  We discuss (1+1)d gapless phases with non-invertible global symmetries, also
referred to as categorical symmetries. This includes gapless phases showing
properties analogous to gapped symmetry protected topological (SPT) phases,
known as gapless SPT (or gSPT) phases; and gapless phases showing properties
analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to
as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped
SPT and SSB phases, into a phase diagram describing possible deformations
connecting them. This phase diagram is partially ordered and defines a
so-called Hasse diagram. Based on these deformations, we identify gapless
phases exhibiting symmetry protected criticality, that we refer to as
intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases.
This includes the first examples of igSPT and igSSB phases with non-invertible
symmetries. Central to this analysis is the Symmetry Topological Field Theory
(SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld
center of the symmetry category. On a mathematical note, gSPT phases are
classified by functors between fusion categories, generalizing the fact that
gapped SPT phases are classified by fiber functors; and gSSB phases are
classified by functors from fusion to multi-fusion categories. Finally, our
framework can be applied to understand gauging of trivially acting
non-invertible symmetries, including possible patterns of decomposition arising
due to such gaugings.
