Morphisms! Time to set the record straight In my limited mathematical reading, I often come across authors that declare functions as isomorphisms, homomorphisms, or homeomorphisms (or any other variety of morphism).  Although I've found definitions of these terms in resources like Wikipedia and Wolfram MathWorld, I still am not able to completely and concretely understand them and their differences.
I assume all these morphisms are related but differentiated by certain properties (an isomorphism has an inverse? a homeomorphism is an isomorphism constrained to topology?) but I don't know exactly what those properties are.
Could someone provide a picture of all types of defined morphisms and how they relate (I'm thinking possibly they fit into a hierarchy?).  Further, providing examples of what is a particular morphism and what is not that same morphism (for example, "f is an endomorphism but not a homeomorphism because...") would be very helpful.  Further, how do these morphisms relate to topology and category theory?
Thanks!
 A: There are two types of term being conglomerated into one.
Terms like morphism ( = homomorphism), isomorphism, endomorphism, automorphism, monomorphism, epimorphism have definitions that are independent of the subject matter (groups, rings, topological spaces, manifolds, etc) and belong to category theory.
When the general definitions are applied in particular categories, such as Topological spaces or Differential manifolds, they are often given additional prefixes or more informative names, such as


*

*diffeomorphism = an arrow in the category of smooth manifolds that is an isomorphism, or more briefly, an "isomorphism of smooth manifolds"

*homeomorphism = an isomorphism (in the category) of topological spaces

*embedding = a monomorphism of spaces


The historical process was the reverse.  Different fields had their own concepts of important functions or relations between objects, and these were often defined by the same pattern.  The category-theoretic definitions abstract or unify some of the patterns.
