I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call the elements of $G$ the reparametrisations. Consider a connected nonempty topological space $X$ containing two distinct points $\alpha$ and $\beta$. Consider a set $P$ of continuous paths $\gamma$ from $[0,1]$ to $X$ such that
- $\forall \gamma\in P,\gamma(0)=\alpha$
- $\forall \gamma\in P,\gamma(1)=\beta$
- $\forall \gamma\in P,\forall \phi \in G,\gamma\circ \phi \in P$
$P$ is equipped with the kelleyfication of the relative topology coming from the cartesian closedness.
Consider the category with one object associated with the group $G$. There is a contravariant diagram $D$ over $G$ defined by taking the unique object of $G$ to the space $P$ and any element $\phi$ to the precomposition by $\phi$, i.e. the continuous map from $P$ to itself defined by $\gamma\mapsto \gamma.\phi$.
Fact 1: The colimit of $D$ is equal to the quotient of $P$ up to reparametrisation.
Fact 2: The homotopy colimit of $D$ is equal to $EG\times_G P$.
Consider for $P$ the space generated by one continuous path from $\alpha$ to $\beta$ up to reparametrisation. $P$ is contractible. In this case, the homotopy colimit of $D$ is $BG$ which is not contractible because $G$ is far from being trivial.
How to fix the behaviour of the homotopy colimit to obtain $P$ up to weak equivalence instead of $EG\times_G P$ ?
As it is formulated, this question is not mathematical... Moreover, there is a way to fix this behavior by considering the translation category $EG$ and the obvious diagram with $P$ for each vertices and the map induced by the reparametrisations. We obtain a diagram $D'$. The colimit of $D'$ is again $P$ up to reparametrisations. And the homotopy colimit of $D'$ is weakly homotopy equivalent to $P$ because $EG$ is contractible and because $D'$ is weakly constant.
I can't work with this fix for reasons I cannot explain in a MathOverflow post (too long...).
I have another fix in mind. In the category $G$, it is missing the information about the homotopies between the reparametrisations, the homotopies between the homotopies between the reparametrisations, etc... It turns out that the group $G$ is naturally equipped with a topology coming from the topology of the space of continuous maps from $[0,1]$ to itself. So $G$ is certainly a topological group. And the underlying space is contractible. In other terms, $G$ can be viewed as a one-object category enriched over topological spaces and the unique space of morphisms is contractible. The diagram $D$ can be also viewed as a one-object category enriched over topological spaces, the set of maps of $D$ from $P$ to itself can be equipped with the topology inherited from the space of all continuous maps from $P$ to itself.
Now I can ask the mathematical question:
Is it true that using the enrichment, some "enriched" homotopy colimit of $D$ is weakly equivalent to $P$ ? And that the "enriched" colimit of $D$ is still $P$ up to reparametrisation ?
Motivation: This question is an attempt to extract a technical problem from a problem which is much too long to be a question in MathOverflow. I hope that I will be successful in this extraction. I know nothing in enriched homotopy colimits so before reading things about them, I would like to know if it is a good direction and I would appreciate to have some pointers.