ScalingStacks

0NYC

Corollary 5.2. Let XX be a perfect stack, and equip QC⁡(X)\qc(X) with its (symmetric) monoidal algebra structure given by tensor product. Then there are canonical equivalences (of symmetric monoidal) ∞\infty-categories

QC⁡(ℒ​X)≃𝒵⁡(QC⁡(X))≃𝒯​r​(QC⁡(X))\qc(\mathcal{L}X)\simeq\mathcal{Z}(\qc(X))\simeq\mathcal{T}r(\qc(X))
0NYD

Proof. By Theorem 4.7, we know that 𝒯​r​(QC⁡(X))\mathcal{T}r(\qc(X)), which is a tensor product, is also calculated by a fiber product

𝒯​r​(QC⁡(X))≃QC⁡(X×X×XX).\mathcal{T}r(\qc(X))\simeq\qc(X\times_{X\times X}X).

On the other hand, by Corollary 4.10, we know that 𝒵⁡(QC⁡(X))\mathcal{Z}(\qc(X)), which consists of functors, is also calculated by a tensor product

𝒵⁡(QC⁡(X))≃𝒯​r​(QC⁡(X)).\mathcal{Z}(\qc(X))\simeq\mathcal{T}r(\qc(X)).

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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