About: Local system

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In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943.

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  • In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943. (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, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943. (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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