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Where in the landscape of many-body phases of matter do we place the Higgscondensate of a gauge theory? On the one hand, the Higgs phase is gapped, hasno local order parameter, and for fundamental Higgs fields is adiabaticallyconnected to the confined phase. On the other hand, Higgs phases such assuperconductors display rich phenomenology. In this work, we propose a minimaldescription of the Higgs phase as a symmetry-protected topological (SPT) phase,utilizing conventional and higher-form symmetries. In this first part, we focuson 2+1D $\mathbb Z_2$ gauge theory and find that the Higgs phase is protectedby a higher-form magnetic symmetry and a matter symmetry, whose meaning dependson the physical context. While this proposal captures known properties of Higgsphases, it also predicts that the Higgs phase of the Fradkin-Shenker model hasSPT edge modes in the symmetric part of the phase diagram, which we confirmanalytically. In addition, we argue that this SPT property is remarkably robustupon explicitly breaking the magnetic symmetry. Although the Higgs and confinedphases are then connected without a bulk transition, they are separated by aboundary phase transition, which we confirm with tensor network simulations.More generally, the boundary anomaly of the Higgs SPT phase coincides with theemergent anomaly of symmetry-breaking phases, making precise the relationbetween Higgs phases and symmetry breaking. The SPT nature of the Higgs phasecan also manifest in the bulk, e.g., at transitions between distinct Higgscondensates. Finally, we extract insights which are applicable to general SPTphases, such as a 'bulk-defect correspondence' generalizing discrete gaugegroup analogs of Superconductor-Insulator-Superconductor (SIS) junctions. Thesequel to this work will generalize 'Higgs=SPT' to continuous symmetries,interpreting superconductivity as an SPT property.

Higgs Condensates are Symmetry-Protected Topological Phases: I. Discrete Symmetries