  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
