Internal and external aspects of continuous logic and categorical logic for sheaves over quantales (2023)
In this text we explore and propose notions of sheaves over commutative, integral quantales, which are based on extensions of results of the theory of sheaves over locales: the interplay of sheaves as valued-sets and the analogy of sheaves as enriched categories. Over these proposals, we define logics that find semantics in these sheaf-like objects, on the one hand, a categorical logic that characterize the notion of sheaves associated to complete valued sets as a model of certain internal construction, and in contrast an externally defined logic whose nature is based on continuous logic for metric spaces which finds in the proposal of sheaves as enriched categories an structure for interpret the semantic. (Texto tomado de la fuente)