Last Updated：2025/11/27

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(category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be over those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.

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presheaf

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Source Word

presheaf

Noun

(category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be "over" those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.

Japanese Meaning

前層：トポロジー空間の開集合を対象、その包含写像を射とする圏からの反変関手として定義される。各開集合に対して、その関手はその集合上の断面（セクション）と呼ばれる要素集合を割り当て、包含写像に対応して制限写像（リストリクション）を与える。

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Google Translation / DeepL Translation

位相空間X上で、前層は各開集合に節の集合を割り当て、包含に対して制限写像を与えます。

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(category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be "over" those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.

Let X be a topological space. (Following tradition, I will switch from my previous convention of using X to denote an object of a topos.) Write Open(X) for its poset of open subsets. A presheaf on X is a functor F: mathbf Open(X)op→ mathbf Set. It assigns to each open subset U a set F(U), whose elements are called sections over U (for reasons to be explained). It also assigns to each open V⊆U a function F(U)→F(V), called restriction from U to V and denoted by s→s|_V. I will write Psh(X) for the category of presheaves on X.