Lawvere Theories and Definable Operations

Abstract

We introduce the inner theory or, more verbosely, isotropy Lawvere theory functor, which generalizes the isotropy group/monoid by assigning a Lawvere theory of coherently extendable arrows to each object of a category with finite powers. Then, we characterize the inner theory for categories of models of an algebraic (or, more generally, quasi-equational) theory, and note its relationship with a notion of definability for morphisms. Finally, we explore a variety of examples.