(Terminology) bijective correspondence between proper classes?

[Fredrik]

I'm just looking for the right words to use to talk about something like the "bijective correspondence" between the class of groups defined as pairs and the class of groups defined as 4-tuples. I'm talking about the "map" $(G,\star,i,e)\mapsto (G,\star)$. It seems to me that it shouldn't be called "map", "function", "bijection" or anything like that, since its "domain" and "codomain" aren't sets. So is there something we can call it?

 

[micromass]

Check Jech's "Set theory" page 11 and 12. He defines functions (and thus bijections, etc.) simply as classes. So a function isn't necessarily between set according to his terminology.

However, he does always seem to be careful to make sure the reader knows it's a class. For example, check the replacement schema on page 13. He says "If the class $F$ is a function then...". In this case, we know $F$ isn't necessarily a set and thus that the domain and codomain aren't necessarily sets. This is different from saying "Take the function $F$ then..." which is more ambiguous.

So if I were you, I would use the same terminology, but I would make it clear we're dealing with classes.

 

[Fredrik]

Thank you, that's exactly what I needed. I don't think I've seen those more general definitions of relation, function and operation before, but now that I have them in front of me, they seem very natural.