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February 28, 2017 2:30 PM
Presented by : Rory Lucyshyn-Wright (Mount Allison)
Topic : Introduction to commutants for algebraic theories
Abstract: In 1963, Lawvere introduced an elegant approach to universal algebra in terms of the notion of algebraic theory, or Lawvere theory. In a 1968 article, Lawvere emphasized that algebraic theories generalize rings, so that certain notions in the theory of rings and modules admit generalizations to universal algebra. In this spirit, Wraith's 1969 lecture notes define a notion of commutant of a set of morphisms in an infinitary algebraic theory. Given instead a morphism of Lawvere theories A : T --> U, one can form an associated Lawvere theory, called the centralizer or commutant of T with respect to A. As a special case, one can take the commutant of a Lawvere theory T with respect to any given T-algebra A. In this talk we will discuss several examples of commutants of Lawvere theories, including theories of modules, affine spaces, convex spaces, and semilattices. Time permitting, we will also comment on generalizations to the realm of enriched algebraic theories and monads.