model category

for ∞-groupoids

Contents

Idea

For $C$ a category with the structure of a model category and

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

an adjunction with $U$ right adjoint, under certain conditions it is possible to transfer the model structure from $C$ to a model structure on $D$ by declaring the fibrations and weak equivalences in $D$ to be precisely those morphisms whose image under $U$ are fibrations or weak equivalences, respectively, in $C$.

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 $F$ is the corresponding free functor.

Definition and Existence

Definition

Let $C$ be a cofibrantly generated model category and

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

an adjunction with right adjoint $U$.

Say a morphism in $D$ is a fibration or weak equivalence precisely if its image under $U$ is, respectively, in $C$.

Proposition

Sufficient conditions for this to define a cofibrantly generated model category structure on $D$ are

1. the functor $F$ preserves small objects

this is the case in particular when $U$ preserves filtered colimits;

2. any sequential colimit of pushouts of images under $F$ of the generating trivial cofibrations in $C$ yields a weak equivalence in $D$;

this is the case in particular if

• $D$ has a fibrant replacement functor;

• and $D$ has functorial path objects for fibrant objects

(meaning: a factorization of the diagonal $\Delta : A \to A \times A$ as a weak equivalence followed by a fibration (under $U$) $\Delta : A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A$, functorial in $A$).

If these conditions are met, then for $I$ (resp. $J$) the set of generating (acyclic) cofibrations in $C$, the image set $F(I)$ (resp. $F(J)$) forms the set of generating (acyclic) cofibrations in $D$.

Proof

One uses the small object argument repeatedly.

The argument goes back to section II.4 of (Quillen). A proof for one set of sufficient conditions in is chapter II of (GoerssJardine). Then (Crans) and (Cisinski).

Properties

General

Observation

If $C$ carries the structure of a right proper model category, then also the 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 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 model structure on algebraic fibrant objects is 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 transferred model structure from the 2-trivial model structure on $K$. (This is proven directly, rather than by appeal to the acyclicity as above.)

Mike Shulman: In addition to lots of examples, I think it would be also nice to include here a non example, of a case where the putative transferred model structure provably doesn’t exist.

• 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 upgrade to a Quillen adjunction.

References

The arguments for transfer of model structures go back to

• 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, Birkh¨auser Verlag, Basel, 1999.

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