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- W4386273183 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 $Fcolon mathcal{C}^{mathrm{op}}to mathcal{V}$. 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 $Fcolon mathcal{C}^{mathrm{op}}to mathcal{V}$ 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." @default.
- W4386273183 created "2023-08-31" @default.
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- W4386273183 date "2023-08-28" @default.
- W4386273183 modified "2023-10-01" @default.
- W4386273183 title "Internal Grothendieck construction for enriched categories" @default.
- W4386273183 doi "https://doi.org/10.48550/arxiv.2308.14455" @default.
- W4386273183 hasPublicationYear "2023" @default.
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