Functors from a mathematical perspectives

Shawn Martin [email protected] HaskellSG meetup, [2014-01-24 Fri 19:34] ** topology Mathematics of super-rubber objects Coffee cup = Human being Poincare invented topology in order to work on differential equations ** how many fundamentally different paths on a torus?

• direct path
• infinitely many paths looping clockwise
• infinitely many paths looping counter clockwise ** A topological space X with a base point x_0 forms a group where
• op is path concatencation
• identity is path from x_0 to x_0
• inverse is path in reverse direction ** A group is a generalization of a set of symmetrics of an object
• integers with addition
• strings of chars, anti-chars with concatenation and annihilation ** Fundamental group of a
• coffeecup is the set of integers
• infinity symbol with base point at the crossing point is strings of a, b, and their anti-chars with concatenation and annihilation
• hollow torus = infinity plane that loops around on all sides is strings of a, b with concatenation ** A category is like a world with its own objects and functions between objects
• the world of topological spaces and continuous functions
• the world of groups and group homomorphism ** A functor is a recipe for taking one category and getting out another
• Fundamental group of a topological space (pi_1) is a functor from topological spaces to groups
• []: a -> [a] is a mapping of the objects, fmap f: [a] -> [b] is a mapping of the function ** natural transformations Consider category Top* (topological spaces with a basepoint) and basepoint-preserving functions, can we do away with the basepoint?

Top*,*’, topological spaces with two base points, and double base point preserving functions. There are two fundamental group functors that are not naturally equivalent. ** monads search “typeclassopedia” functors is a generalization of applicative, which a generalization of monads.