I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $\mathrm{Top}$ of $\Delta$-generated topological spaces where $\mathcal{M}$ has for objects the set of strictly positive real numbers and where the space $\mathcal{M}(L,L')$ is the space of nondecreasing surjective maps from $[0,L]$ to $[0,L']$.
An important class of examples of $\mathcal{M}$-spaces is what I call the quasicofibrant $\mathcal{M}$-spaces. Roughly, an $\mathcal{M}$-space $X$ is quasicofibrant when it really comes from a space of continuous paths in a topological space and if the maps of the diagram $X$ are really reparametrization maps.
Consider a topological space $U$, a set of continuous paths $P$ from $[0,1]$ to $U$ going from $\alpha$ to $\beta$ fixed such that $P$ is closed by reparametrizations by the maps of $\mathcal{M}(1,1)$; then one can define from $P$ an $\mathcal{M}$-space $X(P)$ by $X(P)(L)$ being the space of composites $\gamma\phi$ where $\gamma\in P$ and $\phi\in \mathcal{M}(L,1)$. Note that $X(P)(1)=P$. Such an $\mathcal{M}$-space $X(P)$ is by definition quasicofibrant.
The terminology quasicofibrant comes from the following fact. Let us equip $[\mathcal{M}^{op},\mathrm{Top}]_0$ with the projective q-model structure : the fibrations are the objectwise q-fibrations and the weak equivalences are the objectwise weak homotopy equivalences.
Proposition: Every projective q-cofibrant $\mathcal{M}$-space is quasicofibrant. The converse is false.
There is a biclosed semimonoidal structure on $[\mathcal{M}^{op},\mathrm{Top}]$ defined as follows: $$D\otimes E = \int^{(L_1,L_2)} \mathcal{M}(-,L_1+L_2) \times D(L_1)\times E(L_2).$$
Proposition: Let $f:A\to B$ and $g:C\to D$ be two objectwise weak homotopy equivalences. Assume that $A,B,C,D$ are projective q-cofibrant. Then $f\otimes g$ is an objectwise weak homotopy equivalence.
The proof goes as follows. We write $f \otimes g = (f\otimes 1_D) (1_A \otimes g)$. It then suffices to prove e.g. that $1_A \otimes g$ is a weak equivalence. To conclude, it suffices to observe that the functor $A\otimes -:[\mathcal{M}^{op},\mathrm{Top}]_0\to [\mathcal{M}^{op},\mathrm{Top}]_0$ is a left Quillen adjoint. This relies on Theorem 9.4 of my paper https://doi.org/10.32408/compositionality-3-3.
The question is
Is the tensor product of two objectwise weak homotopy equivalences of $\mathcal{M}$-spaces an objectwise weak homotopy equivalence of $\mathcal{M}$-spaces ?
I am not convinced that it is true in full generality. However, I suspect that it is true by replacing the hypothesis $A,B,C,D$ projective q-cofibrant by the hypothesis $A,B,C,D$ quasicofibrant :
Is the tensor product of two objectwise weak homotopy equivalences between quasicofibrant $\mathcal{M}$-spaces an objectwise weak homotopy equivalence of $\mathcal{M}$-spaces ?
Why do I think that it is true ? When $D$ is quasicofibrant, the canonical map $D(1) \to \mathrm{colim} D$ induced by the universal property of the colimit is the map taking a continuous path to its equivalence class up to reparametrization. In many cases, beyond the projective q-cofibrant case, this map is a weak homotopy equivalence. It is unclear whether it is true in full generality.
Let $D$ be a quasicofibrant $\mathcal{M}$-space ? Is the canonical map $D(1) \to \mathrm{colim} D$ a weak homotopy equivalence ?
Call the quasicofibrant $\mathcal{M}$-spaces $D$ such that $D(1) \to \mathrm{colim} D$ is a weak homotopy equivalence good. All projective q-cofibrant $\mathcal{M}$-spaces are good. Note that all $D(L)$ for $L>0$ are homeomorphic to $D(1)$, so the choice of $1$ does not matter. We obtain the
Proposition: The tensor product of two objectwise weak homotopy equivalences between good $\mathcal{M}$-spaces is an objectwise weak homotopy equivalence again.
The proof goes as follow. By Proposition 5.18 of my paper https://doi.org/10.32408/compositionality-3-3, there is for all $\mathcal{M}$-spaces $D$ and $E$ the homeomorphism $$(\mathrm{colim} D) \times (\mathrm{colim} E) \cong \mathrm{colim} (D \otimes E).$$ The proof is complete because the binary product of two weak homotopy equivalences is again a weak homotopy equivalence.
