Notes

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 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$…		completed
Grothendieck topos	2024/06/14 15:50	Grothendieck topos 1. Introduction This note proceeds the previous note on the topoi of sheaves on topological spaces. The plan at the moment is not to introduce various results of Grothendieck topoi but to covey the readers on how this concept of Grothendieck topology and topos generalizes the ordinary notion of sheaves on topological spaces. The purpose is to provide a stepping stone toward our next topic which is about the Lawvere–Tierney topology.$\mathcal{O}(X)$$U \in \mathcal{O}(X)$$U$$\…	[2. Grothendieck topology]	incomplete