on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
For $C$ a category with the structure of a model category and
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.
Let $C$ be a cofibrantly generated model category and
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 $C$.
Sufficient conditions for this to define a cofibrantly generated model category structure on $D$ are
the functor $F$ preserves small objects
this is the case in particular when $U$ preserves filtered colimits;
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 $D$.
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).
If $C$ carries the structure of a right proper model category, then also the transferred model structure on $D$ is right proper.
Let
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
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 $D$.
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.
Assume the adjunction
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.
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
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
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.
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.
The arguments for transfer of model structures go back to
Proofs can be found in
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