nLab transferred model structure

Idea

For $C$ a category with the structure of a model category and $U:D\to C$ a functor having an adjoint, under certain conditions it is possible to transfer the model structure from $C$ to a model structure on $D$ by declaring the weak equivalences, and the fibrations (if $U$ is a right adjoint) or cofibrations (if $U$ is a left adjoint), in $D$ to be precisely those morphisms whose images under $U$ are such in $C$. If $U$ is a right adjoint this is called the right transferred, right induced, or right lifted model structure, and dually in the left case.

Typically this arises in situations where $D$ consist of the “same” objects as $C$ but equipped with extra stuff, structure, property, and $U$ is the corresponding forgetful functor sending objects in $D$ to their underlying objects in $C$. Then a left adjoint $F$ of $U$ is the corresponding free functor, while a right adjoint $G$ is a cofree functor.

Definition

Right transfer

Definition

(right transferred model structure)
Let $C$ be a model category and

$$(F \dashv U ) \;\colon\; D \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\;\bot\;\;\;} C$$

a pair of adjoint functors with right adjoint $U$. Define a morphism in $D$ to be

• a fibration or weak equivalence precisely if its image under $U$ is so in $C$, respectively

  (by adjointness this means that the fibrations form equivalently the class with the right lifting property against the image under $F$ of the acyclic cofibrations in $C$);

• a cofibration precisely if it has the left lifting property with respect to the fibrations that are also weak equivalences, according to the previous item.

If these classes of maps define a model category structure on $D$, then it is called the right transferred model structure (Crans 1995, §3) or sometimes right induced model structure or right lifted model structure from $C$ along $R$.

In the above situation of Def. it is clear what one means by an acyclic fibration in $D$, but the notion of acyclic cofibration in $D$ is, a priori, ambiguous. For the purpose of stating the following Thm. we declare that:

1. an acyclic fibration in $D$ is (of course) a morphism that is both a fibration and a weak equivalence, or equivalently whose image under $U$ is an acyclic fibration in $C$.

2. a morphism is $D$ is

   1. a cofibration weak equivalence if it is both a cofibration and a weak equivalence in the sense of Def. ;

   2. an anodyne map if it has the left lifting property with respect to all fibrations.

   Of course, if the transferred model structure exists, then these two classes of maps coincide.

Conversely we have:

Theorem

A necessary and sufficient condition for the existence of the right transferred model structure (Def. ) is that:

1. Every morphism in $D$ factors as

   1. a cofibration followed by a trivial fibration,

   2. an anodyne map followed by a fibration;

2. every anodyne map is a weak equivalence.

Proof

Clearly the conditions are necessary. For the converse, the weak equivalences have the 2-out-of-3 property, all the classes of maps are closed under retracts, the lifting properties hold by definition (for the anodyne maps), and we have assumed the factorization properties (for the anodyne maps), so it remains to show that every cofibration weak equivalence is anodyne. This follows by the standard retract argument: if $f \colon A \longrightarrow B$ is a cofibration weak equivalence, factor it as $f = p i$ with $i$ anodyne and $p$ a fibration. Then since anodyne maps are weak equivalences and weak equivalences satisfy 2-out-of-3, $p$ is an acyclic fibration. Thus $f$ has the left lifting property against $p$, hence $f$ is a retract of $i$ and thus also anodyne.

Left transfer

Dually, if $C$ is a model category and we have an adjunction

$$(U \dashv G ) \;\colon\; D \underoverset { \underset{G}{\longleftarrow} } { \overset{U}{\longrightarrow} } {\;\;\;\bot\;\;\;} C$$

with $U$ left adjoint, define a morphism in $D$ to be

• a cofibration or weak equivalence precisely if its image under $U$ is, respectively, in $C$.

• a fibration precisely if it has the right lifting property with respect to the cofibrations that are weak equivalences.

If these classes of maps define a model structure on $D$, it is called the left transferred model structure (or sometimes left induced model structure or left lifted model structure) from $C$.

The above necessary and sufficient conditions dualize directly.

Existence

For right transfer

Constructing factorizations

The most traditional way to obtain the factorization properties in the right-transferred case (Thm. ) is to assume that $C$ is cofibrantly generated.

Proposition

In the right-transferred situation (Def. ), suppose that

1. $C$ is cofibrantly generated,

2. the functor $F$ preserves small objects

   (which is the case, in particular, when $U$ preserves filtered colimits),

then every morphism in $D$ factors as a cofibration followed by a trivial fibration, and as an anodyne map followed by a fibration.

Moreover, if every anodyne map is a weak equivalence so that the transferred model structure on $D$ exists (by Thm. ), then it is itself cofibrantly generated, and for $I$ (resp. $J$) a set of generating cofibrations (resp. acyclic cofibrations) in $C$, the image set $F(I)$ (resp. $F(J)$) forms a set of generating cofibrations (resp. acyclic cofibrations) in $D$.

Proof

By the hom-isomorphism of the adjoint pair $F \dashv U$, a morphism $f$ is a fibration (resp. trivial fibration) if and only if it has the right lifting property against the maps in $F(I)$ (resp. $F(J)$). Since $F$ preserves small objects, the sets $F(I)$ and $F(J)$ permit the small object argument in $D$, which therefore yields (cofibration, trivial fibration) and (anodyne, fibration) factorizations.

Another way to produce these factorizations is to generate them by a functorial factorization:

Lemma

Suppose $F \colon C \rightleftarrows D \colon U$ is an adjunction, where $D$ has pushouts, and that

1. $C$ has a functorial factorization $(L,R)$ with a right weak composition law for factorizations (e.g. if it is algebraic),

2. the induced functorial factorization on $D$ permits Garner’s small object argument (e.g. if $C$ and $D$ are locally presentable and all functors are accessible).

Then there is an induced (algebraic) weak factorization system on $D$ whose right class consists precisely of those maps whose $U$-image admits an $R$-algebra structure.

Proof

Given a morphism $f\colon A\to B$ in $D$, factor it as follows, where the upper commutative square is a pushout:

$$\array{ F U A & \longrightarrow & A \\ \mathllap{ {}^{F ((U f)_L)} } \big\downarrow & & \big\downarrow \\ F E (U f) & & E f \\ \mathllap{ {}^{F ((U f)_R)} } \big\downarrow & & \big\downarrow \\ F U B & \longrightarrow & B }$$

This yields a functorial factorization $(L',R')$ on $D$. By the universal property of pushout, an $R'$-algebra structure on $f$ is determined by a diagonal lifting in the square

$$\array{ F U A & \longrightarrow & A \\ \mathllap{ {}^{F ((U f)_L)} } \big\downarrow & & \big\downarrow \mathrlap{{}^f} \\ F E (U f) & \underset{F ((U f)_R)}{\longrightarrow} & B }$$

which corresponds by adjunction to precisely an $R$-algebra structure on $U f$. Thus, the right weak composition law for $R$-algebras lifts to a right weak composition law for $R'$-algebras. Garner’s small object argument applied to $(L',R')$ then produces a monad over $cod$ with a right strong composition law, which is therefore an algebraic weak factorization system whose algebraic right maps are the $R'$-algebras, and in particular whose underlying right chlass is as desired those maps whose $U$-image is in $R$.

In particular, therefore, the factorizations of any accessible model structure right-lift along any adjunction between locally presentable model categories, whether or not they are cofibrantly generated. But the cofibrantly generated case can also be subsumed, by taking $(L,R)$ to be the one-step factorization produced by the generating left maps.

Left-lifting is generally rather trickier. But by invoking fancier categorical machinery, one can show more generally:

Lemma

Suppose given a functor $U \colon D\to C$ between locally presentable categories and an accessible weak factorization system $(L,R)$ on $C$.

1. If $U$ has a left adjoint $F$, then $(L,R)$ right-lifts along $U$ to an accessible wfs on $D$.

2. If $U$ has a right adjoint $G$, then $(L,R)$ left-lifts along $U$ to an accessible wfs on $D$.

Moreover, if $(L,R)$ is cofibrantly generated, so are the lifted wfs.

Proof

See HKRS15 and GKR18 for the lifting of accessible wfs. Cofibrant generation of a right-lifted wfs follows as in Theorem above, while in the left-lifted case it is Remark 3.8 of Makkai & Rosický 2014.

This sort of result seems to be necessary for the case of left-lifting; the simpler cofibrantly-generated argument does not dualize as directly.

Verifying acyclicity

The “acyclicity condition” in Thm. , i.e. that anodyne maps are weak equivalences, is usually the most difficult to check. Sometimes useful is the observation that for right-lifting in the cofibrantly generated case, it suffices to show that sequential colimits of pushouts of images under $F$ of the generating acyclic cofibrations in $C$ (i.e. forming $F(J)$-cell complexes) yield weak equivalences in $D$. This is because the small object actually factors any map as such an $F(J)$-cell complex followed by a fibration; hence by the retract argument every anodyne map is a retract of an $F(J)$-cell complex.

Another useful sufficient condition is the following, going back roughly to section II.4 of (Quillen).

Proposition

In the situation of right transfer (Def. ), where $U \colon D\to C$ is a right adjoint, suppose that

1. $D$ has a fibrant replacement functor (in fact it suffices for individual objects and morphisms to have fibrant replacements, while functoriality is not necessary),

2. $D$ has path objects for fibrant objects, i.e. a factorization of the diagonal $\Delta : A \to A \times A$ as a weak equivalence followed by a fibration $\Delta \colon A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A$.

Then any anodyne map in $D$ is a weak equivalence. Thus, if the two factorizations exist in $D$, then the transferred model structure exists.

Proof

The notation in the following is mostly that of Rezk 02, Lemma 7.6.
Let $R$ be a fibrant replacement functor, with natural weak equivalence $j \colon Id \to R$, suppose $f \colon X\to Y$ is anodyne, and consider the following commuting diagram:

Both squares are pullbacks, defining the object $P R f$ and the morphism $i$ making the following square commute:

The morphism $\pi$ is the composite $P R f\to R X \times R Y \to R X$ at the top, which is a pullback of $P R Y \to R Y$; but the latter is a fibration (as the composite $P R Y \to R Y \times R Y \to R Y$) and a weak equivalence (as a retraction for $R Y \to P R Y$), so it and hence also $\pi$ are acyclic fibrations. Moreover, since $\pi i = j_X$, by 2-out-of-3 $i$ is also a weak equivalence.

Now the projection $P R f\to P Y$ is a fibration, so since $f$ is anodyne there is a lift $g$ in the second square as shown. Since $i$ and $j_Y$ are weak equivalences, by 2-out-of-6 it follows that $f$ is a weak equivalence.

Note that this condition also dualizes straightforwardly to the left-transferred case.

For left transfer

Proposition

Given a pair of adjoint functors $\mathcal{D} \underoverset {\underset{R}{\longleftarrow}} {\overset{L}{\longrightarrow}} {\;\;\bot\;\;} \mathcal{C}$ such that:

1. $\mathcal{C}$ and $\mathcal{D}$ are locally presentable categories,

2. $\mathcal{C}$ is equipped with the structure of a cofibrantly generated model category (hence a combinatorial model category) with classes of (co-)/fibrations and weak equivalences $Cof, Fib, W \,\subset\, Mor(\mathcal{C})$,

3. $RLP\big( L^{-1} Cof \big) \,\subset\, L^{-1}(W)$ (i.e. co-anodyne maps are weak equivalences),

then the left-transferred model category structure on $\mathcal{D}$ exists (i.e. with cofibrations $L^{-1}(Cof)$ and weak equivalences $L^{-1}(W)$) and is itself cofibrantly generated.

Proof

As observed in the proof of BHKKRS 2015, Thm. 2.23, the existence of the required cofibrantly generated factorization systems follows by Makkai & Rosický 2014, Rem. 3.8 with the comment below Def. 2.3 there, which in turn invokes Lurie 2009, §A.1.5.12. In particular, from inside the proof of Makkai & Rosický 2014, Thm. 3.2 one has that a small set of generating (acyclic) cofibrations of the left transferred model structure is given by those between $\kappa$-presentable objects, for some un-countable regular cardinal $\kappa$.

Properties

General

Lemma

If $C$ carries the structure of a right proper model category, then also a right-transferred model structure on $D$ (Def. ) is right proper.

Proof

Let

$$\array{ A \times_C B &\stackrel{f}{\to}& B \\ \downarrow && \downarrow^{\mathrlap{\in Fib}} \\ A &\stackrel{\simeq}{\to}& C }$$

be a pullback diagram in $D$, with the bottom morphism a weak equivalence and the right morphism a fibration. We need to show that then also the top morphism $f$ is a weak equivalence. By definition of transfer, this is equivalent to $U(f)$ being a weak equivalence in $C$.

Since $U$ is a right adjoint it preserves pullbacks, so that also

$$\array{ U(A \times_{C} B) &\stackrel{U(f)}{\to}& U(B) \\ \downarrow && \downarrow^{\mathrlap{\in Fib}} \\ U(A) &\stackrel{\simeq}{\to}& U(C) }$$

is a pullback diagram in $C$. Since by definition of the transferred model strucure this is still the pullback of a weak equivalence along a fibration, and since $C$ is assumed to be right proper, it follows that $U(f)$ is a weak equivalence in $C$, hence that $f$ is a weak equivalence in $D$.

Enrichment

Often the underlying model category $C$ is an enriched model category over some monoidal model category $S$ and one wishes to transfer also the model enrichment.

Proposition

Assume the adjunction

$$(F \dashv U )\; : \; D \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$$

satisfies the conditions of the above proposition for right transfer, so that the model structure on $C$ is transferred to $D$. Consider the case that $C$ is moreover an $S$-enriched model category and that $D$ can be equipped with the structure of a $S$-enriched category that is also $S$-powered and copowered.

Assume now that the $S$-powering of $D$ is taken by $U$ to the $S$-powering of $C$, in that $U(d^{(s_1 \to s_2)}) = U(d)^{(s_1 \to s_2)}$.

Then the transferred model structure and the $S$-enrichment on $D$ are compatible and make $D$ an $S$-enriched model category.

Proof

By the axioms of enriched model category one sufficient condition to be checked is that for $s \to t$ any cofibration in $S$ and for $X \to Y$ any fibration in $D$, we have that the induced morphism

$$X^t \to X^s \times_{Y^s} Y^{t}$$

is a fibration, which is a weak equivalence if at least one of the two input morphisms is. By the induced model structure, this is checked by applying $U$. But by assumption $U$ commutes with the powering, and since $U$ is a right adjoint it commutes with taking the pullback, so that under $U$ the morphism is

$$U(X)^t \to U(X)^s \times_{U(Y)^s} U(Y)^{t}$$

which is the morphism induced from $U(X) \to U(Y)$. That this is indeed an (acyclic) fibration follows now from the fact that $C$ is an $S$-enriched model category.

Examples

Example

The right-transfer along an adjoint equivalence $L \dashv R$ always exists, and here also the cofibrations are given by the preimage under $R$.

This is straightforward to check directly, even without the above existence theorems (e.g. math.se:a/2220718).

Example

The projective model structure on functors $M^D$ is right-transferred from the product model structure on $M^{ob D}$. Dually, the injective model structure on functors, when it exists, is left-transferred from $M^{ob D}$.

Example

The model structure on algebraic fibrant objects is right-transferred from the underlying model category by forgetting the choice of fillers.

Example

If $T$ is an accessible strict 2-monad on a locally finitely presentable 2-category $K$. then the category $T Alg_s$ of strict $T$-algebras admits a right-transferred model structure from the 2-trivial model structure on $K$. The acyclicity condition is proved by using pseudolimits of arrows for path objects, and the (cofibration, trivial fibration) factorization is constructed using a version of the construction of Lemma (since the 2-trivial model structure on $K$ may not be cofibrantly generated).

Example

A counterexample is provided as Example 3.7 of (Goerss & Schemmerhorn 2007). Let $k$ be a field of characteristic 2 and consider the adjunction

$$(S \dashv U )\; : \; CGA_k \stackrel{\overset{S}{\leftarrow}}{\underset{U}{\to}} Ch_*k$$

of the symmetric algebra functor and the forgetful functor between graded commutative DGAs and chain complexes. One sees that $S$ does not preserve the weak equivalence between 0 and the complex with one copy of $k$ in degrees $n$ and $n-1$. Since all chain complexes are cofibrant this means that $(S \dashv U )$ cannot be upgraded to a Quillen adjunction.

Example

A more category-theoretic counterexample is that the Reedy model structure on directed graphs internal to Cat does not right-transfer to the category DblCat? of double categories along the free-forgetful adjunction. One of the generatic trivial cofibrations for the Reedy model structure “transports” a horizontal arrow (in double-category terminology) along two vertical isomorphisms, which is a levelwise equivalence. But upon pushing out the free double category generated by this along a map into some other double category, new composite horizontal arrows may be generated in the codomain that are not isomorphic to the image of anything in the domain; thus the pushout is no longer a levelwise equivalence. So, the acyclicity condition fails.

References

The explicit notion of right transfer is due to:

• Sjoerd Crans, §3-§4 in: Quillen closed model structure for sheaves, Journal of Pure and Applied Algebra 101 1 (1995) 35-57 [doi:10.1016/0022-4049(94)00033-F, web]

and for the case of cofibrantly generated model categories due to:

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, §9.1 in: Model Categories and More General Abstract Homotopy Theory (1997) [pdf, pdf]

The existence result of right transfer due to Crans 1995, Thm. 3.3 and Dwyer, Hirschhorn & Kan 1997, Thm. 91 (“Crans-Kan transfer theorem”) is reviewed/reworked, with variations, in:

• Paul Goerss, J. F. Jardine, Section II.4 of: Simplicial homotopy theory, Progress Mathematics 174, Birkhäuser (1999) [doi:10.1007/978-3-0346-0189-4]

• Philip Hirschhorn, Thm. 11.3.2 in: Model Categories and Their Localizations, Math. Survey and Monographs 99, AMS (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

• Denis-Charles Cisinski, Prop. 1.4.23 of: Les préfaisceaux comme types d’homotopie|Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 Soc. Math. France (2006), 392 pages [numdam:AST_2006__308__R1_0, pdf]

• Paul Goerss, Kristen Schemmerhorn, §3.2 of: Model Categories and Simplicial Methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436 AMS (2007) [arXiv:math.AT/0609537, doi:10.1090/conm/436]

• Clemens Berger, Ieke Moerdijk, Section 2.5 of: Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 4 [arXiv:math/0206094, doi:10.1007/s00014-003-0772-y, pdf]

These arguments all go back (cf. Goerss & Shemmerhorn 2007, p. 20) to:

• Daniel Quillen, §II.4 in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

The above proof of acyclicity in the presence of path objects is taken from:

• Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology Appl. 119 1 (2002) 65-94 [arxiv:math/0003065]

Discussion of left transfer (left-induced model structures):

• Kathryn Hess, Brooke Shipley, §4 in: The homotopy theory of coalgebras over a comonad, Proceedings of the London Mathematical Society 108 2 (2014) [arXiv:1205.3979, doi:10.1112/plms/pdt038]

• Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kedziorek, Emily Riehl, Brooke Shipley, Left-induced model structures and diagram categories, in: Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics 641 American Mathematical Society (2015) [arXiv:1401.3651, ISBN:978-1-4704-2495-4]

• Kathryn Hess, Magdalena Kedziorek, Emily Riehl, Brooke Shipley, A necessary and sufficient condition for induced model structures, J. Topology 10 2 (2017) 324-369 [arXiv:1509.08154, doi:10.1112/topo.12011]

Beware that HKRS15 contains an error, corrected by:

• Richard Garner, Magdalena Kedziorek, Emily Riehl, Lifting accessible model structures, J. Topology 13 1 (2020) 59-76 [arXiv:1802.09889, doi:10.1112/topo.12123]

Technical lemmas regarding the underlying cellular categories:

• M. Makkai, J. Rosický, Cellular categories, J. Pure Appl. Alg. 218 (2014) 1652-1664 [arXiv:1304.7572, doi:10.1016/j.jpaa.2014.01.005]

• Jacob Lurie, Appendix A of: Higher Topos Theory (2009)

The right-transferred model structure on algebras for a 2-monad is from

• Steve Lack, Homotopy-theoretic aspects of 2-monads, arXiv