Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
A model category is cofibrantly generated if there is a set (meaning: small set, not a proper class) of cofibrations and one of trivial cofibrations, such that all other (trivial) cofibrations are generated from these.
We need the following general terminology
Definition (cells and injectives)
Let be a category with all colimits and let a class of morphisms. We write
for the collection of morphisms with the right lifting property;
for the collection of morphisms with the left lifting property with respect to , the
Moreover, we also write, now for :
A model category is cofibrantly generated if there are small sets of morphisms such that
is precisely the collection of cofibrations of ;
is precisely the collection of acyclic cofibrations in ; and
and permit the small object argument.
Since and are assumed to admit the small object argument the collection of cofibrations and acyclic cofibrations has the following simpler characterization:
In a cofibrantly generated model category we have
And therefore the fibrations are precisely and the acyclic fibrations precisely .
The argument is the same for and . So take .
By definition we have and it is readily checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts (see here for details). So .
For the converse inclusion we use the small object argument: let be in . The small object argument produces a factorization .
Finally we apply the “retract argument”: It follows that has the left lifting property with respect to which yields a morphism in
which exhibits as a retract of
The following theorem allows one to recognize cofibrantly generated model categories by checking fewer conditions.
Let be a category with all small limits and colimits and a class of maps satisfying 2-out-of-3.
If and are sets of maps in such that
both and permit the small object argument;
one of the following holds
then there is the stucture of a cofibrantly generated model category on with
generating cofibrations (i.e. )
generating acyclic cofibrations .
This is originally due to Daniel Kan, reproduced for instance as (Hirschhorn 03, theorem 11.3.1).
We have to show that with weak equivalences setting and defines a model category structure.
The existence of limits, colimits and the 2-out-of-3 property holds by assumption. Closure under retracts of the weak equivalences will hold automatically if we check the rest of the axioms without using it, by an argument of A. Joyal. Closure under retracts of and follows by the general statement that classes of morphisms defined by a left or right lifting property are closed under retracts (e.g. Hirschhorn 03, 7.2.8,).
The factorization property follows by applying the small object argument to the set , showing that every morphism may be factored as
and assumption 3 says that . Similarly applying the small object argument to gives factorizations
and assumption 2 guarantees that .
It remains to verify the lifting axiom. This verification depends on which of the two parts of item 4 is satisfied. Assume the first one is, the argument for the second one is analogous.
Then using the assumption and remembering that we have set we immediately have the lifting of trivial cofibrations on the left against fibrations on the right.
To get the lifting of cofibrations on the left with acyclic fibrations on the right, we show finally that . To see this, apply the factorization established before to an acyclic fibration to get
With assumption 4 a this is
so that we have a lift
which establishes a retract
that exhibits as a weak equivalence.
Transfer along adjunctions
Let be a cofibrantly generated model category, def. 2, with generating (acyclic) cofibrations (and ). Let be any category with all small limits and colimits and consider a pair of adjoint functors
Write and . If
both and admit the small object argument;
takes -relative cell complexes to weak equivalences
then , induce a cofibrantly generated model structure, def. 2, on . Its weak equivalences are the morphisms that are taken to weak equivalences by . Moreover, the above adjunction is a Quillen adjunction for these model structures.
This is due to Daniel Kan, reproduced in (Hirschhorn 03, theorem 11.3.2). See also at transferred model structure.
Presentation and generation
Let be a cofibrantly generated model category which is also left proper. Then there exists a small set of cofibrant objects which detect weak equivalences:
a morphism in is a weak equivalence, precisely if for all the induced morphism of derived hom-spaces
is a weak equivalence.
This appears as (Dugger, prop. A.5).
On topological spaces
The category of (based, compactly generated) topological spaces has a cofibrantly generated model structure (the classical model structure on pointed topological spaces) in which the set of cells is
and the set of acyclic cells is
(Here means disjoint basepoint, not northern hemisphere.)
The category of unbased spaces has a similar cofibrantly generated model structure. (The classical model structure on topological spaces.)
On sequential spectra
The category of sequential prespectra has two cofibrantly generated model structures (the Bousfield-Friedlander model structures).
Let denote a prespectrum whose th space is when , and otherwise. Then the level model structure is generated by cells
(free spectra formed on the standard cells of the classical model structure on pointed topological spaces)
and the set of acyclic cells is
The stable model structure has the same cells, but more acyclic cells, which in turn guarantees that the fibrant spectra are the -spectra.
On structured spectrd
The categories of symmetric spectra and orthogonal spectra have similar cofibrantly generated level and stable model structures (model structure on symmetric spectra, model structure on orthogonal spectra).
(see Mandell, May, Schwede, Shipley: Model categories of diagram spectra.)
The category of diagrams indexed by a fixed small category , taking values in another cofibrantly generated model category .
(The model structure on functors.)
A standard textbook reference is
For the general case a useful reference is for instance the first section of
- Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475.(math.CT/0102087), Sheafifiable homotopy model categories. II, J. Pure Appl. Algebra 164 (2001), no. 3, 307–324.
For the case of a presentable category a useful reference is HTT section A.1.2.
Some useful facts are discussed in the appendix of
- Dan Dugger, Replacing model categories with simplicial ones (pdf)