I use the notation of [this question](https://mathoverflow.net/q/135738/24563). A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $\psi:[0,b]\to U$ are two continuous maps for some topological space $U$ with $\phi(a)=\psi(0)$, 
the map $\phi\*\psi:[0,a+b]\to U$ is the composition of the paths, $\phi$ on $[0,a]$ and $\psi$ on $[a,a+b]$. **All topological spaces are $\Delta$-generated. Therefore all following categories are locally presentable.** 


A **multipointed $d$-space** $X$ is a variant of Marco Grandis' $d$-spaces. It consists of a topological space $|X|$, a subset $X^0$ (of states) of $|X|$ and a set of continuous maps (called execution paths) $\mathbb{P}^{top}X$ from $[0,1]$ to $|X|$ satisfying the following axioms: 


1. for any $\phi\in \mathbb{P}^{top}X$, $\phi(0)$ and $\phi(1)$ belong to $X^0$
2. for any $\phi\in \mathbb{P}^{top}X$, a composite $[0,1] \cong^+ [0,1] \stackrel{\phi}\longrightarrow |X|$ belongs to $\mathbb{P}^{top}X$
3. if $\phi$ and $\psi$ are two execution paths, all composites like 
$[0,1] \cong^+ [0,2] \stackrel{\phi\*\psi}\longrightarrow |X|$ are execution paths.


Tu summarize, a multipointed $d$-space has not only a distinguished set of continuous paths but also a distinguished set of points (the other points are intuitively not interesting). Unlike Grandis' notion, the constant paths are not necessarily execution paths. It is one of the role of the cofibrant replacement of the model category structure constructed in [Homotopical interpretation of globular complex by multipointed d-space](http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html) to remove from a multipointed $d$-space all points which do not belong to an execution path. The cofibrant replacement cleans up the underlying space by removing the useless topological structure.


It turns out that the model structure constructed in [Homotopical interpretation of globular complex by multipointed d-space](http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html) is the left determined model category with respect to the set of generating cofibrations $\mathrm{Glob}(\mathbf{S}^{n-1}) \subset \mathrm{Glob}(\mathbf{D}^{n})$ for $n\geq 0$ and the map $\{0,1\} \to \{0\}$ identifying two points where $\mathbf{S}^{n-1}$ is the $(n-1)$-dimensional sphere, $\mathbf{D}^{n}$ the $n$-dimensional disk, and where $\mathrm{Glob}(Z)$ is the multipointed $d$-space whose definition is explained in the paper (I don't think that it is important to recall it in this post).


Now here is the question. I would be interested in considering the multipointed $d$-spaces $\vec{[0,1]^n}$ defined as follows


1. The underlying space is the $n$-cube $[0,1]^n$
2. The set of distinguished states is the set of vertices $\{0,1\}^n \subset [0,1]^n$
3. The set of execution paths is generated by the continuous maps from $[0,1]$ to $[0,1]^n$ such that of course $0$ and $1$ are mapped to a point of $\{0,1\}^n$ and such that these maps are nondecreasing with respect to each axis of coordinates.


The multipointed $d$-space $\partial\vec{[0,1]^n}$ is defined in the same way by removing the interior of the $n$-cube.


Using Vopenka's principle and a result of Tholen and Rosicky, there exists a left determined model category structure with respect to the set of generating cofibrations $\partial\vec{[0,1]^n} \subset \vec{[0,1]^n}$ with $n\geq 0$ and $R:\{0,1\}\to \{0\}$. 



> 
> How is it possible to remove Vopenka's principle from this statement ?
> 
> 
> 


This question is probably too complicated for a post but if someone could give me a starting point, I would be very grateful. It is the reason why I ask the question anyway. Note: the presence of the map $\{0,1\}\to \{0\}$ in the set of generating cofibrations is not mandatory because I start considering in other parts of my work model structures where I remove this map from the set of generating cofibrations.



Euclidean model structure on multipointed $d$-spaces
As mentioned by David White in the comment, I've recently [proved](https://arxiv.org/pdf/2011.13408.pdf) that left induced model structure exists (without any kind of large cardinal axiom) for any "tractable" class of cofibrations on a locally presentable category.


Tractable means that the class of cofibration is generated by a set of cofibration with cofibrant domain. It seems to me that the $\partial [0,1]^n$ can be obtained by gluing of their faces and so are cofibrant, so I think this applies to your category.


However I should also say that my construction isn't much more explicit than the one using Vopenka's principle, so I'm not sure it is very helphful for your question: it does not tell you anything about the model structure structure itself, like a nice cylinder functor would, except maybe a bound on the size of the generating trivial cofibrations.


A bit more precisely, you consider $W$ the "smallest Cisinski localizer", i.e. the smallest class of maps containing "trivial fibrations" closed under 2-out-of-3 and such that $W$ intersected with your class of cofibration is clsoed under pushout and transfinite composition, and you take for J (the generating trivial cofibrations) all cofibrations that are in $W$ and between $\kappa$-presentable object for some large enough cardinal $\kappa$ (I think in your case $\kappa = 2^\omega$ might work, though it might be $(2^\omega)^+$ depending on what is the presentability rank of the $[0,1]^n$ in your category, which I'm not quite sure about).


Then you have a model structure with $W$ as the weak equivalences, your cofibrations and $J$ as the generating trivial cofibration.


