nLab cell complex

For more see also at CW-complex.

Idea

A 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 $\mathcal{I} \subset Mor(C)$ 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 like this starting from $A$.

A finite cell complex or countable cell complex is a cell complex with a finite set or a countable set of cells, respectively.

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.

• A cell spectrum is a cell complex in the category of topological sequential spectra.

Related concepts

• cellular category

• cellular map

• CW complex

• vertex, edge, face

• dimension of a cell complex

• locally finite cell complex

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits	$\phantom{AAAA}$colimits
$\,$ point space$\,$	$\,$ empty space $\,$
$\,$ product topological space $\,$	$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$	$\,$ quotient topological space $\,$
$\,$ fiber space $\,$	$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$	$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
	$\,$ cell complex, CW-complex $\,$

References

Textbook account:

• Rudolf Fritsch, Renzo Piccinini, Cellular structures in topology, Cambridge University Press (1990) [doi:10.1017/CBO9780511983948, pdf]

A discussion in the context of algebraic model categories is in

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