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I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions:

  1. Why is the question of $\mathbb A^1$-invariance of certain sheaves (for example, motivic homotopy groups $\pi_i^{\mathbb A^1}$, cohomology of a homotopy invariant presheaf with transfers etc) so important in motivic cohomology? Is it because the projection maps $X \times \mathbb A^1 \to X$ are all inverted in the motivic homotopy category?

  2. What difference does it make to motivic homotopy theory if on one hand one takes spaces with base-point and on the other one takes spaces themselves (no base-points)? I can see that Spaces with basepoint are required in order to be able to define the higher $\mathbb A^1$-homotopy groups $\pi_i^{\mathbb A^1}$ of a simplicial sheaf, for $i \geq 1$.

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1. Yes. A¹-invariance of simplicial presheaves guarantees fibrancy in the A¹-localized model structure, which makes it easy to compute homotopy mapping spaces. 2. The situation in the motivic case is no different from the situation in the purely topological case: unpointed spaces give you unpointed spectra, which are the analogs of torsors over abelian groups in the same way that spectra are analogs of abelian groups. –  Dmitri Pavlov Apr 7 at 13:42

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