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 cellular model category is a particularly convenient form of a model category.
It is similar to a combinatorial model category. (For the moment, see there for more details.)
A cellular model category is a cofibrantly generated model category such that there is a set of generating cofibrations $I$ and a set of generating acyclic cofibrations $J$, such that
all domain and codomain objects of elements of $I$ are compact objects;
the domain objects of the elements of $J$ are small objects relative to $I$;
the cofibrations are effective monomorphisms.
Top with the standard model structure on topological spaces (same for pointed topological spaces)
SSet with the standard model structure on simplicial sets (same for pointed simplicial sets).
For $C$ a cellular model category we have that
the functor category $[D,C]$ for any small category $C$ with its projective global model structure on functors is again a cellular model category;
for $c \in C$ any object, the over category $C/c$ is again a cellular model category.
For cellular model categories $C$ that are left proper model categories all left Bousfield localizations $L_S C$ at any set $S$ of morphisms are guaranteed to exist.
A standard textbook reference is section 12 of
In the context of algebraic model categories related discussion is in