stackification commutes with finite limits? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:41:25Z http://mathoverflow.net/feeds/question/112179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112179/stackification-commutes-with-finite-limits stackification commutes with finite limits? Markus Land 2012-11-12T14:40:10Z 2012-11-12T20:27:33Z <p>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.</p> <p>Now the inclusion $Stacks(\mathcal{C}) \to Pre_{grpds}(\mathcal{C})$ has a left adjoint called the stackification functor.</p> <p>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?</p> <p>Precisely, I want to know if and if so, why, the stackification functor commutes with 2-categorical pullbacks, here is the precise setting.</p> <p>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 $$Q = pullback(P_1 \to P \leftarrow P_2).$$</p> <p>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.</p> <p>Any help and comments are appreciated, also if you could give references concerning these questions, that would be great.</p> http://mathoverflow.net/questions/112179/stackification-commutes-with-finite-limits/112220#112220 Answer by Niels for stackification commutes with finite limits? Niels 2012-11-12T20:21:19Z 2012-11-12T20:27:33Z <p>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 :</p> <p>"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."</p> <p>To get there, and get the proof :</p> <p><a href="http://stacks.math.columbia.edu/search?keywords=stackification" rel="nofollow">http://stacks.math.columbia.edu/search?keywords=stackification</a></p> <p>then select the first answer Lemma 52.8.4 and in the .dvi look for Lemma 8.4. Enjoy !</p> <p>ps : the permanent tag is Tag 04Y1</p>