Dear Steve,
Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are.
Of course, they are not.
They are presentations (for theories), and have status similar to other sorts of presentations (for groups, rings, etc.)
I think about a sketch as an alternative to a string of formulae, which represents (axioms of) a theory. I didn't try to compare sketch theory with group theory. I tried to compare sketch theory with the general setting, in which group theory is developed a la Bourbaki (along with many other theories). A classical account of this general setting (which differs at certain points with Bourbaki's version) is Tarski's model theory. The notion of presentation in my understanding implies that what a given presentation is a presentation *of* is somehow given in advance. I try to think of a sketch as a means to build a theory, not to present a ready-made theory. Perhaps *representation* is a better word for it than *presentation*. best, andrei