Joyce recently gave a geometric construction of a vertex coalgebra structure on the cohomology of appropriate moduli stacks of linear objects like quiver representations or coherent sheaves. In appropriate settings, Latyntsev showed that this vertex coalgebra is compatible with cohomological Hall algebras, forming a vertex bialgebra. I will explain how these results generalize to (critical) K-theory, yielding multiplicative refinements of vertex bialgebras. This is relevant to wall-crossing problems -- the original focus of Joyce’s work -- for K-theoretic enumerative invariants.