Abstract:

J. Ayoub recently gave a counter example to my conjecture that (over a field) the sheaf \pi_0^{A^1} of any space (as defined in our joint work with V. Voevodsky) is A^1-invariant.

In this talk I will present a variant of the A^1-homotopy category (at least over a field) in which the Postnikov tower of (strict) A^1-local objects satisfies a priori any of the nice expected properties : the new sheaf \pi_0^{A^1} is always A^1-invariant, and each Postnikov truncation is strictly A^1-local (in particular A^1-local in the original sense and the higher A^1-homotopy sheaves are strongly/strictly A^1-invariant).

This new category is a further localisation of the original A^1-homotopy category but the proof that any space admits a strict A^1-localisation is very explicit and makes this new approach much more computable than the previous one. Most of the classical theorems are then an easy consequence of the construction.

Most of the known A^1-local spaces whose "old" \pi_0^{A^1} is A^1-invariant are in fact strict A^1-local: 0-connected spaces, more generally A^1-h-groups (using a result of Choudhury). In particular this new strict A^1-homotopy category has the same associated stable homotopy category in any sense, in fact this holds after one suspension.