A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance

Abstract

The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of "chimeras" or "heteromorphisms" between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical concepts and some central philosophical themes.