If $X$ is a simply connected space and $\alpha\in\pi\_n(X)$ and $Y=X\cup\_{\alpha} e^{n+1}$.
and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q}
u,d)$ is a commutative model for $Y$,
which we denote by $M\_\alpha$.
If $f\colon X\rightarrow X\_1$
and $Y=X\cup\_{\alpha}e^{n+1}$ and $Y\_1=X\_1\cup\_{f^\*(\alpha)}e^{n+1}$,then we have a canonical map $g\colon X\_1\rightarrow Y\_1$ and the 
following commutative diagram (1)


\begin{array}{lll}
 X & \rightarrow &Y\\
 f\downarrow && \downarrow g\\
 X\_1 & \rightarrow & Y\_1
 \end{array}


Now,if $(\wedge V\_1,d)$ is the minimal model for $X\_1$ and so $(\wedge V\_1\oplus
\mathbb{Q}u,d)$ is a commutative model for $Y\_1$ and if $\phi\colon
(\wedge V\_1,d)\rightarrow (\wedge V,d)$
be the Sullivan representative for $f$,then we have induced map $\psi$ from
 $(\wedge V\_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$
 on $(\wedge V\_1)$ and identity on $u$.


Then we have the following diagram (2)


\begin{array}{lll}
 (\wedge V\_1\oplus\mathbb{Q} u,d) & \rightarrow &(\wedge V\_1,d)\\
 \psi\downarrow & & \downarrow \phi\\
 (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow &(\wedge V,d)\\
 \end{array}


Now,my questions are


a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y\_1$?


b. Is (2) a commutative model for (1)?



Naturality of commutative model for cell attachment in Rational Homotopy Theory
In rational homotopy theory models of homotopy push-outs are homotopy pull-backs in the category of CDGAs. This fact follows from the Quillen adjunction between model categories of spaces and CDGAs. In fact this is just a Mayer-Vietoris type argument.


(1) is a homotopy push-out: $X\rightarrow Y$ is a cofibration and (1) is a push-out.
To get an algebraic model of (1) you should build a surjective model of the map $f:X\rightarrow X\_1$ and then take the pull-back along the model of the map $X\rightarrow Y$ and you will get a model of (1). And that's what you have done.


