It is the motivation of the question Examples of non Quillen-equivalent model categories having equivalent homotopy categories. I did not give at first the motivation because i don't think that people are familiar with the objects.
The two model categories are the one of multipointed $d$-spaces of the paper Homotopical interpretation of globular complex by multipointed d-space, and the category of flows of the paper A model category for the homotopy theory of concurrency.
I will only describe the categorical level. Because I don't even know any zig-zag of categorical adjunctions between them. The fact that the two homotopy categories are equivalent is Theorem V.4.2 of Comparing globular complex and flow.
At first, I give some notations: 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.
1) 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:
- for any $\phi\in \mathbb{P}^{top}X$, $\phi(0)$ and $\phi(1)$ belong to $X^0$
- for any $\phi\in \mathbb{P}^{top}X$, a composite $[0,1] \cong^+ [0,1] \stackrel{\phi}\longrightarrow |X|$ belongs to $\mathbb{P}^{top}X$
- if $\phi$ and $\psi$ are two execution paths, all composites like $[0,1] \cong^+ [0,2] \stackrel{\phi*\psi}\longrightarrow |X|$ are execution paths.
2) A flow is a variant of the notion of small categories enriched over topological spaces : one removes the identity maps. So a flow $X$ consists of a set of states $X^0$, a topological space $\mathbb{P}_{\alpha,\beta}X$ for each pair $(\alpha,\beta)$ of $X^0\times X^0$, and an associative continuous map $*:\mathbb{P}_{\alpha,\beta}X \times \mathbb{P}_{\beta,\gamma}X \to \mathbb{P}_{\alpha,\gamma}X$.
3) The functor $cat$ ("categorification") from multipointed $d$-spaces to flow takes each multipointed $d$-space $(|X|,X^0,\mathbb{P}^{top}X)$ to the flow $(X^0,\mathbb{P}X)$ where $\mathbb{P}X$ is the set $\mathbb{P}^{top}X$ up to strictly increasing reparametrization equipped with a natural topology for this setting (the $\Delta$-kelleyfication of the compact-open topology, the one giving the internal hom of the category of $\Delta$-generated spaces). This functor is described in Section 7 of Homotopical interpretation of globular complex by multipointed d-space. And Theorem 7.2 proves that it has no right-adjoint.
The functor $cat$ does preserve homotopy colimit and seems to be a little bit like a left adjoint up to homotopy. But the deep reason behind Theorem 7.2 is that composition of paths is only associative up to homotopy for multipointed $d$-spaces and is strictly associative for flows. So a possible idea to connect these categories by a zig-zag of categorical adjunctions is to strictify the composition of paths on multipointed $d$-spaces by introducing the multipointed Moore $d$-spaces, in which the execution paths will have a length. Here is the notion I am working with:
A multipointed Moore $d$-space $X$ consists of a topological space $|X|$, a subset $X^0$ (of states) of $|X|$ and a set of continuous maps (called execution paths) $\mathbb{P}^{Moore}X$ from $[0,\ell]$ for some $\ell>0$ to $|X|$ satisfying the following axioms:
- for any $\phi\in \mathbb{P}^{Moore}X$ of length $\ell$, $\phi(0)$ and $\phi(\ell)$ belong to $X^0$
- for any $\phi\in \mathbb{P}^{Moore}X$ of length $\ell$, a composite $[0,\ell] \cong^+ [0,\ell] \stackrel{\phi}\longrightarrow |X|$ belongs to $\mathbb{P}^{Moore}X$
- if $\phi$ and $\psi$ are two execution paths of length $\ell_1$ and $\ell_2$ respectively, the continuous path $[0,\ell_1+\ell_2] \stackrel{\phi*\psi}\longrightarrow |X|$ is an execution path of length $\ell_1+\ell_2$.
4) There is an adjunction between multipointed Moore $d$-spaces and multipointed $d$-spaces. The functor $N$ (for normalization) from multipointed Moore $d$-spaces to multipointed $d$-spaces is defined by keeping the same underlying topological space, the same set of states, and by normalizing all paths. So an execution path $[0,1]\to |X|$ of $N(X)$ is a composite of the form $[0,1]\cong^+[0,\ell]\to |X|$ where the right-hand map is an execution path of length $\ell$ of $X$. The functor $N$ has a right adjoint, that I call the denormalization functor $D$. It also preserves the underlying topological space and the set of states. And the execution paths of $D(Y)$ are all composites of the form $[0,\ell]\cong^+ [0,1] \to |Y|$ where the right-hand map is an execution path of $Y$.
5) There is a functor from multipointed Moore $d$-spaces to flows which consists of forgetting the underlying topological space. It takes the multipointed Moore $d$-space $(|X|,X^0,\mathbb{P}^{Moore}(X))$ to the flow $(X^0,\mathbb{P}^{Moore}(X))$. I believed that this functor would be a left adjoint because on both sides, the composition of paths is now associative. But I cannot find the right adjoint and I suspect that it does not exist. Let me explain why "with my hands". Note that all categories are locally presentable so colimit-preserving and being a left adjoint are equivalent by the dual of the Special Adjoint Functor Theorem. And also, it is important to keep in mind that the forgetful functor taking a multipointed Moore $d$-space $X$ to the underlying set of $|X|$ is topological. So to calculate a colimit of a diagram of multipointed Moore $d$-spaces $X_i$, first we take the colimits of the underlying topological spaces and of the sets of states, and we equip the result with the final structure, which consists of all execution paths which are compositions of execution paths of some $X_i$. This is not the same thing as taking the set of all free compositions, which is how is calculated a colimit in the category like the one of flows, because it is more like an union of paths in a topological space which is a colimit of topological spaces.
