Hello !

If I a have a grothendieck Site (C,J), I can consider :

  • The Stacks on (C,J) : category fibered in groupoid over C which statisfy suitable descent condition with respect to the covering sieve of J...

  • Internal Groupoid in the topos Sh(C,J).

Clearly those two notions are very close. but not exactly equivalent. I found a lot of book/article/thesis who define stacks but none of them were in term of the internal logic of the topos and it appears to me that some properties would be a lot more natural if they were stated in terms of an internal groupoid and the internal logic.

My feeling is that stacks corresponds to internal groupoids "up to weak categorical equivalences" (we invert the functor between groupoid which are internally fully faithfull and essentially surjective). But that this equivalence rely on the (external) axiom of choice. Am I right ? (and if I am, how can we make this more explicit ? )

Oh, and What about Higher Stacks ?

Thank you !


1 Answer 1


The $2$-category of internal groupoids in $Sh(C,J)$ is equivalent to the $2$-category of sheaves of groupoids on $(C,J)$. To see this, given a groupoid $G$ in sheaves,the strict $2$-functor $$C \mapsto \left(G_0\left(C\right) \rightrightarrows G_1\left(C\right)\right)$$ is a sheaf of groupoids. So, there is a $2$-functor

$$F:Gpd(Sh(C,J)) \to Sh_J(C,Gpd) \to St(C,J)$$ which sends $$G \mapsto a(\left(G_0\left(\mspace{3mu}\cdot \mspace{3mu} \right) \rightrightarrows G_1\left(\mspace{3mu}\cdot \mspace{3mu}\right)\right),$$ where $a$ is the stackification $2$-functor.

I claim that $a$ is essentially surjective. Indeed, by the Grothendieck construction, one can see easily that every stack is equivalent to a strict $2$-functor from $C^{op}$ into the $2$-category $Gpd$ of groupoids. Such a strict $2$-functor, is the same as a functor into the $1$-category of groupoids, which is the same as a groupoid object in presheaves (by the same argument). One can sheafify this into a sheaf of groupoids, and then apply $F$ to it, and the result is equivalent to the starting stack.

The fact fact that stacks are equivalent to groupoids in sheaves up to "weak categorical equivalence" can be found in (I believe) in Bunge's Stack completions and Morita equivalence for categories in a topos, but can also be proven directly by examining $F$. As far as the internal logic and the axiom of choice, you are on to something, but I am not a logician. But, what I can say, is that the axiom of choice fails, in the sense that not every epimorphism splits in a topos, and if every epi DID split, then the notion of Morita equivalence would agree with the notion of categorical equivalence of internal groupoid objects.

What about higher stacks? Well, one can see this by the fact that given a site $(C,J),$ there is a model structure on simplicial $J$-sheaves whose associated $(\infty,1)$-category is equivalent to the $\infty$-topos of $\infty$-sheaves. Ok, so how does this answer your question? Well, for simplicity, lets work with the projective model structure. There is the global projective model structure on $Sh_J(C)^{\Delta^{op}}$, which does not yet model $\infty$-sheaves and its associated infinity category corresponds to its fibrant objects which are those simplicial sheaves which are object-wise Kan-complexes, i.e., infinity groupoids. It follows that its associated $(\infty,1)$-category is the $(\infty,1)$-category of $\infty$-groupoid objects in sheaves. One can then left-Bousfield localize this model structure with respect to $J$-covering sieves, and one arrives as the local model strucrure on simplicial sheaves, and its associated $\infty$-category is the $\infty$-topos of $\infty$-sheaves. At the level of $\infty$-categories, this localization corresponds to a left-exact localization $$Sh_\infty(C,J) \leftrightarrows\infty-Gpd\left(Sh(C,J)\right).$$ The left-adjoint corresponds to fibrant replacement at the level of model categories, and $\infty$-stackfiication at the level of $\infty$-categories. So, every higher stack arises from an internal higher groupoid in $Sh(C,J)$.

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    $\begingroup$ Does the local model structure on simplicial sheaves really present the $(\infty,1)$-topos of sheaves on $(C,J)$, or only its hypercompletion? $\endgroup$ Apr 14, 2012 at 21:40
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    $\begingroup$ @Mike: I'm pretty sure your point of confusion is that in the literature, for a while, it was traditional to left-Bousefield localize with respect to those morphisms which induce isomorphisms on homotopy sheaves. This of course will give you the $(\infty,1)$-topos of hypersheaves. However, you can instead localize with respect to the covering sieves, and this will give you the $(\infty,1)$-topos of $\infty$-sheaves $\endgroup$ Apr 15, 2012 at 3:01
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    $\begingroup$ @Mike: Btw, you're totally right of course, it models hypersheaves (I just noticed this old post). However, one can then localize again to obtain $n$-truncated sheaves for any finite $n$. Hypercompletion won't be an issue until $n=\infty$ (and also maybe we'll luck out and be hypercomplete). $\endgroup$ Oct 21, 2013 at 11:10
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    $\begingroup$ (To be more clear: localizing with respect to covering sieves doesn't make sense for simplicial sheaves) $\endgroup$ Oct 21, 2013 at 11:12
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    $\begingroup$ Exactly.* * * * $\endgroup$ Oct 22, 2013 at 14:16

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