nLab
cell complex

Idea

In cell complex is an object in a category which is obtained by successively “gluing cells” via pushouts.

Definition

Let $C$ be a category with colimits and equipped with a set $ℐ\subset \mathrm{Mor}(C)$ of morphisms.

In practice $C$ is usually a cofibrantly generated model category with set $ℐ$ of generating cofibrations and set $𝒥$ of acyclic generating cofibrations.

An $ℐ$-cell complex in $C$ is an object $X$ which is connected to the initial object $\varnothing \to X$ by a transfinite composition of pushouts of the generating cofibrations in $ℐ$.

A relative $ℐ$-cell complex (relative to any object $A$) is any morphism $A\to X$ obtained this starting from $A$.

Examples

• 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.

References

A discussion in the context of algebraic model categories is in

• Emily Riehl, Cellularity in algebraic model structures (blog post)