ScalingStacks

Theorem 1.2. For $X\rightarrow Y\leftarrow X^{\prime}$ maps of perfect stacks, there is a canonical equivalence

$$\qc(X\times_{Y}X^{\prime})\simeq\qc(X)\otimes_{\qc(Y)}\qc(X^{\prime})$$

between the $\infty$-category of sheaves on the derived fiber product and the tensor product of the $\infty$-categories of sheaves on the factors.

There is also a canonical equivalence

$$\operatorname{Perf}(X\times X^{\prime})\simeq\operatorname{Perf}(X)\otimes\operatorname{Perf}(X^{\prime})$$

for $\infty$-categories of perfect complexes.

For $X\to Y$ a perfect morphism to a derived stack $Y$ with affine diagonal, and $X^{\prime}\to Y$ arbitrary, there is a canonical equivalence

$$\qc(X\times_{Y}X^{\prime})\simeq\Fun_{\qc(Y)}(\qc(X),\qc(X^{\prime}))$$

between the $\infty$-category of sheaves on the derived fiber product and the $\infty$-category of colimit-preserving $\qc(Y)$-linear functors.

When $X$ is a smooth and proper perfect stack, there is also a canonical equivalence

$$\operatorname{Perf}(X\times X^{\prime})\simeq\Fun(\operatorname{Perf}(X),\operatorname{Perf}(X^{\prime}))$$

for $\infty$-categories of perfect complexes.