ScalingStacks

0NYN

Corollary 5.12. For a perfect stack XX, consider the stable ∞\infty-category QC⁡(X)\qc(X) equipped with its ℰn\mathcal{E}_{n}-tensor product. Then with XSn=Map⁡(Sn,X)X^{S^{n}}=\Map(S^{n},X), there are canonical equivalences

QC⁡(XSn)≃HHℰn∗⁡(QC⁡(X))≃HH∗ℰn⁡(QC⁡(X))\qc(X^{S^{n}})\simeq\hh^{*}_{\mathcal{E}_{n}}(\qc(X))\simeq\hh_{*}^{\mathcal{E}_{n}}(\qc(X))
0NYP

Proof. The result follows from an inductive application of Theorem 4.7 and Corollary 4.12 to the Cartesian diagrams

XSn\textstyle{X^{S^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}XSn−1\textstyle{X^{S^{n-1}}}

where the two maps X→XSn−1X\to X^{S^{n-1}} assign to a point of XX the corresponding constant map. ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5