🤖 AI Summary
This paper addresses the problem of characterizing when the total category $Sigma_C L$ of the Grothendieck construction is monoidal closed—specifically, when it admits internal hom-objects (i.e., is *monadic closed*). We introduce the novel notion of a *$Sigma$-tractable monad*, which uniformly captures Cartesian closure in fibers, binary products, and wide coproducts, and establish the first sufficient condition for $Sigma_C L$ to be monadic closed. Furthermore, we uncover intrinsic connections between fiberwise limits/colimits and monad algebras in this setting. Our results generalize Gödel’s Dialectica interpretation to indexed categories, providing a categorical semantics for Dialectica formulas in higher-order logic and type theory. This work establishes a systematic, criterion-based theory for monadic closure of Grothendieck constructions, bridging categorical logic, monad theory, and semantics of constructive mathematics.
📝 Abstract
We study the categorical structure of the Grothendieck construction of an indexed category $mathcal{L}:mathcal{C}^{op} omathbf{CAT}$ 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 $mathcal{L}:mathcal{C}^{op} omathbf{CAT}$. 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.