Lemma 85.15.1 (0D85)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.15: The site associated to a semi-representable object
Lemma 85.15.1 (cite)

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Lemma 85.15.1. Let $\mathcal{C}$ be a site.

For $K$ in $\text{SR}(\mathcal{C})$ the functor $j : \mathcal{C}/K \to \mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.
For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $v : \mathcal{C}/K \to \mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.

Proof. Proof of (2). In the notation of the discussion preceding the lemma, the localization functors $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ are continuous and cocontinuous by Sites, Section 7.25 and satisfy $P$ by Sites, Remark 7.25.11. It is formal to deduce $v$ is continuous and cocontinuous and has $P$ . We omit the details. We also omit the proof of (1). $\square$

The Stacks project

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