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Exponentiable virtual double categories and presheaves for double categories
[Submitted on 15 Aug 2025 (v1), last revised 29 May 2026 (this v · 2026-06-01 · via math updates on arXiv.org

Mathematics > Category Theory

arXiv:2508.11611 (math)

[Submitted on 15 Aug 2025 (v1), last revised 29 May 2026 (this version, v3)]

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Abstract:Given a pair of pseudo double categories $\mathbb A$ and $\mathbb B$, the lax functors from $\mathbb A$ to $\mathbb B$, along with their transformations, modules, and multimodulations, assemble into a virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$. We exhibit a universal property of this construction by observing that it arises naturally from the consideration of exponentiability for virtual double categories. In particular, we show that every pseudo double category is exponentiable as a virtual double category, whereby the virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$ of lax functors arises as the virtual double category $\mathbf{\mathbb Mod}(\mathbb B^{\mathbb A})$ of monads and modules in the exponential $\mathbb B^{\mathbb A}$. We explore some consequences of this characterisation, demonstrating that it leads to simple proofs of statements that heretofore required unwieldy computations. For instance, we deduce that the 2-category of pseudo double categories and lax functors is enriched in the 2-category of normal virtual double categories, and demonstrate that several aspects of the Yoneda theory of pseudo double categories - such as the correspondence between presheaves and discrete fibrations - are substantially simplified by this perspective.

Submission history

From: Nathanael Arkor [view email]
[v1] Fri, 15 Aug 2025 17:21:26 UTC (67 KB)
[v2] Tue, 16 Sep 2025 09:41:02 UTC (69 KB)
[v3] Fri, 29 May 2026 17:54:28 UTC (101 KB)

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