Abstract: Fundamental to applications of ordinary persistent homology in one parameter is the reconstruction of a module from the perfect matching between left endpoints and right endpoints of its bar code.  Do these concepts have analogues in multiple parameters?  The answer is largely yes: endpoints can be defined, and the module can be reconstructed from them, though the correspondence is not a perfect matching but rather a more arbitrary linear map.  The algebra needed for these developments will be covered from scratch, followed by a view toward how they might be used for computational purposes.