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We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured spaces.

In part 5 of Lurie's DAG, he describes this notion in terms of (infinity,1)-topoi. However, I don't see myself being able to read DAG any time soon (I'm still going through HTT), so I have a somewhat easier question:

If we consider the same topic in terms of the semiclassical constructions, that is, algebraic stacks and orbifolds, or even schemes and manifolds, can we describe precisely what our "structured spaces" should be to be useful? Why is it standard to talk about the sheaves on a scheme where we talk about bundles on a manifold? Aren't these concepts the same thing? How much of what we talk about in differential geometry and algebraic geometry can be jointly generalized to structured spaces?

If this question is too vague, then just tell me where I can find out.

Edit: To correct my previous vagueness, as Pete pointed out, we want local rings, or their appropriate counterparts for stacks.

Edit 2: To clarify, I'm looking for either a book that's a dumbed-down version of DAG book 5, or someone to dumb down the idea from (infinity,1)-categories to plain categories or sets.

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I do find your question vague at the moment. Let's start at the beginning: by "structured space" do you mean "ringed space" as in en.wikipedia.org/wiki/Locally_ringed_space, or something else? –  Pete L. Clark Dec 11 '09 at 3:51
    
Yes, but the point is that the ring becomes "part of" the space, in that we can consider modules over the structure sheaf. –  Harry Gindi Dec 11 '09 at 4:00
    
I still don't understand why you don't say "ringed space" or "locally ringed space". One can indeed consider modules over a (locally) ringed space. Not coherent modules, in general, but modules. Moreover, I don't understand the first question: precisely what WHAT? –  Pete L. Clark Dec 11 '09 at 4:08
    
Could a moderator close this post? I guess the question wasn't good. –  Harry Gindi Dec 11 '09 at 4:50
    
Do you want this question deleted or just closed? –  Anton Geraschenko Dec 11 '09 at 7:03
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closed as no longer relevant by Harry Gindi, Anton Geraschenko Dec 11 '09 at 6:53

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1 Answer

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I don't quite understand what you want, but here's a shot in the dark: SGA 4 has a notion of ringed topos, and this is generalized in Lurie's work. Schemes and stacks have their "underlying topoi" with respect to whatever topology you put on them. I think you can define notions like smoothness and tangents in this setting, but I haven't looked at it in a while.

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And do these same notions work for orbifolds/manifolds? –  Harry Gindi Dec 11 '09 at 4:23
    
And is there anything on the topic in english? –  Harry Gindi Dec 11 '09 at 4:26
    
I have heard that there is a book by Mac Lane and Moerdijk that is good, but I haven't looked at it. –  S. Carnahan Dec 11 '09 at 4:36
    
Yes, the same notions work for manifolds, see the remark at the very very end of Lurie's "Structured spaces", which points to David Spivak's work on derived smooth manifolds ncatlab.org/nlab/show/derived+smooth+manifold . A summary of key ideas in "Structured Spaces" is, by the way, here: ncatlab.org/nlab/show/… –  Urs Schreiber Dec 11 '09 at 8:01
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