Corollary 3.25 (0NXH) · ScalingStacks

0NXH

Corollary 3.25. Let $X$ be a perfect stack, and let $\Sigma$ be a finite simplicial set. Then the mapping stack $X^{\Sigma}=\Map(\Sigma,X)$ is perfect.

0NXI

Proof. Let $\Sigma_{0}$ be the $0$ -simplices of $\Sigma$ . Since $X$ has affine diagonal, the natural projection $X^{\Sigma}\to X^{\Sigma_{0}}$ is affine. By Proposition 3.24, the product $X^{\Sigma_{0}}$ is perfect, and so by Proposition 3.21 (in the basic case of an affine morphism), the assertion follows. ∎

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5