The mathematical study of categories, which are basically a collection of objects and arrows, and a composition operation, satisfying some basic properties. Lecture notes –Very nice lecture videos
It can be used as a foundation of mathematics, as an alternative to Set theory and has relations to Functional programming, Lambda calculus, Logic, etc.
Functoriality, Naturality, Universality
http://chalkdustmagazine.com/features/an-invitation-to-category-theory/
Duality principle: every property/theorem has a corresponding dual property/theorem, by applying the theorem in the opposite category and interpreting it in the original category. In other words, a statement S is true about C if and only if its dual (i.e. the one obtained from S by reversing all the arrows) is true about C op
Monic morphism, Epic morphism, Isomorphism
Initial object, Terminal object
unique isomorphism property: There is a unique isomorphism between any pair of initial objects; thus initial objects are ‘unique up to (unique) isomorphism’, and we can (and do) speak of the initial object (if any such exists). By Duality, any terminal object also satisfies this, because isommorphism is self-dual.
Universal construction: See Category Theory 4.1: Terminal and initial objects
Curry-Howard-Lambek correspondence
Categorical product generalizing Cartesian products, and their dual, Coproducts.
Pullbacks and Equalizer (category theory)
things which are isomorphic in a category tend to be undistinguishable that is if a property applies to one, it applies to the other too
Representable functor and Yoneda embedding
Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g.
Rel (relations). objects are Sets, and arrow if there is a relation. Composition is Relational composition
..A category in which for each pair of objects A, B there is at most one morphism from A to B is the same thing as a Preorder, i.e. a reflexive and transitive relation.
Category of categories. Product category is the Categorical product in Cat.
Universal mapping property allow one to define the notion of canonica/natural.
Any map which does the same thing as the canonical map, factors through the canonical map in an unique way. For example, Equalizer (category theory)
Category Theory: The Beginner’s Introduction (Lesson 1 Video 1)
Brian Beckman: Don't fear the Monad
Category Theory Applied to Neural Modeling and Graphical Representations (2000)
Modeling cognitive systems with Category Theory
Bayesian machine learning via category theory
What Might Category Theory do for Artificial Intelligence and Cognitive Science?
Theory of Interface: Category Theory, Directed Networks and Evolution of Biological Networks