Category theory: an abstract nonsense? | Tangente

Defining a category requires specifying a collection of objects and, for each pair of objects, a set of morphisms from the first object to the second. For example, consider the collection of sets: if E and F are two sets, the set Mor(E, F) can be the set of maps from E to F; E is then the domain of an element f of Mor(E, F), and F its codomain. We often write f : E → F to express that f is a map from E to F. When F = E, Mor(E, E) has a particular element: the identity map, denoted I_E. Finally, if E, F and G are three sets and we consider f : E → F and g : F → G, two maps such that the codomain of the first is contained in the domain of the second, we then define g o f : E → G, the composite of f and g. This operation o, when defined, is associative. The category thus defined is often denoted Set.

Quite clearly, the notion of the category of sets belongs to a higher level of abstraction than that of a set, as considered by Georg Cantor at the end of the 19^th century. It was introduced by the American mathematicians Samuel Eilenberg and Saunders Mac Lane in the 1940s, in the course of research on group theory and algebraic topology.