homotopy invariant functor and homotopy equivalent spaces
1. Proposition
Let \(\mathcal{C}\) be a category, \(\mathcal{F}: \mathrm{Top} \rightarrow \mathcal{C}\) a functor and \((X, \mathcal{T}_X), (Y, \mathcal{T}_Y)\) be homotopy equivalent spaces.
TFAE:
- \(\mathcal{F}\) is a homotopy invariant functor
- \(\mathcal{F}(X), \mathcal{F}(Y)\) are isomorphic
2. Proof
2.1. 1) \(\implies\) 2)
By assumption, there exists a Homotopy equivalences \(f: X \rightarrow Y\) and homotopy inverse \(g: Y \rightarrow X\). Thus, \(g \circ f \circ \mathrm{id}_{X}\) and by assumption
\begin{align*} \mathcal{F}(g) \circ \mathcal{F}(f) =& \mathcal{F}(g \circ f) \\ =& \mathcal{F}(\mathrm{id}_{\mathcal{F}(X)}) \end{align*}analogously for \(f \circ g \sim \mathrm{id}_{Y}\) and hence \(\mathcal{F}(g), \mathcal{F}(f)\) are isomorphisms