Category Theory: A Primer
Introduction
Category theory is a branch of mathematics that provides a framework for studying mathematical structures and their relationships. It was first introduced in the 1940s by Samuel Eilenberg and Saunders Mac Lane as a way to unify different areas of mathematics by focusing on the relationships between objects and their mappings. Since then, category theory has become an essential tool in many areas of mathematics, including algebra, topology, and logic.
Categories
At its core, category theory is concerned with the study of categories. A category can be thought of as a collection of objects and a collection of arrows (also called morphisms) between those objects. These arrows can be composed, much like functions in calculus, to form new arrows. Additionally, each object in a category has an identity arrow associated with it that serves as a sort of "do nothing" arrow.
For example, we can define a category called Set whose objects are sets and whose arrows are functions between those sets. The composition of two arrows in this category is simply the composition of the corresponding functions, and the identity arrow associated with each set is the identity function on that set.
Functors
One of the key ideas in category theory is the concept of a functor. A functor is a mapping between categories that preserves the structure of the categories. Specifically, it maps objects to objects and arrows to arrows, while preserving composition and identity arrows. This means that if we have two categories C and D and a functor F from C to D, then F will preserve the relationships between objects and arrows in C when we translate them to D.
For example, we can define a functor called P that maps a set to its power set (i.e., the set of all subsets of that set) and maps a function between sets to the function between their power sets that takes a subset of the domain and maps it to the subset of the range that contains all of the images of elements in that subset. It can be shown that P preserves composition and identity arrows, so it is a valid functor.
Natural Transformations
Another important concept in category theory is the natural transformation. A natural transformation is a mapping between functors that preserves the structure of those functors. Specifically, a natural transformation maps objects in the domain category to objects in the codomain category and maps arrows in the domain category to arrows in the codomain category in a way that preserves composition and identity arrows.
For example, suppose we have two functors F and G from a category C to a category D. A natural transformation eta from F to G is a family of arrows eta_X in D for each object X in C such that for any arrow f in C, the diagram
F(X) --------> F(Y) | | | eta_X | eta_Y v v G(X) --------> G(Y)
commutes. This means that if we apply F(f) to an object in F(X) and then apply eta_Y to that result, it is the same as if we had first applied eta_X to the object in F(X) and then applied G(f) to the result.
Conclusion
Category theory provides a powerful framework for studying mathematical structures and their relationships. Its focus on the relationships between objects and their mappings allows us to unify different areas of mathematics and gain new insights into fundamental concepts. While this primer only scratches the surface of the vast field of category theory, it is our hope that it has given you a taste of the beauty and power of this subject.