We study the categorical structure of the Grothendieck construction of an indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$ and characterise fibred limits, colimits, and monoidal structures. Next, we give sufficient conditions for the monoidal closure of the total category $\Sigma_\mathcal{C} \mathcal{L}$ of a Grothendieck construction of an indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$. Our analysis is a generalization of G\"odel's Dialectica interpretation, and it relies on a novel notion of $\Sigma$-tractable monoidal structure. As we will see, $\Sigma$-tractable coproducts simultaneously generalize cocartesian coclosed structures, biproducts and extensive coproducts. We analyse when the closed structure is fibred -- usually it is not.

翻译：我们研究了索引范畴$\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$的Grothendieck构造的范畴结构，并刻画了纤维极限、余极限及幺半结构。进一步，我们给出了索引范畴$\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$的Grothendieck构造的总范畴$\Sigma_\mathcal{C} \mathcal{L}$实现幺半闭包的充分条件。该分析是Gödel的Dialectica解释的推广，并依赖于一种称为Σ-可处理幺半结构的新概念。我们将看到，Σ-可处理余积同时推广了余笛卡尔闭结构、双积以及广延余积。我们特别分析了该闭结构是否具有纤维性——通常情况并非如此。