Category Theory

• a new and fundamental insight about the nature of mathematics.
• a category = mathematical universe
• ingredients = objects, maps, composition of maps
• 'A general theory of natural equivalences’ Eilenberg and Mac Lane
• Spatially, think about the resolution of vectors in space.
  • For Gallelio to accurately talk about projectile motions, he needed to have the following understanding:
  • As time passes, each moment in time has an associated position in space described by some map.
  • In this space there are projections onto planes (shadows) and heights (lines) and given a shadow and a height, the object is completely defined in space.
  • In essence, for each moment in time there is an associated position in space and for each position is in space there is a shadow and a height which completely describe it.
  • Time maps to space which maps to planes and maps to lines.
  • This paints a picture of math being used in a certain "category", a certain mathematical universe where maps and this idea of multiplication of planes and lines into space is defined.