Linguistics is the (scientific) study of natural human language. See a reasonably good page at Wikipedia.
In mathematics and computer science, we are however also interested in artificial analogues of natural languages: formal and computer? languages.
Linguists attempt to specify formal grammars (such as context free grammars) which generate exactly the strings of a human language. The derivation trees of such strings are often interpreted in a formal system such as typed lambda calculus, which is taken to represent a high level description of the meaning the speaker intends. Such a “semantics” should ideally assign meanings to the smallest meaningful subparts of a sentence such that these submeanings compose to assign meanings to sentences.
Categorial grammar (Wikipedia) is a tradition in formal syntax which draws inspiration from category theory.
Derivation trees categorify to string diagrams.
A seminal article, inspired by the structure of a (non-symmetric) biclosed monoidal category is:
The connection to biclosed monoidal categories is made explicit in:
Pregroup grammar (Wikipedia) is a refinement of this, and categorifies to (non-symmetric) rigid monoidal categories.
Preller, A. (2005). Category theoretical semantics for pregroup grammars. In Logical aspects of computational linguistics (pp. 238-254). Springer Berlin Heidelberg.
Preller, A., & Lambek, J. (2007). Free compact 2-categories. Mathematical Structures in Computer Science, 17(2), 309-340.