Strictification of LFDCs with dependent structure

Establish that every left-fibred double category (LFDC) equipped with type-level weakening, function types, and product types is equivalent to a strict left-fibred double category that preserves this structure.

Background

The paper develops left-fibred double categories (LFDCs) as a semantic foundation for substructural dependent type theory. Throughout, strictness—where substitution, Beck–Chevalley, and weakening laws hold on the nose—is highlighted as crucial for interpreting the proposed bidirectional type theory directly in models. While syntactic (term-model) LFDCs are strict, many mathematical examples are not, motivating a strictification result akin to well-known coherence theorems in monoidal category theory.

Proving that any LFDC with the listed type formers admits an equivalent strict LFDC would allow the internal language to be soundly applied in non-strict models by passing to a strict equivalent, thereby extending the applicability and robustness of the theory.