Category theory

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Category theory is a big, complicated, confusing[citation needed] area of mathematics concerned with "Categories".

Introduction

A category can be thought of as—but is not equivalent to—a labeled digraph: a set of nodes containing values with directed edges mapping nodes to other nodes. In the more general way of thought, a category is a class of "objects"—which can be any type of arbitrary thing—with "arrows" (also called "morphisms") between them mapping one to another. An arrow represents some sort of transformation between a "domain" (the object from which the arrow originates) and the "codomain" (the destination object).

Categories also have some sort of operation for composing arrows, written f · g. This leads to the property that, if there is are two arrows f : A → B and g : B → C, then there is another arrow f · g : A → C. This should be reminiscent of function composition in other areas of mathematics, but represents a more general concept. Note that it is required that · is associative—that is, (f · g) · h ≡ f · (g · h).

The final requirement for a category is that every object A has an "identity morphism" idA. For every morphism f : A → B, The identity object has the property that f · idA ≡ f, and for every morphism g : B → A, idA · g ≡ g

Higher Categories

In certain contexts, such as topology, it is natural to define 2-morphisms, things similar to morphisms whose domains and codomains are themselves morphisms. These can be composed in more varied ways, and lead to a generalization of associativity known as a coherence law. There is a hierarchy of k-morphisms parameterized by natural numbers, which may or may not be directed. The study of categories with such higher k-morphisms is known as Higher Category Theory, and is discussed in detail on The nLab.