How to pronounce “presheaf”

Meaning

• [Noun]
• (category theory, sheaf theory) An abstract mathematical construct which associates data to the open sets of a topological space, generalizing the situation of functions, fiber bundles, manifold structure, etc. on a topological space (but not necessarily in such a way as to make the local and global data compatible, as in a sheaf). Formally, A contravariant functor
• F
• {\displaystyle {\mathcal {F}}}
• whose domain is a category whose objects are open sets of a topological space (called the base space or underlying space) and whose morphisms are inclusion mappings. The image of each open set under
• F
• {\displaystyle {\mathcal {F}}}
• is an object whose elements are called sections, and are which are said to be over the given open set; the image of each inclusion map
• A
• →
• B
• {\displaystyle A\to B}
• under
• F
• {\displaystyle {\mathcal {F}}}
• is a morphism
• F
• (
• B
• )
• →
• F
• (
• A
• )
• {\displaystyle {\mathcal {F}}(B)\to {\mathcal {F}}(A)}
• , called the restriction from
• B
• {\displaystyle B}
• to
• A
• {\displaystyle A}
• and denoted
• res
• B
• ,
• A
• {\displaystyle \operatorname {res} _{B,A}}
• or
• |
• B
• ,
• A
• {\displaystyle |_{B,A}}
• .[1]

Practice Nearby Words

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