Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm writing "a structure preserving surjection" way too much when I need to refer a function of the following property:

$$ Y \subseteq Z, X \subseteq Z. g: Z \to A, g \text{ is some fixed function}.$$ $$ \phi : X \to Y, \phi \text{ is a surjection, and } g(\phi(x)) = g(x).$$

Would a category theorist frown if I call $\phi$ a epimorphism?

share|cite|improve this question
I expect they would object mightily, unless the functions with this property are right-cancellable. – Arturo Magidin Jul 1 '12 at 22:01
Regular people might object too ^^ – Olivier Bégassat Jul 1 '12 at 22:40
up vote 2 down vote accepted

If you don't mind making up names (obviously, make sure you define them prominently), then here are two ideas:

  • The preimage under $g$ of each point in $g(Y)$ contains two points, one of which is mapped to the other by $\phi$. Perhaps you could call $\phi$ a preimage map.
  • The image of $X\subset Z$ under $g$ is invariant under $\phi$. Perhaps you could call $\phi$ an invariance, or equivariance.

(I like the second idea better than the first. In fact, I see some nonce uses of invariance in what appears to be this sense on Google Books (example).)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.