Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
Consider the free cdga on one generator $\Lambda(t, dt)$, where \linebreak$\deg{t}=0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. Notice that this cdga is isomorphic to \linebreak$\Lambda(D(0))$ as defined in the previous section. We define two \emph{endpoint maps} as follows:
this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t)=0$ and $d_1(1-t)=1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt)\tensor X \to\k\tensor X \tot{\iso} X$.
In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt)\tensor X$ is a path object for $X$. We can see as follows that it is a very good path object (\DefinitionRef{path_object}). First note that $\Lambda(t, dt)\tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y)\in X \oplus X$ take $t \tensor x +(1-t)\tensor y$). Secondly we note that $\Lambda(t, dt)=\Lambda(D(0))$ and hence $\k\to\Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to\Lambda(t, dt)\tensor X$ is a (necessarily trivial) cofibration.
The results from model categories immediately imply the following results. Here we use Lemma \ref{lem:left_homotopy_properties}, \ref{lem:right_homotopy_properties} and \ref{lem:weak_strong_homotopy}.
\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X]\to[A, X]$ is a bijection.
\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X]\to[A, Y]$ is a bijection.
\item Let $A$ and $X$ both be cofibrant, then $f: A \we X$ is a weak equivalence if and only if $f$ is a strong homotopy equivalence. Moreover, the two induced maps are bijections:
By \RemarkRef{cdga-mc5a-left-inverse} we can generalize the second item to arbitrary weak equivalences: If $A$ is cofibrant and $f : X \to Y$ a weak equivalence, then the induced map $f_\ast : [A, X]\to[A, Y]$ is a bijection, as seen from the following diagram:
\[\xymatrix{
& [A, X \tensor C] \ar[dl]_{\overline{\phi}_\ast}^\iso\ar[dr]^{\psi_\ast}_\iso&\\
Now we know that $H(d_0)= H(d_1) : H(\Lambda(t, dt))\to\k$ as $\Lambda(t, dt)$ is acyclic and the induced map sends $1$ to $1$. So the two bottom maps in the diagram are equal as well. Now we conclude $H(f)= H(d_1)H(h)= H(d_0)H(h)= H(g)$.