  We present a new language for persistent homology in terms of Galois
connections. This language has two main advantages over traditional approaches.
First, it simplifies and unifies central concepts such as interleavings and
matchings. Second, it provides access to Rota's Galois connection theorem -- a
powerful tool with many potential applications in applied topology. To
illustrate this, we use Rota's Galois connection theorem to give a
substantially easier proof of the bottleneck stability theorem. Finally, we use
this language to establish relationships between various notions of
multiparameter persistence diagrams.
