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
An cell complex is an object in a category which is obtained by successively “gluing cells” via pushouts.
Let $C$ be a category with colimits and equipped with a set $\mathcal{I} \subset Mor(C)$ of morphisms.
In practice $C$ is usually a cofibrantly generated model category with set $\mathcal{I}$ of generating cofibrations and set $\mathcal{J}$ of acyclic generating cofibrations.
An $\mathcal{I}$-cell complex in $C$ is an object $X$ which is connected to the initial object $\emptyset \to X$ by a transfinite composition of pushouts of the generating cofibrations in $\mathcal{I}$.
A relative $\mathcal{I}$-cell complex (relative to any object $A$) is any morphism $A \to X$ obtained this starting from $A$.
A CW-complex is a cell complex in Top with respect to the generating cofibrations in the standard model structure on topological spaces.
Every simplicial set is a cell complex with respect to the generating cofibrations in the standard model structure on simplicial sets.
A Sullivan model is a cell complex with respect to the generating cofibrations in the standard model structure on dg-algebras.
A cell spectrum is a cell complex in the category of topological sequential spectra.
A discussion in the context of algebraic model categories is in