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Thea Li: The internal language of sheaves and applications to algebraic geometry

Bachelor thesis presentation

Time: Tue 2022-06-07 09.00 - 10.00

Location: Kräftriket, house 6, room 306

Respondent: Thea Li

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This thesis has two purposes. The first one is to introduce sheaves on spaces and present the Kripke-Joyal semantics for the internal language of sheaves. We will do this by first laying the ground work for the internal language, i.e. defining unary and binary operators on the subobject classifier corresponding to logical connectives and quantifiers. Then we define a forcing relation that translates internal properties to external properties and give a recursive way to unwind internal statements to external statements. In the second part we illustrate some translations of common properties and internal proofs of results from algebraic geometry presented in [Ble21], an example being proving that a sheaf of modules of finite type is internally a finitely generated module. The main portion of this part is dedicated to □-operators and proving that for a geometric formula the □’d version is equivalent to the □-translated version.