Isotropy Groups of Quasi-Equational Theories

Abstract

To every small category or Grothendieck topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. In this thesis, we investigate this invariant in the particular context of quasi-equational theories, which are multi-sorted equational theories in which operations may be partially de fined. It is known that every such theory T has a classifying topos, which is a topos that classi fies all topos-theoretic models of the theory, and that this classifying topos is in fact equivalent to the covariant presheaf category Sets^fpTmod, with fpTmod being the category of all finitely presented, set-based models of T. We then investigate the isotropy group of this classifying topos of T, which will therefore be a presheaf of groups on fpTmod, and show that it encodes a notion of inner automorphism for the theory. The main technical result of this thesis is a syntactic characterization of the isotropy group of a quasi-equational theory, and we illustrate the usefulness of this characterization by applying it to various concrete examples of quasi-equational theories.