The Enriched Grothendieck Construction

TL;DR

This paper generalizes the classical Grothendieck correspondence to $V$-enriched categories, establishing an equivalence between $V$-enriched opfibrations over free $V$-categories and pseudofunctors into $V$-categories, for extensive monoidal categories $V$.

Contribution

It extends the Grothendieck construction to enriched categories over extensive monoidal categories, broadening its applicability.

Findings

Established an equivalence of 2-categories for $V$-enriched opfibrations and pseudofunctors.

Generalized the classical Grothendieck correspondence beyond $Set$-enrichment.

Applicable to categories like sets, simplicial sets, and locally cartesian closed categories.

Abstract

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category $B$, there is an equivalence of 2-categories between $V$-enriched opfibrations over the free $V$-category on $B$, and pseudofunctors from $B$ to the 2-category of $V$-categories. This generalizes the classical ($Set$-enriched) Grothendieck correspondence.