Since definitions form the second step in the development of any theory, with the first step being the listing of axioms, one naturally questions when a definition is good, or as to how to make a good definition.
It turns out that a philosophical and logical answer is available naturally from
the viewpoint of Category Theory involving the formulation of a
suitable Universal Property.
Thus for example, one may define a polynomial ring in certain variables
using neither the notion of variable nor of polynomial,
and a basis of a vector space using neither the notion of linear combinations
(linear dependence / independence) nor of spanning sets.
In this talk we shall try to explain using some examples the nexus between good definitions and
universal properties, with applications to Algebra, Topology etc.
