This page provides the links to my notes.
| Page | Date | Description | Edit summary | Tags |
|---|---|---|---|---|
| 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… | completed | |
| Grothendieck topos | 2024/06/14 15:50 | Grothendieck topos 1. Introduction This note proceeds from the previous note on the toposes of sheaves on topological spaces. The plan at the moment is not to introduce various results of Grothendieck toposes but to guide the reader on how this concept of Grothendieck topology and toposes generalizes the ordinary notion of sheaves on topological spaces. The purpose is to provide a stepping stone towards our next topic, which is about the Lawvere–Tierney topology.$\mathcal{O}(X)$$U \in \mathcal… | [3. Sheaves on site] | incomplete |
| Lawvere–Tierney topology | 2024/12/21 21:02 | Lawvere–Tierney topology incomplete | created | incomplete |