MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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. – S. Carnahan Nov 12 '12 at 23:08
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? – Markus Land Nov 14 '12 at 22:58
up vote 7 down vote accepted

The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online : this is Lemma 52.8.4 in De Jong's stack project. I quote :

"Lemma 8.4. Let C be a site. Let f : X → Y and g : Z → 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."

To get there, and get the proof :

then select the first answer Lemma 52.8.4 and in the .dvi look for Lemma 8.4. Enjoy !

ps : the permanent tag is Tag 04Y1

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.