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I was reading https://math.dartmouth.edu/~jvoight/notes/moduli-red-harvard.pdf, introduction of some notes on algebraic geometry (stacks, I do not know what it is) and came across following statement :

In beginning of algebraic geometry, one starts with varieties over the complex numbers, a set of points with a Zariski topology in which all points are closed points. In generalizing this to schemes, one asks for a locally ringed topological space equipped with a structure sheaf, allowing for closed and nonclosed points. To generalize this further, we define an object called a stack which will allow “points” equipped with nontrivial automorphisms: it will be a category with a Grothendieck topology.

I understand generalisation to schemes, spaces where we allow non closed points as well. But I do not understand what does it mean to say non trivial automorphisms and I am not at all aware of what a Grothendieck topology is.

Any information regarding that non trivial automorphism is welcome. I tried reading definition of Grothendieck topology but did not understand the definition.

1 Answers1

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A readable reference is David Carchedi's thesis, he deals with differentiable stacks but the discussion of Grothendieck topologies and stacks are general. I can answer parts of your question in terms of differentiable stacks, and the intuition should carry over.

Non-trivial automorphisms: If you have a Lie group $G$ acting on a manifold $M$, the quotient $M/G$ won't in general be smooth. You can always define the "stacky quotient" $[M/G]$ which is really some categorical nonsense (see below) but behaves (in some ways) like a smooth manifold. For instance, $[M/G]$ has a complex of differential forms. To think of why $[M/G]$ might have points with nontrivial automorphisms, imagine $M=\{-1,0,1\}\subset \mathbb{R}$ and $G$ is the group with two elements acting by reflections over $0$. As a manifold, $M/G$ is just two points, but in the stack quotient the point $0$ "remembers" it was the fixed point of the action and it has a nontrivial automorphism.

Grothendieck topologies: A stack is a generalization of a sheaf, and this is where Grothendieck topologies come in. You've seen sheaves defined over topological spaces, but here we actually mean sheaves over a category. In order to make sense of the sheaf (gluing) condition, one needs to define the analogue of a topology on your category, which is the Grothendieck topology. It more or less specifies which morphisms in your category you should think of as open covers.

Finally, you should look up the functor of points/Yoneda lemma to see where sheaves over a category come in to the whole story.

  • Thanks for your answer.. that thesis looks good for my purpose. I want to know what exactly is the non trivial automorphism that he is talking about. It is still not clear. –  Apr 30 '17 at 03:33
  • For the nontrivial automorphism, consider the following: Maps from a point to a manifold form a set; elements of the set are points of your manifold. Maps from a point to a stack form a groupoid $G$. You can think of $\pi_0(G)$ as the set of points of of your stack. Given an object $g$ of $G$, you have $\pi_1(G,g)$, which is the group of automorphisms of the element of $\pi_0(G)$ associated with $g$. –  May 04 '17 at 13:28