Essentially Algebraic Theories and Localizations in Toposes and Abelian Categories

UoM administered thesis: Phd

  • Authors:
  • Philip Bridge

Abstract

The main theme of this thesis is the parallel between results in topos theory and the theory of additive functor categories.In chapter 2, we provide a general overview of the topics used in the rest of the thesis. Locally finitely presentable categories are introduced, and their expression as essentially algebraic categories is explained. The theory of localization for toposes and abelian categories is introduced, and it is shown how these localizations correspond to theories in appropriate logics.In chapter 3, we look at conditions under which the category of modules for a ring object R in a topos E is locally finitely presented, or locally coherent. We show that if E is locally finitely presented, then the category of modules is also; however we show that far stronger conditions are required for the category of modules to be locally coherent.In chapter 4, we show that the Krull-Gabriel dimension of a locally coherent abelian category C is equal to the socle length of the lattice of regular localizations of C. This is used to make an analogous definition of Krull-Gabriel dimension for regular toposes, and the value of this dimension is calculated for the classifying topos of the theory of G-sets, where G is a cyclic group admitting no elements of square order.In chapter 5, we introduce a notion of strong flatness for algebraic categories (in the sense studied by Adamek, Rosickey and Vitale). We show that for a monoid M of finite geometric type, or more generally a small category C with the corresponding condition, the category of M-acts, or more generally the category of set-valued functors on C, has strongly flat covers.

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Original languageEnglish
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Award date31 Dec 2012