related by the Dold-Kan correspondence
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
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 with respect to ;
for the collection of morphisms with the left lifting property with respect to .
Moreover, we also write, now for :
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 checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts. So .
For the converse inclusion we use the small object argument: let be in . The small object argument produces a factorization .
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.
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 Dan Kan, reproduced for instance as theorem 11.3.1 in ModLoc .
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. 7.2.8 in ModLoc ).
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.
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).
The category of (based, compactly generated) topological spaces has a cofibrantly generated model structure 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 category of prespectra has two cofibrantly generated model structures. Let denote a prespectrum whose th space is when , and otherwise. Then the level model structure is generated by cells
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. The categories of symmetric and orthogonal spectra have similar cofibrantly generated level and stable model structures (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 .
A standard textbook reference is section 11 of
For the general case a useful reference is for instance the first section of
Some useful facts are discussed in the appendix of