Re: Name for a concept

Is there a standard name for a square A ----> B | | | | | | v v C ----> D in which the canonical map A ---> B x_D C is epic?

These are called "quasi-pullbacks" by Joyal, and they form a class of "open maps" in the category of squares. The pullbacks form the corresponding class of etal maps. These two classes are essential for the development of the theory (etal class and open class in the sense of Joyal). There are published articles by Joyal and Moerdijk on the subject.