28 October 2026 — Padova

When Lawvere Meets Noether: A Logical Approach to Categorical Algebra and Isomorphism Theorems

Davide Trotta, University of Padova

Abstract

Lawvere doctrines are a fundamental notion in categorical logic, providing a unified framework for the study of logical systems and their categorical semantics. Several important classes of categories, including regular categories and elementary toposes, admit characterizations in terms of doctrines satisfying suitable properties. However, the potential of this framework in categorical algebra remains largely unexplored.

This talk will give an accessible introduction to Lawvere doctrines and explore their applications to categorical algebra. After explaining how familiar algebraic notions, such as kernels, images and exact sequences, can be generalized to the setting of Lawvere doctrines, we will show how the first Noether isomorphism theorem can be stated and proved under suitable doctrinal assumptions.

From a logical perspective, this approach identifies the logical principles underlying the isomorphism theorem, while from an algebraic perspective, this result provides the key to characterizing abelian and semi-abelian categories in terms of doctrines.

We will also discuss the connection with Janelidze’s theory of Noetherian forms and conclude with potential applications and directions for further research.

The results presented are based on joint work with J. Lanfranco and Z. Janelidze.