  The aim of this work is to build a theory of homomorphisms of Hecke and Artin
monoids with possible applications to representation theory. Our original
motivation was the striking observation that parabolic projections of Hecke
monoids respect all parabolic elements. We found other classes of homomorphisms
of Hecke monoids with the same property and discovered that many of them lift
to homomorphisms of covering Artin monoids with a similar property. It turned
out that they belong to a much larger class (in fact, a category) of
homomorphisms of Artin monoids, most of which appear to be new.
