Dagger Categories in Riemannian Geometry
Abstract
A dagger on a category assigns to every morphism f : X → Y a morphism f† : Y → X, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.
Related
- Models for dagger (∞,1)-categories III: Univalent flagged dagger (∞,1)-categories and unitarily complete dagger Segal anima
- Models for dagger (∞,1)-categories I: Dagger simplicial sets and unitary cores
- Models for dagger (∞,1)-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence
- Characterizing (Co)Free Dagger Categories
- Retro-synthetic Riemannian geometry in Rocq I : Taylor series of a metric
BibTeX
@misc{garðarsson2026dagger,
title = {{Dagger Categories in Riemannian Geometry}},
author = {Jón Hákon Garðarsson and Paolo Perrone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02257},
eprint = {2610.02257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02257},
}
RIS
TY - UNPB
TI - Dagger Categories in Riemannian Geometry
AU - Jón Hákon Garðarsson
AU - Paolo Perrone
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02257
UR - https://arxiv.org/abs/2610.02257
AB - A dagger on a category assigns to every morphism f : X → Y a morphism f^† : Y → X, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.
ER -