homotopy invariant functor and interval

1. Proposition

Let \(\mathcal{C}\) be a category and \(\mathcal{F}: \mathrm{Top} \rightarrow \mathcal{C}\) a functor from category Top

TFAE:

\(\mathcal{F}\) is homotopy invariant.
for a topological space \(X\) and the product topology, it follows, that the projection under the functor \(\mathcal{F}(X) \sim \mathcal{F}(X \times [0,1])\)

\begin{align*} \mathcal{F}(\pi): \mathcal{F}(X \times I) \rightarrow \mathcal{F}(X) \end{align*}

is an isomorphism

2. Proof

2.1. 1) \(\implies\) 2)

see: product with the interval homotopy equivalent

2.2. 2) \(\implies\) 1)

Let

\begin{align*} \iota_t: X \rightarrow& X \times I \\ x \mapsto& (x,t) \end{align*}

Then

\begin{align*} \pi \circ \iota_t = \mathrm{id} \end{align*}

and hence

\begin{align*} \mathcal{F}(\mathrm{id}) =& \mathcal{F}(\pi \circ \iota_t) \\ =& \mathcal{F}(\pi) \circ \mathcal{F}(\iota_t) \\ \end{align*}

Here by assumption, \(\mathcal{F}(\pi)\) is an isomorphism, thus by right inverse of an isomorphism as left inverse it follows, that \(\mathcal{F}(\iota_t)\) is an inverse.

By assumption, there exists a homotopy

\begin{align*} H: X \times I \rightarrow Y \end{align*}

where

\begin{align*} H \circ \iota_0 =& f \\ H \circ \iota_1 =& g \end{align*}

Hence

\begin{align*} \mathcal{F}(f) =& \mathcal{F}(H \circ \iota_0) \\ =& \mathcal{F}(H) \circ \mathcal{F}(\iota_0) \\ =& \mathcal{F}(H) \circ \mathcal{F}(\iota_1) \\ =& \mathcal{F}(H \circ \iota_1) \\ =& \mathcal{F}(g) \end{align*}