# 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!

• This is pretty much the definition of too broad. Dec 20, 2015 at 0:19
• Fairly related: See many of the good answers to What does “homomorphism” require that “morphism” doesn't? Dec 20, 2015 at 0:28
• I've always assumed that the reason topologists have stuck with "homeomorphism" rather than "isomorphism" is just that if you are specifically referencing which category you are in - diffeomorphism and homeomorphism can apply to a lot of the same spaces. It's also possibly the case that "isomorphism" is too strongly viewed as an algebraic term. Dec 20, 2015 at 0:33
• @Matt, would constraining this question to requesting a picture representing the each of the morphisms in category theory be more productive? Dec 20, 2015 at 0:44
• Category theory is a morphism factory. As long as composition of morphisms is associative and every object has an identity morphism, anything you want can be a morphism. What do you mean by "each of the morphisms"? Dec 20, 2015 at 0:49

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.

• Thomas Andrews' comment explains well why additional words are needed. Sometimes the same space can be considered in more than one category (the circle is a group, a topological space, and a manifold) and there can be a need to distinguish different notions of isomorphism applied to the same objects.
– zyx
Dec 20, 2015 at 0:44