# Stacks and Grothendieck topology.

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.

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.
• 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 $g$. – user171326 May 4 '17 at 13:28