Internal Grothendieck construction for enriched categories

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.