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 have? Is it standard to talk about the tangent bundle as a sheaf? Is it standard to talk about the structure sheaf as the espace etale? 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.

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.

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 have? Is it standard to talk about the tangent bundle as a sheaf? Is it standard to talk about the structure sheaf as the espace etale? 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.

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 have? Is it standard to talk about the tangent bundle as a sheaf? Is it standard to talk about the structure sheaf as the espace etale? 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.