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A bstract In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry. 

A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions. 

Cell death is intrinsically linked with immunity. Disruption of an immune-activated MAPK cascade, consisting of MEKK1, MKK1/2, and MPK4, triggers cell death and autoimmunity through the nucleotide-binding leucine-rich repeat (NLR) protein SUMM2 and the MAPK kinase kinase MEKK2. In this study, we identify aCatharanthus roseusreceptor-like kinase 1-like (CrRLK1L), named LETUM2/MEDOS1 (LET2/MDS1), and the glycosylphosphatidylinositol (GPI)-anchored protein LLG1 as regulators ofmekk1-mkk1/2-mpk4cell death. LET2/MDS1 functions additively with LET1, anotherCrRLK1L, and acts genetically downstream of MEKK2 in regulating SUMM2 activation. LET2/MDS1 complexes with LET1 and promotes LET1 phosphorylation, revealing an intertwined regulation between differentCrRLK1Ls. LLG1 interacts with the ectodomain of LET1/2 and mediates LET1/2 transport to the plasma membrane, corroborating its function as a co-receptor of LET1/2 in themekk1-mkk1/2-mpk4cell death pathway. Thus, our data suggest that a trimeric complex consisting of twoCrRLK1Ls LET1, LET2/MDS1, and a GPI-anchored protein LLG1 that regulates the activation of NLR SUMM2 for initiating cell death and autoimmunity.