In algebraic geometry a family is essentially a flat morphism $X\to Y$, between some kind of schemes. I am interested in the analytic counterpart. When in the litereture one writes "An analytic family" or "family of analytic sets\spaces", what exactly is meant?
More specifically, is an analtic family given by the fibres of any moprhism $X\to Y$ between analytic sets\spaces? Or do we need to add something to this morphism which would be the counterpart of flatness? Say it must by submersive? Some kind of stabilty conditions?
Why am I asking this? Well I am writing a paper, where I have a result for "analytic families" given by an arbitrary analytic map $X\to Y$. When stating the result, shoud I write "analytic family" or should I specifically mention that I mean the fibres of an arbitrary analytic map?
Thank you very much!
Even in the complex analytic world I would call a proper flat morphism of complex spaces an analytic family.
If your result holds for an arbitrary morphism of complex spaces I would definitely state it that way to avoid confusion.