  Where in the landscape of many-body phases of matter do we place the Higgs
condensate of a gauge theory? On the one hand, the Higgs phase is gapped, has
no local order parameter, and for fundamental Higgs fields is adiabatically
connected to the confined phase. On the other hand, Higgs phases such as
superconductors display rich phenomenology. In this work, we propose a minimal
description of the Higgs phase as a symmetry-protected topological (SPT) phase,
utilizing conventional and higher-form symmetries. In this first part, we focus
on 2+1D $\mathbb Z_2$ gauge theory and find that the Higgs phase is protected
by a higher-form magnetic symmetry and a matter symmetry, whose meaning depends
on the physical context. While this proposal captures known properties of Higgs
phases, it also predicts that the Higgs phase of the Fradkin-Shenker model has
SPT edge modes in the symmetric part of the phase diagram, which we confirm
analytically. In addition, we argue that this SPT property is remarkably robust
upon explicitly breaking the magnetic symmetry. Although the Higgs and confined
phases are then connected without a bulk transition, they are separated by a
boundary phase transition, which we confirm with tensor network simulations.
More generally, the boundary anomaly of the Higgs SPT phase coincides with the
emergent anomaly of symmetry-breaking phases, making precise the relation
between Higgs phases and symmetry breaking. The SPT nature of the Higgs phase
can also manifest in the bulk, e.g., at transitions between distinct Higgs
condensates. Finally, we extract insights which are applicable to general SPT
phases, such as a 'bulk-defect correspondence' generalizing discrete gauge
group analogs of Superconductor-Insulator-Superconductor (SIS) junctions. The
sequel to this work will generalize 'Higgs=SPT' to continuous symmetries,
interpreting superconductivity as an SPT property.
