  The subject of persistent homology has vitalized applications of algebraic
topology to point cloud data and to application fields far outside the realm of
pure mathematics. The area has seen several fundamentally important results
that are rooted in choosing a particular algebraic foundational theory to
describe persistent homology, and applying results from that theory to prove
useful and important results.
  In this survey paper, we shall examine the various choices in use, and what
they allow us to prove. We shall also discuss the inherent differences between
the choices people use, and speculate on potential directions of research to
resolve these differences.
