Some interesting Points: An exploration of the concept in algebraic geometry and category theory

Abstract

The concept of a ’point’ has been a part of mathematics since its conception,
it’s an intuitive idea that every person implicitly knows. Nonetheless, if the
concept is formalized and abstracted there appear new intrinsic properties that are
more complex than the euclidean one of being ’that which has no parts’. Two paths
are explored: the first one using Algebraic Geometry and structures that come from
algebraic objects and the second one using the language of Category Theory.