Monoidal Grothendieck construction

TL;DR

This paper extends the classical equivalence between fibrations and indexed categories to the monoidal setting, exploring the relationship between global and fibrewise monoidal structures and providing relevant examples.

Contribution

It establishes an equivalence between monoidal fibrations and monoidal indexed categories, and investigates the interplay between global and fibrewise monoidal structures.

Findings

Lifts of functors correspond to lax monoidal structures on the functor

Examples include fundamental fibrations, family fibrations, and network models

The framework unifies various monoidal structures in fibrations

Abstract

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.