My research is on Category Theory and most of it concerns adjunctions.
It has been said that categories are what one has to define in order to
define functors and these, in turn, are what one has to define in order to
define natural transformations. To this conventional wisdom can be added
the assertion that natural transformations are what one has to define in
order to obtain a first, reasonably full, view of adjunctions. Just as
natural transformations were recognized by mathematicians --- but not
defined --- before Samuel Eilenberg and Saunders Mac Lane provided the
framework of Category Theory, adjunctions too have been an implicit part of
Mathematics for many, many years. Lawvere showed us that the rules of
inference of logic itself provide instances of adjunctions and that viewed
as such admit profound generalizations.