Isotropy of Algebraic Theories

To every small category or topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. We investigate this invariant in the particular situation of algebraic theories, thus obtaining a group-theoretic invariant of algebraic theories. This invariant encodes a notion of inner automorphism relative to the theory. Our main technical result is a syntactic characterization of the isotropy group of an algebraic theory, and we illustrate the usefulness of this characterization by applying it to various concrete examples of algebraic theories.