# Homotopy groups of categories of elements as higher colimits

Given a diagram of sets $$D\colon\mathcal{C}\to\mathsf{Set}$$, we have a bijection (Proof) $$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$

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?

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 $$p\in P$$.

1. I think the very first part of my question answers this one.
• Thanks, Maxime, this is really great! Could I ask two follow-up questions? 1) Why is $|\int_CD|$ the colimit of $C\xrightarrow{D}\mathsf{Set}\xrightarrow{\iota}\mathsf{Spaces}$? 2) We have $\pi_k|S_\bullet|\cong\pi_k\mathrm{Ex}^\infty(S_\bullet)$ for a simplicial set $S_\bullet$, so it doesn't matter whether we use geometric realisations or simplicial homotopy groups with $\mathrm{Ex}^\infty$ for this, right? Commented Apr 15, 2023 at 19:05
• The answer to 2) is yes. It is a bit more complicated to answer 1), although I can give you references and a name. The name is "Thomason's theorem" ; cf. e.g. these exercises : math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf; and for an $\infty$-category perspective, this is Corollary 3.3.4.6. in HTT Commented Apr 15, 2023 at 19:26
• Thank you so much Maxime, this is perfect :) Commented Apr 15, 2023 at 19:47