2023/08/28 by Lyne Moser, Moser, Lyne, Maru Sarazola +3 · 1 citation
Mathematics · #18A05 #18A30 #18B50 #18D15 #18D20 #18D30 #18D40 #18N10 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2308.14455
openalex publication_date 2023/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a cartesian closed category V, we introduce an internal category of elements ∫C F associated to a V-functor F\colon Cop→ V. When V is extensive, we show that this internal Grothendieck construction gives an equivalence of categories between V-functors Cop→ V and internal discrete fibrations over C, which can be promoted to an equivalence of V-categories. Using this construction, we prove a representation theorem for V-categories, stating that a V-functor F\colon Cop→ V is V-representable if and only if its internal category of elements ∫C F has an internal terminal object. We further obtain a characterization formulated completely in terms of V-categories using shifted V-categories of elements. Moreover, in the presence of V-tensors, we show that it is enough to consider V-terminal objects in the underlying V-category Und∫C F to test the representability of a V-functor F. We apply these results to the study of weighted V-limits, and also obtain a novel result describing weighted V-limits as certain conical internal limits.