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cofibrantly generated model category

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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;

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

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

                                      Definition

                                      A model category 𝒞\mathcal{C} is cofibrantly generated if there are small sets of morphisms I,JMor(𝒞)I, J \subset Mor(\mathcal{C}) 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 readily checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts (see here for details). 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.

                                      Finally we apply the “retract argument”: 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.

                                      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.

                                      This is originally due to Daniel Kan, reproduced for instance as (Hirschhorn 03, theorem 11.3.1).

                                      Proof

                                      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. Closure under retracts of the weak equivalences will hold automatically if we check the rest of the axioms without using it, by an argument of A. Joyal. 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. Hirschhorn 03, 7.2.8,).

                                      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 inj(I) f Y = Y = Y \array{ X &\to& Z &\stackrel{\sigma}{\to}& X \\ \downarrow^f && \downarrow^{\mathrlap{\in inj(I)}} && \downarrow^f \\ Y &\stackrel{=}{\to}& Y & \stackrel{=}{\to} & Y }

                                      that exhibits ff as an element of inj(I)inj(I) as this is closed under retracts.

                                      Transfer along adjunctions

                                      Theorem

                                      Let 𝒞\mathcal{C} be a cofibrantly generated model category, def. , with generating (acyclic) cofibrations II (and JJ). Let 𝒟\mathcal{D} be any category with all small limits and colimits and consider a pair of adjoint functors

                                      (FU):𝒟UF𝒞. (F \dashv U) \;\colon\; \array{ \mathcal{D} \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} } \,.

                                      Write FI{F(i)|iI}F I \coloneqq \{F(i) | i \in I\} and FJ{F(j)|jJ}F J \coloneqq \{F(j) | j \in J\}. If

                                      1. both FIF I and FJF J admit the small object argument;

                                      2. UU takes FJF J-relative cell complexes to weak equivalences

                                      then FIF I, FJF J induce a cofibrantly generated model structure, def. , on 𝒟\mathcal{D}. Its weak equivalences are the morphisms that are taken to weak equivalences by UU. Moreover, the above adjunction is a Quillen adjunction for these model structures.

                                      This is due to Daniel Kan, reproduced in (Hirschhorn 03, theorem 11.3.2). See also at transferred model structure.

                                      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

                                      On simplicial sets

                                      The classical model category structure on simplicial sets is cofibrantly generated:

                                      Proposition

                                      sSet QuillensSet_{Quillen} is a cofibrantly generated model category with

                                      • generating cofibrations the boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n];

                                      • generating acyclic cofibrations the horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n].

                                      On topological spaces

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

                                      I={S + n1D + n} n0 I = \{S^{n-1}_+ \rightarrow D^n_+\}_{n\geq 0}

                                      and the set of acyclic cells is

                                      J={D + n(D n×I) +} n0. J = \{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 classical model structure on topological spaces.)

                                      On sequential spectra

                                      The category of sequential prespectra has two cofibrantly generated model structures (the Bousfield-Friedlander 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,n0 I = \{F_d S^{n-1}_+ \rightarrow F_d D^n_+\}_{d,n\geq 0}

                                      (free spectra formed on the standard cells of the classical model structure on pointed topological spaces)

                                      and the set of acyclic cells is

                                      J={F dD + nF d(D n×I) +} d,n0 J = \{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.

                                      On structured spectra

                                      The categories of symmetric spectra and orthogonal spectra have similar cofibrantly generated level and stable model structures (model structure on symmetric spectra, model structure on orthogonal spectra).

                                      (see Mandell, May, Schwede, Shipley: Model categories of diagram spectra.)

                                      On diagrams

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

                                      (The model structure on functors.)

                                      References

                                      A standard textbook reference is

                                      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)

                                      Last revised on November 9, 2017 at 09:00:53. See the history of this page for a list of all contributions to it.