For more details see also at CW-complex.
model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A 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 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.
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.
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 $\,$ |
Textbook account:
A discussion in the context of algebraic model categories is in