Snopi nad okoliši : delo diplomskega seminarja

Kobe, Ivan (author), Simpson, Alexander Keith (mentor)

V pričujočem delu diplomskega seminarja bomo predstavili kategorijo okolišev, ki je z vidika teorije kategorij naraven pristop k topologiji in posploši mnoge topološke konstrukcije, med drugim snope. Nato bomo pokazali, da lahko snope nad okoliši obravnavamo tudi z drugačne perspektive, namreč kot poseben primer snopov nad lokacijami, tj. delnimi ureditvami, opremljenimi z Grothendieckovo topologijo. Z uporabo te karakterizacije bomo podali primer presenetljive aplikacije snopov nad nekim okolišem, ki ovrže hipotezo kontinuuma, če slednjo interpretiramo kot trditev o kategoriji snopov nad danim okolišem. Ta argument vsebuje matematično jedro Cohenovega dokaza združljivosti negacije hipoteze kontinuuma z aksiomi Zermelo-Fraenklove teorije množic.

• Language: Slovenian
• Keywords: okoliš, lokacija, snop, topos, hipoteza kontinuuma
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-150545
• UDC: 510.6
• COBISS.SI-ID: 165230595
• Publication date in RUL: 20.09.2023

KOBE, Ivan, 2023, Snopi nad okoliši : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 8 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=150545

Secondary language

• Language: English
• Title: Localic sheaves

Abstract

In our thesis, we study the category of locales, a natural categorical approach to topology which generalises many topological constructions, e.g. sheaves. We show that sheaves on a locale also have an alternative perspective, since we can view them as a special case of sheaves on a site, i.e. a partially ordered set with a Grothendieck topology. Using this latter formulation we will give a striking application of the category of sheaves on a locale which refutes the continuum hypothesis if the latter is interpreted as a statement about that category. This argument contains the matematical core of Cohens proof of the compatibility of the negation of the continuum hypothesis with the axioms of Zermelo-Fraenkl set theory.

• Keywords: locale, site, sheaf, topos, continuum hypothesis