notes
Notes
This page provides the links to my notes.
Page | Date | Description | Edit summary | Tags |
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Sheaf on a topological space | 2024/06/13 17:17 | Sheaf semantics on topological spaces 1. Introduction In this short note, I provide a tutorial on sheaf semantics, aiming to invalidate the dichotomy of real numbers on the unit interval $(0,1)$: $$(0,1)\not\Vdash \forall x : \mathbf{R}.\;x \leq 0 \lor x \geq 0$$ while validating it on the discrete topology of the booleans $\{0, 1\}$$$\{0,1\}\Vdash \forall x : \mathbf{R}.\;x \leq 0 \lor x \geq 0.$$$X$$\mathcal{O}(X)$$X$$U \to V$$U \subseteq V$$X$$P$$X$$P : \mathcal{O}(X)^{op} \to \mathbf{Set… | [3. Presheaves and sheaves] | completed |
Grothendieck topos | 2024/06/14 15:50 | Grothendieck 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] | incomplete |
notes.txt · Last modified: 2024/06/14 18:02 by sewon