Is $B\mathbb{G}_m$ strongly $A^1$-invariant? I have just seen the definition of strongly ${A}_1$ invariance:

A sheaf of group $G$ is strongly $A_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A_1$-invariant. 

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?
 A: 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.
