CW-structure on an anima

Definition 3.3.5 (CW-structure). A CW-structure on an anima \(X\) consists of a sequence of animae \[ \emptyset = X^{-2}=X^{-1} \to X^0 \to X^1 \to X^2 \to \cdots \] called the skeleta of \(X\), together with an equivalence \(X \simeq \colim _n X^n\), satisfying:

(1)

The anima \(X^0\) is a set;

(2)

For each \(n \geq 1\), there exists a set \(I_n\) of \(n\)-cells and a pushout square of animae

The map \(\phi \) is called the attaching map for the \(n\)-cells.

We say that \(X\) is finite-dimensional if there are no \(n\)-cells for sufficiently large \(n\), and finite if it has finitely many cells in total.