Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes. Let us denote the category of stacks over $\mathcal{C}$ by $Stacks(\mathcal{C})$. This is naturally a full subcategory of the category $Pre_{grpds}(\mathcal{C})$ of presheaves in groupoids over $\mathcal{C}$. This is equivalent to formulating stacks as being categories fibered in groupoids, fulfilling effective descent.

Now the inclusion $Stacks(\mathcal{C}) \to Pre_{grpds}(\mathcal{C})$ has a left adjoint called the stackification functor.

It is a classical fact that the sheafification functor on presheaves of sets commutes with finite limits. Is this also true for the mentioned stackification functor?

Precisely, I want to know if and if so, why, the stackification functor commutes with 2-categorical pullbacks, here is the precise setting.

Suppose we are given Hopf-algebroids $P_1,P_2,P$ and view them as presheaves in groupoids. Suppose we are given two morphisms $P \to P_i$ and we construct the 2-pullback of functors $$Q = pullback(P_1 \to P \leftarrow P_2).$$

In my situation I can show that Q is equivalent to an affine scheme, and I want to conclude that the pullback of the stackified diagram is also equivalent to an affine scheme. If I knew that stackification commuted with finite limits, this would be ok, since the (co)unit (I always mix them up) of the adjunction is an equivalence, i.e., if $X$ is a stack, and $st(X)$ is its stackification, then the natural map $X\to st(X)$ adjoint to the identity of $st(X)$ is an equivalence of stacks.

Any help and comments are appreciated, also if you could give references concerning these questions, that would be great.

  • 1
    $\begingroup$ Minor note: I think stacks naturally form a full subcategory of presheaves of groupoids only if you assume your stacks are equipped with a splitting. $\endgroup$
    – S. Carnahan
    Commented Nov 12, 2012 at 23:08
  • $\begingroup$ I am not sure I know what you mean. Following Charles Rezk I wanted to define a stack over a site as a presheaf in groupoids such that it is a "homotopy sheaf" i.e. the functor evaluated on an object is given by a homotopy limit over a simplicial diagram induced by a covering in the site. Do you know what the splitting is in this viewpoint? $\endgroup$ Commented Nov 14, 2012 at 22:58

1 Answer 1


The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online: Tag04Y1 in the Stacks Project. I quote:

Lemma 8.4. Let $C$ be a site. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of fibred categories over $C$. In this case the stackification of the 2-fibre product is the 2-fibre product of the stackifications.


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