cof(I)cof(I) is precisely the collection of cofibrations of CC;

  • cof(J)cof(J) is precisely the collection of acyclic cofibrations in CC; and

  • II and JJ permit the small object argument.

Since II and JJ are assumed to admit the small object argument the collection of cofibrations and acyclic cofibrations has the following simpler characterization:

Proposition

In a cofibrantly generated model category we have

  • cof(I)=llp(rlp(I))cof(I) = llp(rlp(I))

  • cof(J)=llp(rlp(J))cof(J) = llp(rlp(J)).

And therefore the fibrations are precisely rlp(J)rlp(J) and the acyclic fibrations precisely rlp(I)rlp(I).

Proof

The argument is the same for II and JJ. So take II.

By definition we have Illp(rlp(I))I \subset llp(rlp(I)) and it is checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts. So cof(I)llp(rlp(I))cof(I) \subset llp(rlp(I)).

For the converse inclusion we use the small object argument: let f:XZf : X \to Z be in llp(rlp(I))llp(rlp(I)). The small object argument produces a factorization f:Xfcof(I)Yfrlp(I)Zf : X \stackrel{f' \in cof(I)}{\to} Y \stackrel{f''\in rlp(I)}{\to} Z.

It follows that ff has the left lifting property with respect to ff'' which yields a morphism σ\sigma in

X f Y f σ f Z = Z \array{ X &\stackrel{f'}{\to}& Y \\ \downarrow^{\mathrlap{f}} &{}^\sigma\nearrow& \downarrow^{\mathrlap{f''}} \\ Z &\stackrel{=}{\to}& Z }

which exhibits ff as a retract of ff'

X = X = X f f f Z σ Y f Z. \array{ X &\stackrel{=}{\to}& X &\stackrel{=}{\to}& X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f''}} && \downarrow^{\mathrlap{f}} \\ Z &\stackrel{\sigma}{\to}& Y &\stackrel{f''}{\to}& Z } \,.

Therefore fcof(I)f \in cof(I).

Properties

Recognition theorem

The following theorem allows one to recognize cofibrantly generated model categories by checking fewer conditions.

Theorem

Let CC be a category with all small limits and colimits and WW a class of maps satisfying 2-out-of-3 and closed under retracts (in the arrow category).

If II and JJ are sets of maps in CC such that

  1. both II and JJ permit the small object argument;

  2. cof(J)cof(I)Wcof(J) \subset cof(I) \cap W;

  3. inj(I)inj(J)Winj(I) \subset inj(J) \cap W;

  4. one of the following holds

    1. cof(I)Wcof(J)cof(I) \cap W \subset cof(J)

    2. inj(J)Winj(I)inj(J) \cap W \subset inj(I)

then there is the stucture of a cofibrantly generated model category on CC with

  • weak equivalences W C:=WW_C := W

  • generating cofibrations II (i.e. cof C:=llp(rlp(I))cof_C := llp(rlp(I)))

  • generating acyclic cofibrations JJ.

Proof

This is originally due to Dan Kan, reproduced for instance as theorem 11.3.1 in ModLoc .

We have to show that with weak equivalences WW setting cof C:=cof(I)cof_C := cof(I) and fib C:=inj(J)fib_C := inj(J) defines a model category structure.

The existence of limits, colimits and the 2-out-of-3 property holds by assumption, as does closure under retracts of WW.

Closure under retracts of fibfib and cofcof follows by the general statement that classes of morphisms defined by a left or right lifting property are closed under retracts (e.g. 7.2.8 in ModLoc ).

The factorization property follows by applying the small object argument to the set II, showing that every morphism may be factored as

cof(I)inj(I)fib C \stackrel{\in cof(I)}{\to} \stackrel{\in inj(I) \subset fib_C}{\to}

and assumption 3 says that inj(I)Winj(I) \subset W. Similarly applying the small object argument to JJ gives factorizations

cof(J)inj(J)=fib C \stackrel{\in cof(J)}{\to} \stackrel{\in inj(J) = fib_C}{\to}

and assumption 2 guarantees that cof(J)Wcof(J) \subset W.

It remains to verify the lifting axiom. This verification depends on which of the two parts of item 4 is satisfied. Assume the first one is, the argument for the second one is analogous.

Then using the assumption cof(I)Wcof(J)cof(I) \cap W \subset cof(J) and remembering that we have set inj(J)=fib Cinj(J) = fib_C we immediately have the lifting of trivial cofibrations on the left against fibrations on the right.

To get the lifting of cofibrations on the left with acyclic fibrations on the right, we show finally that inj(J)Winj(I)inj(J) \cap W \subset inj(I). To see this, apply the factorization established before to an acyclic fibration f:XYf : X \to Y to get

X = X cof(I)W finj(J)W Z inj(I) Y. \array{ X &\stackrel{=}{\to}& X \\ \downarrow^{\mathrlap{\in cof(I) \cap W}} && \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y } \,.

With assumption 4 a this is

X = X cof(J) finj(J)W Z inj(I) Y \array{ X &\stackrel{=}{\to}& X \\ \downarrow^{\mathrlap{\in cof(J) }} && \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y }

so that we have a lift

X = X cof(J) σ finj(J)W Z inj(I) Y \array{ X &\stackrel{=}{\to}& X \\ {}^{\mathllap{\in cof(J) }} \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y }

which establishes a retract

X Z σ X f W f Y = Y = Y \array{ X &\to& Z &\stackrel{\sigma}{\to}& X \\ \downarrow^f && \downarrow^{\mathrlap{\in W}} && \downarrow^f \\ Y &\stackrel{=}{\to}& Y & \stackrel{=}{\to} & Y }

that exhibits ff as a weak equivalence.

Presentation and generation

Proposition

Let CC be a cofibrantly generated model category which is also left proper. Then there exists a small set SObj(C)S \subset Obj(C) of cofibrant objects which detect weak equivalences:

a morphism f:ABf : A \to B in CC is a weak equivalence, precisely if for all sSs \in S the induced morphism of derived hom-spaces

Hom(s,f):Hom(s,A)Hom(s,B) \mathbb{R}Hom(s,f) : \mathbb{R}Hom(s,A) \to \mathbb{R}Hom(s,B)

is a weak equivalence.

This appears as (Dugger, prop. A.5).

Examples

  • The category of (based, compactly generated) topological spaces has a cofibrantly generated model structure in which the set of cells is

    I={S + n1D + n} n0I = \{S^{n-1}_+ \rightarrow D^n_+\}_{n\geq 0}

    and the set of acyclic cells is

    J={D + n(D n×I) +} n0J = \{D^n_+ \rightarrow (D^n \times I)_+\}_{n\geq 0}

    Here ++ means disjoint basepoint, not northern hemisphere. The category of unbased spaces has a similar cofibrantly generated model structure.

  • The category of prespectra has two cofibrantly generated model structures. Let F dAF_d A denote a prespectrum whose nnth space is Σ ndA\Sigma^{n-d} A when ndn \geq d, and ** otherwise. Then the level model structure is generated by cells

    I={F dS + n1F dD + n} d,n0I = \{F_d S^{n-1}_+ \rightarrow F_d D^n_+\}_{d,n\geq 0}

    and the set of acyclic cells is

    J={F dD + nF d(D n×I) +} d,n0J = \{F_d D^n_+ \rightarrow F_d (D^n \times I)_+\}_{d,n\geq 0}

    The stable model structure has the same cells, but more acyclic cells, which in turn guarantees that the fibrant spectra are the Ω\Omega-spectra. The categories of symmetric and orthogonal spectra have similar cofibrantly generated level and stable model structures (see Mandell, May, Schwede, Shipley, Model Categories of Diagram Spectra.)

  • The category of diagrams indexed by a fixed small category DD, taking values in another cofibrantly generated model category CC.

References

A standard textbook reference is section 11 of

  • ModLoc, Hirschhorn, Model categories and their localization .

For the general case a useful reference is for instance the first section of

  • Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475.(math.CT/0102087), Sheafifiable homotopy model categories. II, J. Pure Appl. Algebra 164 (2001), no. 3, 307–324.

For the case of a presentable category a useful reference is HTT section A.1.2.

Some useful facts are discussed in the appendix of

  • Dan Dugger, Replacing model categories with simplicial ones (pdf)

Revised on January 7, 2014 01:34:32 by Tim Porter (2.26.23.142)