# Why does the name "epimorphism" refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are very much welcome though).

In group theory, an epimorphism is a surjective homomorphism. Where does this name come from? Why have we chosen it?

As a comparison, a monomorphism is an injective homomorphism. Some authors like to call injective functions "one-to-one", and so one can see the rationale behind the "mono" part of the name. But for an epimorphism, I'm not really sure where the terminology has come from (despite reading the wikipedia section on it).

• Greek instead of Latin. Maths draws on both. May 14, 2016 at 18:12
• The prefix epi- has many meanings. According to the American Heritage Dictionary, one is "on, upon." So one could see a connection between being onto and being epimorphic. May 14, 2016 at 18:13
• In analogy to mono-, "The prefix epi-, or ep- if followed by a vowel or the letter "h", is derived from the Greek preposition ἐπί meaning... upon..." en.wikipedia.org/wiki/EPI May 14, 2016 at 18:14
• Since you mentioned category theory I want to point out: If a morphism (in a concrete category) is injective / surjective, then it is mono / epi (because faithful functors reflect monos and epis), but in general the converse does not hold (it does for groups, it does not for rings with $1$). Some authors say a ring morphism is mono, iff it is injective. This is bad practice. May 15, 2016 at 11:50

## 3 Answers

The prefix "epi-" in Greek has several meanings, but a common one is "upon, over". This is similar to the meaning of the prefix "sur-" in French, which was the origin of the term "surjective", introduced by Bourbaki. As such, both give the meaning that the function/morphism "covers" all of its range.

• It also corresponds to the term often used in English, "onto". E.g., a bijection is "one-to-one and onto". May 14, 2016 at 18:35
• Of course. I thought the OP was already familiar with the meaning of "surjective" so I did not mention that. May 14, 2016 at 18:55
• Well, you might be right but one can know the mathematical meaning but not the expression "onto". Since none of the answers mentioned it, I thought I'd throw it in. May 14, 2016 at 19:05

The prefix "epi-" in Greek means "on top of, above". Surjection is a map onto its codomain, and hence the name.

To give another example, the epigraph of a function is the part above the graph.

Also with function between sets sometimes the terms "epic" and "monic" are used instead of "surjective" and "injective"

It comes from the fact that the prefix "epi" is Greek for "upon", "over", or "at". The prefix is also used in, epidemic, epidermis, or epicenter to indicate these meanings. Thus an onto homomorphism is said to be an epimorphism, i.e. a morphism which maps over/upon/onto the range of the function.