Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
This entry discusses model category structures on categories of algebras over an operad in the category of chain complexes (unbounded).
This is a special case of the general discussion at model structure on algebras over an operad, but the category of (unbounded) chain complexes warrents some special attention.
For the operad of ordinary (commutative) algebras see also model structure on dg-algebras.
Let be a commutative ring. Write for the category of unbounded chain complexes of -modules.
An operad over is called -split if (…)
A quasi-isomorphism between such operads is said to be compatible with -splitting if (…)
If contains the ring of rational numbers, , then every -operad is -split and every quasi-isomorphism of operads is compatible with -splitting.
The associative operad is -split for all .
This is (Hinich, example 4.2.5).
This appears as (Hinich, theorem 4.1.1).
Invariance under equivalence and rectification
If is a quasi-isomorphism of -split operads compatible with splittings, then there is an induced Quillen equivalence
between the corresponding model structures on their algebras, as above.
This is (Hinich, theorem 4.7.4).
We discuss how the above model structure on is almost enhanced to a simplicial model category structure.
First we recall the standard definition of polynomial differential forms on simplices:
For let be the commutative dg-algebra of polynomial differential forms on the -simplex:
as a graded algebra it is
with the differential the usual de Rham differential under the embedding .
For a morphism in the simplex category let
be the morphism of dg-algebras given on generators by
This yields a simplicial commutative dg-algebra
or equivalently a cosimplicial object in the opposite category .
By the general definition of differential forms on presheaves this extends by left Kan extension to a functor
where on the right be have the coend over the copowering of over Set.
For a dg-operad as above, define sSet-hom-objects between objects by
This is (Hinich, lemma 4.8.4).
For a degreewise finite simplicial set, we have a natural isomorphism
This is (Hinich, lemma 4.8.3).
The homotopy category of is given by
where is a cofibrant resolution of .
This appears as (Hinich, section 4.8.10).
The model structure on dg-algebras over an operad is discussed in
based on results in
- A.K. Bousfield, V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type , Memoirs AMS, t.8, 179(1976)