Polymorphic Automorphisms and the Picard Group

TL;DR

This paper explores the structure of automorphisms in algebraic and categorical theories, revealing that the isotropy group of a monoidal category is its Picard group and providing explicit descriptions for presheaf categories.

Contribution

It extends the characterization of automorphism groups from algebraic theories to quasi-equational theories and links isotropy groups to the Picard group in monoidal categories.

Findings

Isotropy group of a strict monoidal category equals its Picard group.

Provides explicit description of covariant isotropy group of presheaf categories.

Extends syntactical characterization of automorphisms to broader classes of theories.

Abstract

We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply this characterization to prove that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore, we obtain an explicit description of the covariant isotropy group of a presheaf category.