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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?

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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

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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).)

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