Categorial grammar is a tradition in formal linguistics which draws inspiration from category theory.
Syntactic “calculus” for categorial grammars was introduced by Lambek formalizing the function type constructors.
Derivation trees are encoded by 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.
A contemporary thesis in this area:
Matteo Capelletti, The non-associative Lambek calculus, Logic, Linguistic and Computational Properties, thesis, Bologna, pdf
Wikipedia: categorial grammar, combinatory categorial grammar
Mati Pentus, Lambek calculus and formal grammars, Amer. Math. Soc. Transl. 1997
Michael Moortgat, Categorial type logics, pdf (ch. 2 in Handbook of logic and language, Elsevier 1997)
Research along similar lines is sometimes described as typelogical grammar:
Daniel J. Dougherty, Closed Categories and Categorial Grammar , Notre Dame Journal of Formal Logic 34 no.1 (1993) pp.36-49. (pdf)
Philippe de Groote, Towards Abstract Categorial Grammars, In: Association for Computational Linguistics, 39th Annual Meeting and 10th Conference of the European Chapter, Proceedings of the Conference, pages 148–155, Toulouse, France, 2001. (pdf)