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

References

A discussion in the context of algebraic model categories is in

Revised on December 30, 2013 00:03:00 by Tim Porter (90.24.132.252)