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# Contents

## Idea

The Tate sphere, in the strict sense, is the object in a sheaf topos (or (∞,1)-sheaf (∞,1)-topos, or its stabilization) over a gros site of algebraic varieties, which is given by

$S^1_{Tate} \;\coloneqq\; \mathbb{A}^1 / (\mathbb{A}^1 \setminus \{0\}) \,.$

This denotes the (homotopy-)cofiber of the inclusion morphism into the sheaf represented by the affine line $\mathbb{A}^1$ from the subobject $\mathbb{A}^1 \setminus \{0\}$ represented by the affine line with origin removed. The latter is equivalently the object underlying the multiplicative group $\mathbb{G}_m$.

This construction is typically understood by default in the context of motivic homotopy theory working over the Nisnevich site (VRO 07, Remark 2.22). But the construction principle is clearly more general.

## References

Created on June 14, 2020 at 11:27:31. See the history of this page for a list of all contributions to it.