nLab
cell complex

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

Idea

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

Definition

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 this starting from AA.

Examples

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\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

A discussion in the context of algebraic model categories is in

Revised on May 2, 2017 13:18:00 by Urs Schreiber (131.220.184.222)