Internal Grothendieck construction for enriched categories

TL;DR

This paper develops an internal Grothendieck construction for enriched categories over a cartesian closed category, establishing equivalences and representation theorems that connect enriched functors, internal categories, and limits.

Contribution

It introduces an internal category of elements for enriched functors and proves an equivalence with internal discrete fibrations, extending classical results to enriched category theory.

Findings

Establishes an equivalence between enriched functors and internal discrete fibrations.

Provides a representation theorem linking $ ext{V}$-functors and internal terminal objects.

Characterizes weighted $ ext{V}$-limits as internal conical limits.

Abstract

Given a cartesian closed category $\mathcal{V}$, we introduce an internal category of elements $\int_\mathcal{C} F$ associated to a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$. When $\mathcal{V}$ is extensive, we show that this internal Grothendieck construction gives an equivalence of categories between $\mathcal{V}$-functors $\mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ and internal discrete fibrations over $\mathcal{C}$, which can be promoted to an equivalence of $\mathcal{V}$-categories. Using this construction, we prove a representation theorem for $\mathcal{V}$-categories, stating that a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ is $\mathcal{V}$-representable if and only if its internal category of elements $\int_\mathcal{C} F$ has an internal terminal object. We further obtain a characterization formulated completely in terms of $\mathcal{V}$-categories using shifted $\mathcal{V}$-categories of elements. Moreover, in the presence of $\mathcal{V}$-tensors, we show that it is enough to consider $\mathcal{V}$-terminal objects in the underlying $\mathcal{V}$-category $\mathrm{Und}\int_\mathcal{C} F$ to test the representability of a $\mathcal{V}$-functor $F$. We apply these results to the study of weighted $\mathcal{V}$-limits, and also obtain a novel result describing weighted $\mathcal{V}$-limits as certain conical internal limits.