I have just seen the definition of strongly ${A}\_1$ invariance:



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> A sheaf of group $G$ is strongly $A\_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A\_1$-invariant. 
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I haven't got too much about the point of this definition and just try to think about some examples. Can someone give me some simple examples? Is $B\mathbb{G}\_m$ strongly $A\_1$-invariant where $\mathbb{G}\_m$ is the multiplicative group?



Is $B\mathbb{G}_m$ strongly $A^1$-invariant?
The point of the notion is that for a strongly $\mathbb{A}^1$-invariant sheaf of groups $G$ the classifying space $BG$ is $\mathbb{A}^1$-local. The characterization in terms of $H^0$ and $H^1$ is equivalent to that since $H^0(-,G)$ and $H^1(-,G)$ are the homotopy presheaves $\pi\_1$ and $\pi\_0$ of $BG$ in the homotopy theory of simplicial Nisnevich sheaves, with all other homotopy presheaves trivial.


The sheaf $\mathbb{G}\_m$ is a simple example of a strongly $\mathbb{A}^1$-invariant sheaf of groups (since $\mathbb{G}\_m$ and ${\rm Pic}$ are $\mathbb{A}^1$-invariant, assuming we are over a regular base). Over a field, other examples of strongly $\mathbb{A}^1$-invariant sheaves are the sheaves $\mathbf{K}^{\rm M}\_n$ of Milnor K-groups. The space $B\mathbb{G}\_m$ in the question isn't a sheaf of groups, so strictly speaking the notion doesn't apply. 


Of course, the tautological examples are the $\mathbb{A}^1$-fundamental group sheaves, by the theory in Morel's book "$\mathbb{A}^1$-algebraic topology over a field". More examples can be found there. Section 7.3 of the book also contains an example of a strongly $\mathbb{A}^1$-invariant sheaf of groups which is not abelian, namely $\pi\_1^{\mathbb{A}^1}(\mathbb{P}^1)$. Most of the examples of strongly $\mathbb{A}^1$-invariant sheaves are actually abelian, but that's mostly due to our lack of knowledge of how to compute $\mathbb{A}^1$-fundamental groups of things.


