9
$\begingroup$

In the groundbreaking paper Champs Affines (DOI), Toen constructs a generalisation of rational homotopy types which he calls schematic homotopy types. This is part of a larger programme of a theory developed in Toen's Homotopy types of algebraic varieties (different to motivic homotopy theory), which, to my knowledge, has not been carried out in full.

One particular direction suggested in sections 3.5.2 and 3.5.3 of Champs Affines is the development of a theory of crystalline and $\ell$-adic homotopy types.

The first has been developed by Ollson in $F$-isocrystals and homotopy types (DOI).

Has any work been published in the $\ell$-adic direction?

More generally has any more of Toen's programme been worked out in the literature?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Étale homotopy type of scheme is rather classical object, see original Artin–Mazur paper or/and Quillen works on Adams conjecture with brilliant lectures of Sullivan "geometric topology" on the same topic, where localization and completion of homotopy type were first described.

Improve this answer
$\endgroup$
2
  • 6
    $\begingroup$ This begs the question of what is the relation between etale and schematic homotopy types. The original question is about the latter. $\endgroup$
    – Donu Arapura
    Commented Jul 22, 2023 at 15:06
  • $\begingroup$ I was sure that they coincide, and that may be the case. There are papers by Gereon Quick on section conjecture, where the same statement as in Toen's one made. $\endgroup$
    – Cyril
    Commented Aug 24, 2023 at 7:41

You must log in to answer this question.

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