The paper discusses (1+1)d gapless phases with non-invertible global symmetries, also known as categorical symmetries. These phases include gapless SPT (gSPT) and gapless SSB (gSSB) phases, which exhibit properties analogous to gapped SPT and gapped SSB phases, respectively. The authors fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram described by a Hasse diagram, which is a partially ordered set that defines the possible deformations connecting these phases. This framework allows for the identification of intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases, which cannot be deformed into gapped SPT or SSB phases. The analysis is based on Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. The paper provides examples of igSPT and igSSB phases for specific symmetries, such as $\mathbb{Z}_4$ and $\text{Rep}(D_8)$, and discusses the classification of phases using functors between fusion categories. The framework also addresses the gauging of trivially acting non-invertible symmetries and the decomposition patterns that arise from such gaugings.The paper discusses (1+1)d gapless phases with non-invertible global symmetries, also known as categorical symmetries. These phases include gapless SPT (gSPT) and gapless SSB (gSSB) phases, which exhibit properties analogous to gapped SPT and gapped SSB phases, respectively. The authors fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram described by a Hasse diagram, which is a partially ordered set that defines the possible deformations connecting these phases. This framework allows for the identification of intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases, which cannot be deformed into gapped SPT or SSB phases. The analysis is based on Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. The paper provides examples of igSPT and igSSB phases for specific symmetries, such as $\mathbb{Z}_4$ and $\text{Rep}(D_8)$, and discusses the classification of phases using functors between fusion categories. The framework also addresses the gauging of trivially acting non-invertible symmetries and the decomposition patterns that arise from such gaugings.