# Naturality of minimal model of a fibre bundle

$$\require{AMScd}$$ For rational fibrations $$F \rightarrow E \rightarrow B$$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$\begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD}$$ where all vertical arrows are minimal models. This is sometimes called a $$\wedge$$-minimal $$\wedge$$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $$G$$-structure'.

Suppose that you have two fibrations and a morphism of fiber sequences: $$\require{AMScd} \begin{CD} F @>>> E @>p>> B \\ @VhVV @VgVV @VfVV \\ F' @>>> E' @>p'>> B' \end{CD}$$ where for example you have base points $$x_0 \in B$$, $$x_0' \in B'$$ such that $$f(x_0) = x_0'$$, $$F = p^{-1}(x_0)$$ and $$F' = (p')^{-1}(x_0')$$.
Suppose that you are given a model for the fibration $$\begin{CD} (\Lambda W, \bar{d}) @<<< (\Lambda V \otimes \Lambda W, D) @<<< (\Lambda V, d) \\ @V{\bar{m}}VV @VMVV @VmVV \\ A_{PL}(F) @<<< A_{PL}(E) @<{p^*}<< A_{PL}(B) \end{CD}$$ and the same thing with primes everywhere. Suppose also that you have a model $$\psi : (\Lambda V', d') \to (\Lambda V, d)$$ for $$f$$, i.e. $$m\psi = f^* m'$$.
Then you get a map from the pushout: $$\xi : (\Lambda V, d) \otimes_{(\Lambda V', d')} (\Lambda V' \otimes \Lambda W', D) \to A_{PL}(E).$$ You can rewrite the domain of this pushout as $$(\Lambda V \otimes \Lambda W', \delta)$$ for a differential $$\delta$$. You moreover have a commutative diagram: $$\begin{CD} (\Lambda V, d) @>>> (\Lambda V \otimes \Lambda W', \delta) @>>> (\Lambda W', \bar{d'} \\ @VVV @VVV @VVV \\ A_{PL}(B') @>>> A_{PL}(E) @>>> A_{PL}(F) \end{CD}$$ The proposition is that if $$h^* : H^*(F') \to H^*(F)$$ is an isomorphism (with rational coefficients), then this last diagram is a model for the fibration $$F \to E \to B'$$.