Type-amalgamation properties of a theory are about when certain kinds of finite directed systems of types are consistent, and when such systems have an essentially unique solution. A fundamental property of stable and simple theories is that triples of complete types p(x,y), q(y,z), and r(x,z) over a model which satisfy natural coherence and independence conditions can always be amalgamated -- this is called 3-existence over models. Similarly, the n-existence property is about amalgamating systems of types indexed by proper subsets of {1, ..., n}. Hrushovski recently gave an interesting characterization of which stable theories have the 4-existence property which has to do with the definability of certain kinds of groupoids in the theory. In ongoing work with Alexei Kolesnikov, we are trying to generalize these results to characterize the n-existence property using higher-dimensional groupoids, and I will try to give an idea of our methods.