The Department of Mathematics & Statistics invites students, faculty, and staff to attend the upcoming colloquium with speaker Dr. Basil Nanayakkara on Thursday, August 6th, from 2:00 PM to 3:00 PM in MCJ 404.  The talk is entitled What is a natural map?

Abstract

Mathematicians use the term ‘natural map’ informally.  For example, if A is a ring and I is an ideal of A, then the map A → A/I given by a → a + I is said to be natural.  As another example, if V is a vector space, then there is a natural map from V to its double dual V ∗∗. (Thus, V and V∗∗ are naturally isomorphic when V is finite dimensional.)  Also, if X is a topological space and Y is a subspace of X, then the quotient map X → X/Y is natural. There are many other examples one can give.

We will discuss the definition of natural transformations in category theory, and show that the maps mentioned above are indeed natural. If time permits we will show that the dual of the Zariski cotangent space at a point of a smooth manifold is naturally isomorphic to the tangent space at that point.