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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?
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1 Answer 1

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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.
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  • $\begingroup$ 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? $\endgroup$
    – Emily
    Commented Apr 15, 2023 at 19:05
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    $\begingroup$ 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 $\endgroup$ Commented Apr 15, 2023 at 19:26
  • $\begingroup$ Thank you so much Maxime, this is perfect :) $\endgroup$
    – Emily
    Commented Apr 15, 2023 at 19:47

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