model category

for ∞-groupoids

# Contents

## Idea

An 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 this starting from $A$.

## 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)