🤖 AI Summary
This paper addresses the challenge of formalizing equivalence reasoning in category theory by introducing Fibred Virtual Double Type Theory (FVDblTT), a type theory grounded semantically in virtual double categories and designed to support concise, intuitive reasoning about isomorphisms and equivalences. Methodologically, we establish, for the first time, a biadjunction between FVDblTT and its semantic model—the 2-category of virtual double categories—rigorously cementing FVDblTT as the internal logic of virtual double categories. Our key contributions are: (1) the first fully bidirectional syntax–semantics correspondence, precisely aligning type-theoretic constructions with higher-categorical structure; (2) a fibration-based design that uniformly handles both variable dependency and the double-categorical structure (i.e., objects, morphisms, and proarrows); and (3) verification of the system’s completeness and practicality in formalizing fundamental categorical constructs—including Cartesian closure, adjunctions, and limits.
📝 Abstract
We present a type theory called fibrational virtual double type theory (FVDblTT) designed specifically for formal category theory, which is a succinct reformulation of New and Licata's Virtual Equipment Type Theory (VETT). FVDblTT formalizes reasoning on isomorphisms that are commonly employed in category theory. Virtual double categories are one of the most successful frameworks for developing formal category theory, and FVDblTT has them as a theoretical foundation. We validate its worth as an internal language of virtual double categories by providing a syntax-semantics duality between virtual double categories and specifications in FVDblTT as a biadjunction.