nLab cell complex


For more details see also at CW-complex.


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




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


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

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

An \mathcal{I}-cell complex in CC is an object XX which is connected to the initial object X\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 AA) is any morphism AXA \to X obtained like this starting from AA.

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


examples of universal constructions of topological spaces:

\, 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

Last revised on August 17, 2022 at 13:53:32. See the history of this page for a list of all contributions to it.