Motivic homotopy theory

Remarks

By construction the projection $\mathbb{A}^1_S \to S$ becomes an isomorphism in $\mathcal{H}(S)$, so the affine line is contractible.

There are two basic spheres: the simplicial circle $S^1$ and the punctured line $\mathbb{G}_m = \mathbb{A}^1 \setminus \{0\}$. The projective line $\mathbb{P}^1_S$ is isomorphic in $\mathcal{H}(S)$ to the smash product $S^1 \wedge \mathbb{G}_m$ (with suitable base points).

Non-example: over a field $k$ with more than two elements, $\mathbb{G}_m$ is not contractible. The sheaf of units is a Nisnevich sheaf with $\mathcal{O}(X \times \mathbb{A}^1)^\times = \mathcal{O}(X)^\times$ for reduced $X$, so it is $\mathbb{A}^1$-local. Hence the morphisms $\operatorname{Spec} k \to \mathbb{G}_m$ in $\mathcal{H}(k)$ are the elements of $k^\times$, which is not a single point.

For a regular base $S$, algebraic K-theory of smooth $S$-schemes is representable in $\mathcal{H}(S)$ by $\Z \times \mathrm{BGL}_\infty$.