Kategorička logika

master's thesis

• Title (english): Categorical logic
• Author: Teo Šestak
• Mentor: Ivan Tomašić (mentor)
• Mentor: Mladen Vuković (sumentor)
• Committee member: Mladen Vuković (predsjednik povjerenstva)
• Committee member: Ivan Tomašić (član povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu započinje se s osnovnim pojmovima teorije kategorija i teorije toposa s ciljem da se definira Grothendieckov topos. Također se definiraju klase kategorija koje koristimo za konstrukciju modela formalnih teorija. Uvodi se sekventni račun kao formalizacija izvoda i dokaza te se pomoću njega daje konstrukcija sintaktičke kategorije za geometrijske teorije. Takva kategorija sadrži generički model sa svojstvom da se ostali modeli mogu dobiti kao njegova slika. Rad završava konstrukcijom klasifikacijskog toposa, dokazom njegovog univerzalnog svojstva te se navodi kako konstruirati univerzalni model geometrijske teorije u njenom klasifikacijskom toposu.

Abstract (english)

In this thesis we begin by introducing some basics of category theory and topos theory in order to define the Grothendieck topos. We also define several classes of categories which are used to construct models of formal theories. Sequent calculus is introduced in order to formalize concepts of derivations and proofs. Using sequent calculus we also describe the construction of syntactic categories for geometric theories. Syntactic category contains the generic model such that other models can be expressed as images of the generic model. At the very end, the construction of classifying topos is is given, along with the proof of its universal property. We also show how to construct the universal model of a geometric theory inside its classifying topos.

• Keywords: teorija kategorija; teorija toposa; Grothendieckov topos
• Keywords (english): category theory; topos theory; Grothendieck topos
• Language: croatian
• URN:NBN: urn:nbn:hr:217:467030
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-11-08 11:18:56