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- In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. (en)
- 在數學中，局部系統或稱局部係數是源於代數拓撲的一種觀念，它是常係數的同調或上同調理論的推廣。這個觀念也能應用於代數幾何 。 用層論的語言來講，局部系統是局部上同構於的阿貝爾群層。若此層整體來看也同構於常數層，則就回到了傳統的常係數理論。例子包括了帶有的向量叢，基本群的線性表示則給出了局部同構於向量空間常數層的局部系統。 (zh)
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- Take local system and a loop at x. It's easy to show that any local system on is constant. For instance, is constant. This gives an isomorphism , i.e. between L and itself.
Conversely, given a homomorphism , consider the constant sheaf on the universal cover of X. The deck-transform-invariant sections of gives a local system on X. Similarly, the deck-transform-ρ-equivariant sections give another local system on X: for a small enough open set U, it is defined as
:
where is the universal covering. (en)
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- Proof of equivalence (en)
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- In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. (en)
- 在數學中，局部系統或稱局部係數是源於代數拓撲的一種觀念，它是常係數的同調或上同調理論的推廣。這個觀念也能應用於代數幾何 。 用層論的語言來講，局部系統是局部上同構於的阿貝爾群層。若此層整體來看也同構於常數層，則就回到了傳統的常係數理論。例子包括了帶有的向量叢，基本群的線性表示則給出了局部同構於向量空間常數層的局部系統。 (zh)
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- Local system (en)
- 局部系統 (zh)
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