Mohammad Jabbari: Algebraic Theories and Algebraic Categories
Tid:
Fr 2012-02-24 kl 13.15 - 15.00
Plats:
Room 16, building 5, Kräftriket, Department of mathematics, Stockholm university
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Algebra is a subject dealing with variables, operations and equations. This viewpoint - present in the Tarski-Birkhoff's tradition in universal algebra - was the scene before the birth of categorical logic in F. W. Lawvere's 1963 thesis. In his thesis, among other things, he objectified algebraic "theories" as special kind of categories and algebraic structures as special set-valued functors on them. He learned to do universal algebra by category theory! This was the start of a fruitful line of discoveries which culminated in the creation of (elementary) topos theory by Lawvere and M. Tierney in late 1970.
At about the same time, an alternative categorical approach to general algebra emerged out of the collective efforts of some homological algebraists (Godement, Huber, Eilenberg, Beck, ...) which centers around the notion of "monads (triples)". Among other intuitions, this machinery enables us to grasp the algebraic part of an arbitrary category. This approach later found applications in descent theory and computer science. The highpoint of this area is Beck's monadicity theorem.
This seminar is divided into three parts. At first we review some of Lawvere's ideas in algebraic theories and state his characterization theorem for algebraic categories. Then we shortly describe the monadic approach to algebra. Finally we state some theorems about the equivalences between these three approaches into algebraic categories.
Main References.
[1] Lawvere, F. W., "Functorial semantics of algebraic theories", PhD Thesis, Columbia, 1963.
[2] Adamek, J., et all, "Algebraic A Categorical Introduction to General Algebra", Cambridge University Press, 2011.
