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

On 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.

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?


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