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, to be called the generating cofibrations.

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

Now:

Definition

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 (cell attachments) of (coproducts of) the generating cofibrations in $\mathcal{I}$.

More generally, 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.

Remark

Definition is often stated without explicitly allowing forming coproducts of generating cofibrations (formed in the arrow category) before taking their pushouts (e.g. Hovey 1999 Def. 2.1.9 and Hirschhorn 2002 Def. 10.5.8) while in practice authors often do consider this (e.g. Hirschhorn 2015 Rem. 4.4).

The two variants are equivalent assuming the axiom of choice in the form of the well-ordering theorem: With that theorem/assumption, the set of coproducts to be pushed out may be equipped with a well-order and with that the pushout of the coproduct is equivalently the transfinite composition of pushouts of summands of the coproduct in that order.

(Note that the axiom of choice is used already for basic statements in this context, like the very existence of the classical model structure on topological spaces, see there this Lemma).

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 in general topology:

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

and in the context of the classical model structure on topological spaces:

• Philip Hirschhorn, §4 in: The Quillen model category of topological spaces, Expositiones Mathematicae 37 1 (2019) 2-24 [arXiv:1508.01942, doi:10.1016/j.exmath.2017.10.004]

and in the general context of (cofibrantly generated) model category theory:

• Mark Hovey, Def. 2.1.9 in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, Google books]

• Philip Hirschhorn, Def. 10.5.8 in: Model Categories and Their Localizations, Math. Survey and Monographs 99, AMS (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]

A discussion in the context of algebraic model categories:

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