Основные понятия
The theory of institutions encodes satisfaction as the core architecture in the indexed aspect, and logical environments enrich this truth architecture by axiomatizing the truth adjunction in the fibered aspect.
Аннотация
The content discusses the theory of institutions as an indexed/fibered duality, where the indexed aspect specifies the fibered aspect. It frames the representation of truth in terms of a satisfaction relation, with the theory of institutions encoding satisfaction as its core architecture in the indexed aspect.
The key insights are:
Logical environments enrich the truth architecture by axiomatizing the truth adjunction in the fibered aspect.
The truth architecture is preserved by morphisms of logical environments.
The theory frames structures as hypergraphs of classifications, with a reference semidesignation and a signature designation.
Frames are used to represent objects, events, and relationships, with the frame name as a relation type, roles as variables, and role fillers as entity instances.
Datasets can be represented as set-functors, with the database-as-category example demonstrating the connections between the various components.
The content introduces the indexed categories of specifications, structures, logics, and sound logics, along with their homogenizations and the relationships between them, particularly the intent indexed functor that captures the invariance of truth under change of notation.