Fibrational linguistics: First concepts

We define a general mathematical framework for linguistics based on the theory of fibrations, called FibLang. We start by modelling the interaction between linguistics and cognition in the most general way possible, with a heavy focus on conceptually motivating any assumption we make. The advantage is that FibLang remains agnostic with respect to any particular axiomatization of grammar one may choose. As such, it is compatible with already existing categorical models of language (such as for example, DisCoCat), providing a formally sound framework to apply mathematical tools developed in the context of category theory, mainly categorical logic, to the study of language