We propose a concrete surface representation of abstract categorial grammars in the category of word cobordisms or cowordisms for short, which are certain bipartite graphs decorated with words in a given alphabet, generalizing linear logic proof-nets.
They can be seen as a surface representation of abstract categorial grammars ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages.
This category is symmetric monoidal closed and compact closed and thus is a model of linear $\lambda$-calculus and classical, as well as intuitionistic linear logic.
A well-known approach to treating syntactic island constraints in the setting of Lambek grammars consists in adding specific bracket modalities to the logic.
We propose a categorial grammar based on classical multiplicative linear logic.