All of Basic Category Theory • Mark Hopkins • YOW! 2018

Categories are best thought of as networks or systems, with nodes (objects) and edges (arrows/morphisms) that preserve structure and relationships

The three key pillars of category theory are:

• Universal properties (products, sums, limits)
• Functors (mappings between categories)
• Natural transformations (morphisms between functors)

Representable functors are fundamental building blocks that act like “basis vectors” in the pre-sheaf world, with each representing a data structure of fixed shape

Category theory connects multiple domains through the Curry-Howard-Lambek correspondence:

• Logic (propositions)
• Programming (types)
• Category theory (objects)

Adjoint functors provide “best approximations” when exact inverses don’t exist, making them widely applicable in mathematics and programming

Limits and co-limits generalize the concepts of products and sums, with limits being like “generalized products” and co-limits like “generalized sums”

Natural transformations emerge from examining how functors interact, with polymorphism in programming languages approximating naturality

CAN (Kan) extensions allow extending functors to larger categories, providing powerful tools for working with categorical structures

Category theory proved essential in physics (particularly quantum mechanics) despite initial resistance to its abstract nature

The field unifies seemingly different mathematical concepts through universal constructions, making it applicable across mathematics, computer science, and physics