Euclidean model structure on multipointed $d$-spaces

Discussion in 'Mathematics' started by Philippe Gaucher, Aug 2, 2020 at 2:25 AM.

1. Philippe GaucherGuest

I use the notation of this question. 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 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 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.