Contents

model category

for ∞-groupoids

# Contents

## 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

Let $C$ be a model category and

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

an adjunction with right adjoint $U$. Define a morphism in $D$ to be

• a fibration or weak equivalence precisely if its image under $U$ is, respectively, in $C$.
• a cofibration precisely if it has the left lifting property with respect to the fibrations that are weak equivalences.

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

Of course, in the above situation by a trivial fibration in $D$ we mean a morphism that is both a fibration and a weak equivalence, or equivalently whose image under $U$ is a trivial fibration in $C$. The term “trivial cofibration”, however, is a priori ambiguous: for the nonce let us call a morphism in $D$ a cofibration weak equivalence if it is both a cofibration and a weak equivalence and 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

Necessary and sufficient conditions for the existence of the transferred model structure are:

1. Every morphism in $D$ factors as a cofibration followed by a trivial fibration, and as 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 $f:A\to B$ 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 a trivial 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

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

with $U$ left adjoint, we 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

### Constructing factorizations

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

###### Proposition

In the right-transferred situation, suppose that

1. $C$ is cofibrantly generated, and

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, then it is is cofibrantly generated, and for $I$ (resp. $J$) a set of generating cofibrations (resp. trivial cofibrations) in $C$, the image set $F(I)$ (resp. $F(J)$) forms a set of generating cofibrations (resp. trivial cofibrations) in $D$.

###### Proof

By the adjunction $F\dashv U$, 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 $F: C \rightleftarrows D : U$ is an adjunction, where $D$ has pushouts, and that

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

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

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

$\array{ F U A & \to & A \\ ^{F ((U f)_L)}\downarrow & & \downarrow^f \\ F E (U f) & \xrightarrow{F ((U f)_R)} & B }$

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 $U:D\to C$ 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 MR13.

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” 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 colimit of pushouts of images under $F$ of the generating trivial cofibrations in $C$ (i.e. an $F(J)$-cell complex) yields a weak equivalence 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, where $U:D\to C$ is a right adjoint, suppose that

• $D$ has a fibrant replacement functor (in fact it suffices for individual objects and morphisms to have fibrant replacements; functoriality is not required).

• $D$ 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 $\Delta : A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A$.

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

## Properties

### General

###### Observation

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

###### Observation

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

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

• 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 $M^{ob D}$.

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

• 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).

• A non-example is provided as Example 3.7 of (GoerssSchemmerhorn). 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.

• A more category-theoretic non-example 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.

The arguments for (right-)transfer of model structures go back to

• Dan Quillen, Homotopical Algebra , Lecture Notes in Math. 43, Springer-Verlag, Berlin-eidelberg-New York, 1967.

Proofs can be found in

• Paul Goerss, Jardine, J. F., Simplicial homotopy theory , Progress Mathematics 174, Birkhäuser Verlag, Basel, 1999.

The explicit study of (right-)transfer of model structures (on categories of sheaves) is apperently originally due to