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This page provides the links to my notes.

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Sheaf on a topological space2024/06/13 17:17Sheaf semantics on topological spaces 1. Introduction In this short note, I give a tutorial on sheaf semantics heading toward invalidating the dichotomy of real numbers on a unit interval $$(0,1)\not\Vdash \forall x : \mathbf{R}.\;x \leq 0 \lor x \geq 0$$ but validating it on the discrete topology of the booleans $$\{0,1\}\Vdash \forall x : \mathbf{R}.\;x \leq 0 \lor x \geq 0.$$$X$$\mathcal{O}(X)$$X$$U \to V$$U \subseteq V$$X$$P$$X$$P : \mathcal{O}(X)^{op} \to \mathbf{Set}$$U \subseteq X$$PU$… 
Grothendieck topos2024/06/14 15:50Grothendieck topos 1. Introduction This note proceeds the previous note on the topoi of sheaves on topological spaces. The plan at the moment is not to introduce various results of Grothendieck topoi but to covey the readers on how this concept of Grothendieck topology and topos generalizes the ordinary notion of sheaves on topological spaces. The purpose is to provide a stepping stone toward our next topic which is about the Lawvere–Tierney topology.$\mathcal{O}(X)$$U \in \mathcal{O}(X)$$U$$\…[2. Grothendieck topology]
notes.txt · Last modified: 2024/06/14 18:02 by sewon