The analysis of context-sensitivity should not be constrained by preconceived ideas about the available mathematical tools to represent it. In that perspective, Chapter 2 (together with the appendices) introduces the mathematical theory of “fibrations” and “descent theory,” so as to show their fundamental concepts to match the ingredients of the philosophical analysis of context-sensitivity carried out in Chapter 1: Contextualization corresponds to a localization; semantic contents, to local data; context-shift, to base-change; transposition, to reindexing; and the collation of contents underlying a linguistic meaning, to the process of gluing local data. Actually, fibrations have been devised in the first place as an abstract version of a family of contexts, emphasizing how contextual data can be moved across contexts and bound together. The further conception of a context as the envelope of a family of specified versions of it is introduced at the end of the chapter, which also establishes the framework of fibrations as a natural enrichment of both dynamic semantics and Tarski’s logical semantics.