In the first part of the 20th century people concentrated on the study of individual spaces, such as groups, topological spaces, and various kinds of linear spaces. Interesting associations between spaces were found: homology and homotopy groups associated with topological spaces, topological spaces associated with Boolean algebras, etc. However, it was eventually realized that maps between spaces are equally important, and that many properties of maps are best viewed in a more general framework.
Some examples that such a framework should cover:
(1) the class of groups and homomorphisms between them;
(2) the class of topological spaces and continuous functions between them;
(3) the class of partially ordered sets and isotone maps between them.