Functorial Semiotics for Creativity in Music and Mathematics

This book presents a new semiotic theory based upon category/topos theory, and applied to a classification (three types) of creativity in music and mathematics. The purpose is a first functorial approach to a mathematical semiotics that will be applied to AI implementations for creativity. It is the only semiotic approach to creativity in music and mathematics that uses topos theory and its applications to music theory. Of particular interest is the generalized Yoneda embedding in the bidual of the category of categories (Lawvere) parametrizes semiotic units and enables a Čech cohomology of manifolds of semiotic entities. It opens up a conceptual mathematics as initiated by Grothendieck and Galois. It enables a precise description of musical and mathematical creativity, including a classification thereof in three types. This approach is new, as it connects topos theory, semiotics, creativity theory, and AI objectives for a missing link to HI (Human Intelligence). The readers can apply creativity research using our classification, cohomology theory, generalized Yoneda embedding, and Java implementation of the presented functorial display of semiotics, especially generalizing the Hjelmslev architecture. The intended audience are academic, industrial, and artistic researchers in creativity.