An Introduction To Category Theory In Four Easy Movements

Description

In many parts of mathematics we study 'structures' of various kinds. These may be algebraic, topological, geometric, or a mixture of of each. As well as looking at each structure in isolation we also consider how two structure can be compared using morphisms, maps, translations, or whatever. In fact, whenever we have a collections of structures of a like kind we should always try to isolate the appropriate comparison gadgets. Category theory codifies these general matters. Thus a category consists of objects to take the role of structures, and arrows to take the role of the comparison gadgets. However, if this codification was all it did then category theory would be rather a superficial subject. Each category is itself a structure, so how should two categories be compared? We match objects against objects and arrows against arrows. The resulting comparison gadget is a functor. It turns out that many constructions used in mathematics are functorial, and have been around since before category theory was developed. Furthermore, once we see that a construction is functorial we begin to understand it in a better way. In the first instance category theory was devised (around 1945) to explain why certain manipulations are 'natural' and others are not. This uncovered the notion of a natural transformation which is the appropriate comparison gadget between functors. Later category theory uncovered the idea of an adjunction which helps to unify many different results in mathematics.