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I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. they talk about "homotopy invariant presheaves with transfers", and it seems to be fairly important (perhaps only technically? I can't tell yet.) to show that various objects are homotopy invariant. Eg, the Ninscevich sheaf associated to a homotopy invariant presheaf with transfers is also homotopy invariant.

There are lots of results in this language in that book. I assume that philosophically this language has something to do with "homotopy theory of schemes" - a subject on which there is plenty written. I was wondering if someone could direct me towards a good reference or better yet give me a good example from the theory of homotopy in schemes (or a reference to one).

I'd like to know how the theory could be useful, a bit of direction would be well appreciated.

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Look at the pieces of work that granted Voevodsky the Fields medal. – Fernando Muro Jun 23 '13 at 8:06
There are already several good questions and answers on MO itself, e.g:…… – Simon Pepin Lehalleur Jun 23 '13 at 11:46 Homotopy theory of schemes, F Morel. – Matt Carmona G May 14 at 20:31

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