Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for singlet vertex algebra M(p), p∈Z>1, is equal to OM(p) C1-cofinite M(p)-modules, and this admits algebraic braided tensor structure Huang-Lepowsky-Zhang. Since includes uncountably many typical which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend previous work categories atypical M(p)-modules. also introduce a subcategory OM(p)T, graded by an torus T, has enough projectives conjecturally equivalent finite-dimensional weight unrolled restricted quantum group sl2 at 2pth root unity. compute all products involving projective we prove both OM(p)T rigid thus ribbon. As application, use operator extension theory representation finite cyclic orbifolds triplet algebras W(p) non-semisimple modular categories, confirm conjecture Adamović-Lin-Milas classification these orbifolds.