Indexed Grothendieck construction

TL;DR

This paper extends the classical Grothendieck construction to an indexed, 2-dimensional setting, establishing an equivalence between opfibrations and copresheaves that generalizes known slice-copresheaf relationships.

Contribution

It introduces an indexed version of the Grothendieck construction, providing a 2-dimensional equivalence that encompasses all bonds between different indexes.

Findings

Establishes an equivalence between opfibrations over a fixed base and copresheaves on the Grothendieck construction.

Proves the pseudonaturality of the equivalence in the base category.

Generalizes the classical slice-copresheaf equivalence to a 2-dimensional context.

Abstract

We produce an indexed version of the Grothendieck construction. This gives an equivalence of categories between opfibrations over a fixed base in the 2-category of 2-copresheaves and 2-copresheaves on the Grothendieck construction of the fixed base. We also prove that this equivalence is pseudonatural in the base and that it restricts to discrete opfibrations with small fibres and copresheaves. Our result is a 2-dimensional generalization of the equivalence between slices of copresheaves and copresheaves on slices. We can think of the indexed Grothendieck construction as a simultaneous Grothendieck construction on every index that takes into account all bonds between different indexes.