Contents

model category

for ∞-groupoids

# Contents

## 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.

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.

## Examples

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

A discussion in the context of algebraic model categories is in

Last revised on March 21, 2021 at 04:12:01. See the history of this page for a list of all contributions to it.