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Stack entry says: "The notion of stack is the one-step vertical categorification of a sheaf." In Grothendieck's main works, like pursuing stacks and in the following works of French schools, stack is any-times categorification of a sheaf, and the one-step case is called more specifically 1-stack. We can talk thus about stack in narrow sense or 1-stacks and stacks in wider sense as n-stacks for all n. Topos literature mainly means that the stack is the same as internal 1-stack.
Okay, maybe you could add that remark.
I added an MO link to the connection between groupoids and stacks. Is this the best reference?
The entry on stacks contains the request:
Somebody should turn this here into a coherent entry on stacks.
(This was already in Revision 4, from 2009!)
I was looking down a down to earth discussion and noted the old sketch that Todd gave. This is useful but does someone have the time / inclination to edit to get something more in line with the other entries. I really wanted something looking at base change for stacks that could be used in introductory notes, i.e. more approachable to non-categorists. Any thoughts?
What sort of base change? Like having a stack on a space $X$ and then pulling back along $Y\to X$? Or do you mean the functors between fibres induced by maps in the site?
My aim was the first of those. The point is to put some flesh on the description of constant and locally constant (and eventually construcible ) stacks that is given by Treumann in his paper on exit paths. (My direction after that would be towards the Lurie version but I wanted the locally constant and constructible cases of 1-stacks done in a way that would be understandable to someone without enormous $\infty$-categorical knowledge.) One of Treumann’s discussions needed ’stalks’ and that if well done using pullbacks etc. Filling in details of the induced adjointness between Stacks(X) and Stacks(y) seemed one way to proceed. Any thought could be useful. The end result is to try to incorporate ‘defects’ into TQFTs and HQFTs with a bit more clarity (for me!) than at present exists in the literature. (This is a follow on to things talked on at the Lisbon meeting.)
Yeah, like the sheaf condition in terms of the category of descent data for a stack. I have a copy of Giraud’s original book where I believe they were introduced if you want a classic presentation.
Giraud is available on Numdam.
That is of course not the famous Giraud’s book Cohomologie non abelienne but his earlier Memoirs SMS article on descent, which is as a size of a small book as well and which contributed in its content to the later book. I will teach descent theory and nonabelian cohomology in a graduate course the next academic year.
This entry has loads of room to be (re-)written. Volunteers are welcome.
But (2,1)-sheaves are in particular 2=(2,2)-sheaves, and not the other way around, and that’s what the first two sentences refer to.
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