# 自由阿贝尔群层　`def.free-abelian-sheaf`
定义：$\mathbb{Z}\cdot X$，遗忘函子的左伴随
layer 17 · 定义 · 阿贝尔层 · 同调代数
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设 $\mathcal{T}$ 是拓扑斯。遗忘函子

$$\mathcal{T}(\mathbf{Ab}) \longrightarrow \mathcal{T}$$

（把阿贝尔层忘成底集合层）有**左伴随**，记作

$$\mathbb{Z}\cdot(-) : \mathcal{T} \longrightarrow \mathcal{T}(\mathbf{Ab}), \qquad X \longmapsto \mathbb{Z}\cdot X \quad (\text{也记 } \mathbb{Z}[X]),$$
> 陈述续见 [nodes/def.free-abelian-sheaf.2.md](./def.free-abelian-sheaf.2.md)


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

## 它能推出什么 / 谁在用它
- …另有出边，续页见 [nodes/def.free-abelian-sheaf.4.md](./def.free-abelian-sheaf.4.md)

refs: Le Stum, An introduction to condensed mathematics, Exercise 5.18, Exercise 5.24

> 说明见 `notes/def.free-abelian-sheaf.md`

引用：`def.free-abelian-sheaf`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)