on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal 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 :
for the class of morphisms obtained by transfinite composition of pushouts of elements in ;
for the class of retracts (in the arrow category ) of elements in
for the class of morphisms with the right lifting property with respect to .
A model category with all colimits is cofibrantly generated if there is a small set and a small set such that
is precisely the collection 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 pushouts. 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
Therefore .
The following theorem allows 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 and closed under retracts (in the arrow category).
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
weak equivalences
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, as does closure under retracts of .
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 cofibrantly generated model category that is also a locally presentable category is called a combinatorial model category.
A cofibrantly generated model category for which the doamins of the morphisms in and are compact objects or small objects is a cellular model category.
A standard textbook reference is section 11 of
For the general case a useful reference is for instance the first section of
For the case of a presentable category a useful reference is HTT section A.1.2.