Conceptua: Institutions in a Topos

Abstract: Tarski's semantic definition of truth is the composition of its extensional and intensional aspects. Abstract satisfaction, the core of the semantic definition of truth, is the basis for the theory of institutions (Goguen and Burstall). The satisfaction relation for first order languages (the truth classification), and the preservation of truth by first order interpretations (the truth infomorphism), form a key motivating example in the theory of Information Flow (IF) (Barwise and Seligman). The concept lattice notion, which is the central structure studied by the theory of Formal Concept Analysis (FCA) (Ganter and Wille), is constructed by the polar factorization of derivation. The study of classification structures (IF) and the study of conceptual structures (FCA) provide a principled foundation for the logical theory of knowledge representation and organization. In an effort to unify these two areas, the paper "Distributed Conceptual Structures" (Kent arXiv:1810.04774) abstracted the basic theorem of FCA in order to established three levels of categorical equivalence between classification structures and conceptual structures. In this paper, we refine this approach by resolving the equivalence as the category-theoretic factorization of the Galois connection of derivation. The equivalence between classification and conceptual structures is mediated by the opposite motions of factorization and composition. Abstract truth factors through the concept lattice of theories in terms of its extensional and intensional aspects.