Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by functors only up to isomorphism, whereas syntactic substitution commutes strictly with all type theoretic structure. Our solution involves a suitable presentation of the higher category of lcc categories as model category. To that end, we construct a model category of lcc sketches, from which we obtain by the formalism of algebraically (co)fibrant objects model categories of strict lcc categories and then algebraically cofibrant strict lcc categories. The latter is our model of dependent type theory.

中文翻译：

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局部笛卡尔闭范畴模型范畴中依赖类型理论的解释

局部笛卡尔封闭 (lcc) 类别是外延依赖类型理论的自然分类模型。本文介绍了 lcc 类别中的“gros”语义：我们不是在给定的单个 lcc 类别中构建解释，而是表明所有 lcc 类别的类别也可以被赋予依赖类型理论模型的结构. 然后可以通过切片恢复单个 lcc 类别中的原始解释。正如在最初的解释中一样，我们面临着连贯性的问题：范畴结构通常由函子保留直到同构，而句法替换与所有类型理论结构严格交换。我们的解决方案涉及将 lcc 类别的较高类别适当地表示为模型类别。为此，我们构建了一个 lcc 草图的模型范畴，从中我们通过代数 (co)fibrant 对象的形式主义获得了严格 lcc 范畴的模型范畴，然后是代数共纤的严格 lcc 范畴。后者是我们的依赖类型理论模型。