# Representation of fundamental groupoid as $2$-sheaf

By https://arxiv.org/abs/1406.4419 (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $$X$$, the $$2$$-functor $$\text{Top}(X)\rightarrow \text{Gpd}, \quad (U\rightarrow X)\mapsto \Pi_1(U)$$ is a $$2$$-cosheaf, in fact the terminal one. In particular, it follows that $$\text{Top}(X)^{op}\rightarrow \text{Gpd}, \quad (U\rightarrow X)\mapsto \text{Hom}(\Pi_1(U),\mathcal{G})$$ is a $$2$$-sheaf, where $$\mathcal{G}$$ is a groupoid. However, if we assume $$X$$ is a manifold (probably locally simply connected does the job, but I want to be cautious), then for any point $$x\in X$$, open simply connected sets $$U\ni x$$ are final in all open neighborhoods of $$x$$. Hence the stalk of this $$2$$-sheaf is $$\text{colim}_{U\ni x}\text{Hom}(\Pi_1(U),\mathcal{G})\cong \text{colim}_{U\ni x,\text{ simp. conn}}\text{Hom}(\Pi_1(U),\mathcal{G})\cong \text{colim}_{U\ni x,\text{ simp. conn}}\text{Hom}(\{x\},\mathcal{G})$$ where I denote by $$\{x\}$$ the groupoid whose objects are $$\{x\}$$ and has no automorphism (I think this is called the trivial groupoid). If we assume that $$\mathcal{G}$$ is a group $$G$$, then we get that the stalk consits only of the trival morphism.

However, having trivial stalks means that $$\text{Hom}(\Pi_1(-),G)$$ is trivial $$2$$-sheaf. But surely this can not be true, as that would mean $$\text{Hom}(\Pi_1(X),\text{Gl}_n(\mathbb{C})$$ is trivial, which is not true. Hence there is a gap/flaw in my reasoning, but I don't see it.

Basically, as $$\Pi_1(-)$$ is the terminal $$2$$-cosheaf, $$\text{Hom}(\Pi_1(-),\mathcal{G})$$ is the initial $$2$$-sheaf, but that would mean that there are almost no representations of the fundamental group, which seems wrong to me.

• Do you mean $\hom$ with values in $Gpd$ or with values in $Set$ ? (note, by the way, that you never use, in your "proof", that $\Pi_1(-)$ is the terminal $2$-cosheaf) Sep 23 '20 at 19:40
• I mean $\text{hom}$ in $Gpd$ to get the sheaf property. Sep 23 '20 at 19:45
• But then $\hom(\{x\},\mathcal G)\cong \mathcal G$. So you probably want $\hom$ to take values in $Set$, so the thing you're after is the fact that $\Pi_1(-)$ is a $1$-cosheaf (which is also claimed in the paper) Sep 23 '20 at 19:52
• Shouldn't $\{x\}$ be the initial groupoid? So in partiuclar $\text{hom}(\{x\},\mathcal{G})$ would be a singleton as a set. Sep 23 '20 at 20:01
• So you're looking at $\hom$ valued in $Set$, which is what I first asked (it's not the initial groupoid, it's the terminal one -in particular $\hom(\{x\}, \mathcal G) \cong Ob(\mathcal G)$ in general, although that doesn't change your question when $\mathcal G= BG$ has only one object) Sep 23 '20 at 20:03

First of all, note that you haven't used the fact that $$\Pi_1(-)$$ was the terminal $$2$$-cosheaf, just that it was a ($$2$$-)cosheaf.

Then, as I pointed out in the comments, there's a question of whether you're considering the usual $$\hom$$ functor into $$Set$$, or the internal $$\hom$$ with values in $$Gpd$$.

• If you're considering the usual, external $$\hom$$, then the question is more about whether $$\Pi_1(-)$$ is a cosheaf than a $$2$$-cosheaf. But note that if you're considering $$\hom$$ into $$Set$$, then $$\hom$$ is not equivalence invariant. In particular, $$\hom(\Pi_1(U),\mathcal G) \not\cong \hom(\{x\},\mathcal G)$$ even when $$U$$ is simply-connected. For instance if you take for $$\mathcal G$$ an indiscrete groupoid on a set $$S$$ of objects, the former is $$\hom_{Set}(U,S)$$, while the latter is $$\hom_{Set}(\{x\},S)$$. So your stalk computation doesn't work anymore.

(to perhaps get a better grasp on this, compare $$\hom(2^{in},BG)$$ and $$\hom(\{x\},BG)$$ where $$2^{in}$$ is the indiscrete groupoid on $$2$$ objects, and $$BG$$ the one-object groupoid associated to the group $$G$$)

• If you're considering the internal $$\hom$$, and looking at stacks/$$2$$-sheaves (so that you do have invariance under equivalence), then $$\hom(\{x\},\mathcal G)\cong \mathcal G$$, so it's definitely nontrivial.

(moreover, note that in this context of stacks, or $$2$$-sheaves, you need to be more careful with colimits: this would probably be a $$2$$-colimit, so you'd need to check that the subcategory of simply-connected neighbourhoods of $$x$$ is final in the $$2$$-categorical sense in the category of neighbourhoods. This is a rather stronger condition than just finality in the $$1$$-categorical sense, and here I'm not sure whether it would be satisfied in general - I'm definitely not an expert on stacks and $$2$$-categories so I can't say much more)

• Directed sets have the wonderful property that cofinal = homotopy cofinal, so it isn't a problem. Sep 23 '20 at 22:17
• I'm still confused by your statement that $hom_{gpd}(\{x\},\mathcal{G})\cong \mathcal{G}$. To define a functor $\{x\}\rightarrow \mathcal{G}$ corresponds to a choice of $\mathcal{G}$, so therefore shouldn't $hom_{gpd}(\{x\},\mathcal{G})\cong \text{Ob}(\mathcal{G})$? Sep 23 '20 at 23:59
• You should read more carefully : there are two flavours of $\hom$, one which is set-valued (for this one, you are correct, and I explain in the first bullet point why the argument doesn't work for this one), and one which is groupoid-valued : this is a different functor Sep 24 '20 at 7:09
• @ZhenLin : the poset of simply-connected neighbourhoods if $x$ isn't directed in general, is it ? Ah maybe up to a cofinal subset, in manifolds ? Sep 24 '20 at 7:36
• The poset of open neighbourhoods, ordered by reverse inclusion, is directed. The simply connected open neighbourhoods are cofinal in this poset, for a manifold or more generally a locally simply connected space. Sep 24 '20 at 7:55