On Ribbon Categories for Singlet Vertex Algebras

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $${\mathcal {M}}(p)$$ , $$p\ge 2$$ . The first category consists all finite-length -modules with atypical composition factors, while the second is subcategory that induce to local triplet {W}}(p)$$ show every irreducible module has a projective cover in these categories, although not first, and we compute fusion products involving their covers.