nLab
transferred model structure

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

For CC a category with the structure of a model category and

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

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

Typically this arises in situations where DD consist of the “same” objects as CC but equipped with extra stuff, structure, property, and UU is the corresponding forgetful functor sending objects in DD to their underlying objects in CC. Then FF is the corresponding free functor.

Definition and Existence

Definition

Let CC be a cofibrantly generated model category and

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

an adjunction with right adjoint UU.

Say a morphism in DD is a fibration or weak equivalence precisely if its image under UU is, respectively, in CC.

Proposition

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

  1. the functor FF preserves small objects

    this is the case in particular when UU preserves filtered colimits;

  2. any sequential colimit of pushouts of images under FF of the generating trivial cofibrations in CC yields a weak equivalence in DD;

    this is the case in particular if

    • DD has a fibrant replacement functor;

    • and DD has functorial path objects for fibrant objects

      (meaning: a factorization of the diagonal Δ:AA×A\Delta : A \to A \times A as a weak equivalence followed by a fibration (under UU) Δ:AP(A)fibA×A\Delta : A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A, functorial in AA).

If these conditions are met, then for II (resp. JJ) the set of generating (acyclic) cofibrations in CC, the image set F(I)F(I) (resp. F(J)F(J)) forms the set of generating (acyclic) cofibrations in DD.

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 CC carries the structure of a right proper model category, then also the transferred model structure on DD is right proper.

Proof

Let

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

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

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

U(A× CB) U(f) U(B) Fib U(A) U(C) \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 CC. Since by definition of the transferred model strucure this is still the pullback of a weak equivalence along a fibration, and since CC is assumed to be right proper, it follows that U(f)U(f) is a weak equivalence in CC, hence that ff is a weak equivalence in DD.

Enrichment

Often the underlying model category CC is an enriched model category over some monoidal model category SS and one wishes to transfer also the model enrichment.

Observation

Assume the adjunction

(FU):DUFC (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 CC is transferred to DD. Consider the case that CC is moreover an SS-enriched model category and that DD can be equipped with the structure of a SS-enriched category that is also SS-powered and copowered.

Assume now that the SS-powering of DD is taken by UU to the SS-powering of CC, in that U(d (s 1s 2))=U(d) (s 1s 2)U(d^{(s_1 \to s_2)}) = U(d)^{(s_1 \to s_2)}.

Then the transferred model structure and the SS-enrichment on DD are compatible and make DD an SS-enriched model category.

Proof

By the axioms of enriched model category one sufficient condition to be checked is that for sts \to t any cofibration in SS and for XYX \to Y any fibration in DD, we have that the induced morphism

X tX s× Y sY t 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 UU. But by assumption UU commutes with the powering, and since UU is a right adjoint it commutes with taking the pullback, so that under UU the morphism is

U(X) tU(X) s× U(Y) sU(Y) t U(X)^t \to U(X)^s \times_{U(Y)^s} U(Y)^{t}

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

Examples

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.

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

See also prop. 1.4.23 of

A summary of the result is on p. 20 of

and on p. 6 of

The dual notion of transfer, “left induced” instead of “right induced”, is discussed in

Revised on February 9, 2014 01:46:27 by Urs Schreiber (89.204.139.50)