  We propose a categorial grammar based on classical multiplicative linear
logic.
  This can be seen as an extension of abstract categorial grammars (ACG) and is
at least as expressive. However, constituents of {\it linear logic grammars
(LLG)} are not abstract ${\lambda}$-terms, but simply tuples of words with
labeled endpoints, we call them {\it multiwords}. At least, this gives a
concrete and intuitive representation of ACG.
  A key observation is that the class of multiwords has a fundamental algebraic
structure. Namely, multiwords can be organized in a category, very similar to
the category of topological cobordisms. This category is symmetric monoidal
closed and compact closed and thus is a model of linear $\lambda$-calculus and
classical linear logic. We think that this category is interesting on its own
right. In particular, it might provide categorical representation for other
formalisms.
  On the other hand, many models of language semantics are based on commutative
logic or, more generally, on symmetric monoidal closed categories. But the
category of {\it word cobordisms} is a category of language elements, which is
itself symmetric monoidal closed and independent of any grammar. Thus, it might
prove useful in understanding language semantics as well.
