Formal Category Theory: Adjointness for Two Categories (Repost)

The purpose of category theory is to try to describe certain general aspects of the structure of athematics. Since category theory is also part of mathematics, this categorical type of description should apply to it as well as to other parts of mathematics. When I first conducted a seminar on this subject during the Bowdoin Summer Session on Category Theory in 1969, Saunders Mac Lane suggested the name "Formal Category Theory" for this study. The basic idea is that the category of small categories, Cat, is a 2-category with properties and that one should attempt to identify those properties that enable one to do the "structural parts of category theory". The results of this present study suggest the following analogy with homological algebra; Cat corresponds to the category of abelian groups; the categories, Cat x , of category objects in a category x with pullbacks correspond to categories of modules; and representable 2-categories correspond to abelian categories .