John Baez

Lectures at Geometry of Computation 2006

January 30 - February 3, 2006

Universal Algebra and Diagrammatic Reasoning

Since the introduction of category theory, the old subject of "universal algebra" has diversified into a large collection of frameworks for describing algebraic structures. These include "monads" (formerly known as "triples"), the "algebraic theories" of Lawvere, and the "PROPs" of Adams, Mac Lane, Boardman and Vogt. We give an overview of these different frameworks, which are closely related, and explain how one can reason diagrammatically about algebraic structures defined using them. Our treatment of monads focuses on the abstract "bar construction". Our treatment of algebraic theories and PROPs explains how the latter are related to Feynman diagrams, and leads up to an adjunction between algebraic theories and PROPs which is analogous to the relation between classical and quantum physics. We conclude with some reflections on how features of our physical universe have influenced our notions of universal algebra.
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On a more touristic note, you can also see some pictures I took near Marseille around this time.

© 2006 John Baez