  We present an expository overview of the monoidal structures in the category
of linearly compact vector spaces. Bimonoids in this category are the natural
duals of infinite-dimensional bialgebras. We classify the relations on words
whose equivalence classes generate linearly compact bialgebras under shifted
shuffling and deconcatenation. We also extend some of the theory of
combinatorial Hopf algebras to bialgebras that are not connected or of finite
graded dimension. Finally, we discuss several examples of quasi-symmetric
functions, not necessarily of bounded degree, that may be constructed via
terminal properties of combinatorial bialgebras.
