8
$\begingroup$

There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.

I would like to know the reference in which Grothendieck did it, and some references on subsequently development.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
13
$\begingroup$

Not sure about later developments, but the idea is mentioned in a famous passage of Grothendieck's Récoltes et Semailles. I quote from Roy Lisker's translation:

Thus, the motive presents itself as the deepest "form invariant" which one has been able to associate up to the present moment with an algebraic variety, setting aside its "motivic fundamental group". For me both invariants represent the "shadows" projected by a "motivic homotopy type" which remains to be discovered (and about which I say a few things in the footnote: "The tower of scaffoldings- or tools and vision" (R&S IV, #178, see scafolding 5 (Motives), and in particular page 1214)).

It is the latter object which appears to me to be the most perfect incarnation of the elusive intuition of "arithmetic form" ( or "motivic"), of an arbitrary algebraic variety.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .