A F . B N  C T

e concept of adjoint functors was first defined by Daniel Kan in  [K-], and since then, it has proven to be quite useful. e history leading up to Kan’s breakthrough is concisely described in [ML-, p. ]. e motivation for the study of adjoint functors in this paper arose out of the importance of universal constructions su as products, coproducts, free objects, etc. whi pervade all of mathematics in a fundamental way. A aracterization of universal constructions in terms of adjoint functors and an associated morphism (the unit of the adjunction) is presented, following [H-]. However, we present examples of universal construction whi, despite their importance and usefullness, seem to la sufficient “universality”. A section of this paper is focused on developing a stronger definition of universal construction—one whi we claim is more “universal”. is definition also gives us a good tool to understanding when a given functor cannot have a le or right adjoint. Another focus of this paper is to demonstrate the limitations of adjoint functors to explain certain phenomena in mathematics. Our major example is that of the de Rahm cohomology functor H∗ dR. Conversely, we also demonstrate how the categorical perspective on certain problems can lead to elegant methods whi would be considerably more difficult to describe without the categorical language. In particular, we explain a categorical approa to determining if certain universal objects exist in a category. I would like to thank Dr. Alves for mentoring me during this semester, and Dr. Curtis, Dr. Clifford and Dr. Hingston for helpful discussions about the topology and analysis going on in the last sections. I am also indebted to the wonderful websitemathoverflow.net for the useful discussions when I was geing lost in the abstraction. I would especially like to thank Andrew Stacey, Reid Barton and Chris Sommer-Pries.