Category Theory

Category theory is a set of tools and ways of thinking for exploring relations between abstract structures in maths. It's focus is not on the objects, but on the relations between them and the rules they operate under.

An alternative way of thinking about category theory could be a view on the patterns of human thought, rather than the patterns in nature. We can't help but impose these patterns everywhere we go.

At the core of category theory is, unsurprisingly, the category. A category is a collection of objects and morphisms, where each object has at least an 'identity morphism'. A morphism is an arrow pointing from one object to another. Objects exist as named points to give the morphisms context. Category theory concerns itself with the composition of morphisms within different categories and the different states that are possible.

References

• Video Series: Category Theory
• nLab

Backlinks

• Topic: Set Theory
  • Category theory for programmers[^1] touches on set theory in order to expand into category theory
• CT: Morphism
  • In category theory, a morphism is a transformation from one object to another. This can be within a category or between categories.
• CT: Object
  • In Category Theory, an object is a part of a category. It exists to define each end of a morphism.
• CT: Categories
  • In Category Theory, A category is a group of objects and morphisms.
• Topic: Maths
  • category theory
• Video Series: Category Theory
  • category theory
• CT: Isomorphism
  • In category theory, an isomorphism is a morphism which can be inverted. $f$
• The singleton set and the unit type
  • The singleton set from set theory corresponds to the Unit type from type theory. Also to true in logic, and to a 1-category in category theory.
• Typeclass
  • These patterns are available to be constructed in user space. Some of the more common patterns are taken from category theory, such as Functors, Monoids etc, etc.
• The empty set and the void type
  • The empty set in set theory corresponds to the Void type in type theory or to false in logic. There's also the 0-category in category theory.
• A set-theory monoid and a category-theory monoid are the same thing
  • The set theory definition and the category theory definition of monoids are different views over the same concept. Category theory defines a monoid as any category with a single object. Set theory defines a monoid as a set with an associative binary operation and a unit element.
• Notes
  • Category Theory