Cell Complex

Idea

In algebraic topology, a CW complex is a way of building up types through an iterative process of gluing cells (disks) of increasing dimension.

Defintion

Let $\mathcal{C}$ be a category closed under colimits with a set $\mathcal{J}\subseteq \mathrm{map}\mathcal{C}$ ¹ of generating cofibrations.

A $\mathcal{J}$-cell complex in $\mathcal{C}$ is an object $X$ generated by a colimit of any bounded ordinal-indexed diagram, $D:{\mathrm{Ord}}_{<\alpha }\Rightarrow \mathcal{C}$

Examples

A CW-complex is a cell complex in Top with $\mathcal{J}$ coming from the Model Structure on topological spaces.

Footnotes

1. Map (Category Theory) ↩