A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are developed, the latter in terms of (strict) indexed symmetric monoidal categories with comprehension. Various optional type formers are treated in a modular way. In particular, we see that the historically much-debated multiplicative quantifiers and identity types arise naturally from categorical considerations. These new multiplicative connectives are further characterised by several identities relating them to the usual connectives from dependent type theory and linear logic. Finally, one important class of models, given by families with values in some symmetric monoidal category, is investigated in detail.

翻译：提出了一种结合（直觉主义）线性类型与类型依赖的类型理论，从而恰当地推广了直觉主义依赖类型理论和完整线性逻辑。本文给出了该理论的语法和完备范畴语义，后者通过带有理解的（严格）索引对称幺半范畴来表述。多种可选类型构造子以模块化方式处理。特别地，我们看到历史上备受争议的乘法量词和同一类型可以从范畴考量中自然产生。这些新的乘法联结词进一步通过若干恒等式刻画，这些恒等式将其与依赖类型理论和线性逻辑中的常见联结词联系起来。最后，详细研究了由某个对称幺半范畴中取值族给出的一类重要模型。