. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for ….

. . . in one sense a foundation is a security blanket: If you meticulously follow the rules laid down, no paradoxes or contradictions will arise. In ….

... the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand..

The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of s….