# 层化　`thm.sheafification`
层化：层范畴是预层范畴的反射子范畴
layer 18 · 定理 · 层与拓扑 · 范畴论
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层范畴是预层范畴的**反射子范畴**，反射

$$\sharp : \widehat{\mathcal{C}} \longrightarrow \widehat{\mathcal{C}}, \qquad T \mapsto T^{\sharp}$$

叫 **sheafification**（层化）：对每个层 $F$，

$$\operatorname{Hom}_{\widehat{\mathcal{C}}}\bigl(T^{\sharp},\ F\bigr) \;\cong\; \operatorname{Hom}_{\widehat{\mathcal{C}}}\bigl(T,\ F\bigr)$$
> 陈述续见 [nodes/thm.sheafification.2.md](./thm.sheafification.2.md)


## 它能推出什么 / 谁在用它
- ⇒ [`prop.site-properties`](../nodes/prop.site-properties.md) site 的层范畴的好性质

- …另有出边，续页见 [nodes/thm.sheafification.4.md](./thm.sheafification.4.md)

## 为什么成立（入边，证明在 proofs/）
- …另有入边，续页见 [nodes/thm.sheafification.3.md](./thm.sheafification.3.md)

refs: Le Stum, An introduction to condensed mathematics, §3.1–3.2

> 说明见 `notes/thm.sheafification.md`

引用：`thm.sheafification`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)