Fibrational linguistics I: first concepts
FibLang is an approach to mathematical linguistics rooted in the theory of categorical fibrations. Since Lambek, a language is a certain category L; we postulate that a speaker p of the language L is a category p: E -> L fibred over L, thus representing the fibre of p# over l in L as “what p thinks l means”, i.e. the totality of interpretations that p can give to l.
From this assumption we derive consequences in both ways (properties of fibrations have an interpretation as features of how speakers behave/interact; reasonable properties that one might want to equip a “speaker” with reflect into mathematical properties). This first piece of work is primarily intended to lay a foundation for a general theory, describing how speakers interact (“speak with each other”, through e.g. a fibration map) to form new fibrations from old ones.