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.

• Universal Algebra and Diagrammatic Reasoning - PDF and Postscript versions.

  Even better - PDF with hyperlinks.

For more details, try these:

• John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory
• John Baez and James Dolan, Categorification
• John Baez, Quantum quandaries: a category-theoretic perspective
• Michael Barr and Charles Wells, Toposes, Triples and Theories
• Stanley Burris and H.P. Sankappanavar, A Course in Universal Algebra
• Aaron Lauda, Frobenius algebras and ambidextrous adjunctions
• Aaron Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras
• F. William Lawvere, Functorial Semantics of Algebraic Theories
• Robert Rosebrugh, Nicoletta Sabadini and Robert Walters, Generic commutative separable algebras and cospans of graphs
• Todd Trimble, Bar constructions

On a more touristic note, you can also see some pictures I took near Marseille around this time.

home