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
A model category $C$ 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 $C$ be a category with all colimits and let $S \subset Mor(C)$ a class of morphisms. We write
$rlp(S)$ for the collection of morphisms with the right lifting property with respect to $S$;
$llp(S)$ for the collection of morphisms with the left lifting property with respect to $S$.
Moreover, we also write, now for $I \subset Mor(C)$:
$cell(I)$ for the relative cell complexes, the class of morphisms obtained by transfinite composition of pushouts of elements in $I$;
$cof(I)$ for the class of retracts (in the arrow category $Arr(C)$) of elements in $cell(I)$
$inj(I) := rlp(I)$ for the class of morphisms with the right lifting property with respect to $I$.
A model category with all colimits is cofibrantly generated if there is a small set $I$ and a small set $J$ such that
$cof(I)$ is precisely the collection of cofibrations of $C$;
$cof(J)$ is precisely the collection of acyclic cofibrations in $C$; and
$I$ and $J$ permit the small object argument.
Since $I$ and $J$ 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
$cof(I) = llp(rlp(I))$
$cof(J) = llp(rlp(J))$.
And therefore the fibrations are precisely $rlp(J)$ and the acyclic fibrations precisely $rlp(I)$.
The argument is the same for $I$ and $J$. So take $I$.
By definition we have $I \subset llp(rlp(I))$ and it is checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts. So $cof(I) \subset llp(rlp(I))$.
For the converse inclusion we use the small object argument: let $f : X \to Z$ be in $llp(rlp(I))$. The small object argument produces a factorization $f : X \stackrel{f' \in cof(I)}{\to} Y \stackrel{f''\in rlp(I)}{\to} Z$.
It follows that $f$ has the left lifting property with respect to $f''$ which yields a morphism $\sigma$ in
which exhibits $f$ as a retract of $f'$
Therefore $f \in cof(I)$.
The following theorem allows one to recognize cofibrantly generated model categories by checking fewer conditions.
Let $C$ be a category with all small limits and colimits and $W$ a class of maps satisfying 2-out-of-3 and closed under retracts (in the arrow category).
If $I$ and $J$ are sets of maps in $C$ such that
both $I$ and $J$ permit the small object argument;
$cof(J) \subset cof(I) \cap W$;
$inj(I) \subset inj(J) \cap W$;
one of the following holds
$cof(I) \cap W \subset cof(J)$
$inj(J) \cap W \subset inj(I)$
then there is the stucture of a cofibrantly generated model category on $C$ with
weak equivalences $W_C := W$
generating cofibrations $I$ (i.e. $cof_C := llp(rlp(I))$)
generating acyclic cofibrations $J$.
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 $W$ setting $cof_C := cof(I)$ and $fib_C := inj(J)$ 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 $W$.
Closure under retracts of $fib$ and $cof$ 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 $I$, showing that every morphism may be factored as
and assumption 3 says that $inj(I) \subset W$. Similarly applying the small object argument to $J$ gives factorizations
and assumption 2 guarantees that $cof(J) \subset W$.
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 $cof(I) \cap W \subset cof(J)$ and remembering that we have set $inj(J) = fib_C$ 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 $inj(J) \cap W \subset inj(I)$. To see this, apply the factorization established before to an acyclic fibration $f : X \to Y$ to get
With assumption 4 a this is
so that we have a lift
which establishes a retract
that exhibits $f$ as a weak equivalence.
Let $C$ be a cofibrantly generated model category which is also left proper. Then there exists a small set $S \subset Obj(C)$ of cofibrant objects which detect weak equivalences:
a morphism $f : A \to B$ in $C$ is a weak equivalence, precisely if for all $s \in S$ 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 $F_d A$ denote a prespectrum whose $n$th space is $\Sigma^{n-d} A$ when $n \geq d$, 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 $\Omega$-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 $D$, taking values in another cofibrantly generated model category $C$.
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 domains of the morphisms in $I$ and $J$ 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.
Some useful facts are discussed in the appendix of