Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection ([Proof](https://math.stackexchange.com/a/3838971))
$$\operatorname{colim}(D) \cong \pi\_0 (\textstyle\int\_\mathcal{C}D).$$


1. Is there any known application or significance of the homotopy groups of $\int\_\mathcal{C}D$ (meaning the homotopy groups of the geometric realisation of $\int\_\mathcal{C}D$ at a point $x\in\pi\_0(\textstyle\int\_\mathcal{C}D)$)?
2. What about its "fundamental monoid" $\pi\_0(\mathrm{Hom}\_{\mathrm{N}\_{\bullet}(\int\_\mathcal{C}D)}(x,x))$ at a point $x\in\pi\_0(\textstyle\int\_\mathcal{C}D)$? (Is the fundamental group of the geometric realisation of $\textstyle\int\_\mathcal{C}D$ at $x$ just the groupification of this?)
3. Can these be described in terms of other well-known notions in ($\infty$-)category theory?



Homotopy groups of categories of elements as higher colimits
To answer these questions, the best is to note that $|\int\_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, *viewed as a functor with values in the $\infty$-category of spaces*, namely, along the inclusion $Set\to Spaces$.


So


1. Insofar, as you believe that this colimit has significance, so do its homotopy groups. For example, if $C= \Delta^{op}$, $D$ is a simplicial set and $|\int\_{\Delta^{op}} D|$ is the homotopy type of this simplicial set; if $C= BG$ for some group $G$, $D$ is a set with a $G$-action, and this homotopy type is $D\_{hG}$, the homotopy orbits of $D$ under this action, etc.
2. The "fundamental monoid" is maybe not so relevant, in a sense because it is easily computed. First, note that it depends on more than a class in $\pi\_0$ ($\pi\_0$ inverts all morphisms), but really on an object in $\int\_C D$. Second, given such an object $x$, it lives in a fiber over some $c\in C$, and consequently, $\hom\_{\int\_CD}(x,x) = \coprod\_{f\in \hom\_C(c,c)} \hom\_{D(c)}(f\_!x,x)$ where $f\_! : D(c)\to D(c)$ is $D(f)$. The monoid structure is "fiberwise". But in a sense it can be easily read off of $D$, whereas understanding $|\int\_C D|$ is typically more complicated.


In particular, note that the group-completion of this monoid is *not* (in general) $\pi\_1(|\int\_CD|, x)$, because the latter also involves "visiting" other objects than $x$. For example if $D$ is the constant functor with value a point, then $\int\_C D=C$, and you'd be claiming that for an arbitrary $C$, $\pi\_1(|C|,c) = $ the goup-completion of $\hom\_C(c,c)$. But any group can be realized as $\pi\_1(|P|)$ for a *poset* $P$ where, in particular, $\hom\_P(p,p) = $ a point for any $p\in P$.


3. I think the very first part of my question answers this one.


