The Category Theory

Overview of functors

Functors have, or are at least spoken of in such a way as to convey, an air of complexity. In practice, however, they are rather powerful artifacts at the heart of which is a map operation not too different from the one described earlier. Functors are efficient chaining structures from which monads are derived. Understanding the concepts that make them the efficient objects they are, requires combining knowledge of category theory with a practical understanding of the functional programming paradigm as a whole. The misconception that functors are exceedingly arcane could not be further from the truth: this chapter attempts to disprove it.

A crash course in category theory

In mathematics, problem solving usually takes the form of either generalization or abstraction. Generalizing means starting with a solution adapted for a few contexts and expanding its scope and making it viable for other problems. PHP’s programming paradigm has been generalized to include functional, object-oriented, and even asynchronous approaches, making it a viable multi-purpose or general-purpose language.

Abstraction, on the other hand, involves tidying away a pattern adaptable to similar situations. It renders a solution that’s usable for different invariants. Abstractions and blueprints have a point in common. That is, they’re both reusable resources. Numbers are an abstraction that allows us to count in different bases.

Category theory is a system of useful generalizations and abstractions in mathematics. It takes a taxonomical approach to mathematics, with many similarities to programming.

Defining category theory

The programmatic and mathematical paradigms overlap in many places. One overlap is noticeable through category theory. A category is a set of objects and the arrows between them. Object contents aren’t the primary focus of category theory, because they’re abstractions. An object could be a representation of any number of mathematically viable ideas. Arrows, on the other hand, are the mappings between objects (also referred to as morphisms). They represent abstract object-to-object transformations.