Lie groups in tangent join restriction categories
Robin Cockett, Florian Schwarz
TL;DR
The paper develops a purely categorical framework for principal bundles by unifying tangent categories with join restriction categories, enabling a consistent description across differential geometry, topology, and algebraic geometry. It proves the tangent bundle of a group object is trivial, $T(G) \cong G \times T(G)_u$, and furnishes an external Lie algebra structure via left-invariant vector fields, together with a Lie bracket defined on these fields. Principal bundles are then formulated as G-bundles in tangent join restriction categories, with a torsor interpretation in the restriction-slice, and classical notions recovered in appropriate settings (differential geometry and topology). The framework also provides a coherent treatment of the vertical bundle and shows equivalences between fibre bundles, atlases, and gluings, paving the way for future extensions to algebraic geometry and connections to differential bundles and groupoids. The approach promises a versatile, algebraic handle on symmetry and local triviality across multiple geometric contexts.
Abstract
Principal bundles have at least three different definitions, depending on the category of geometric objects studied. In Differential Geometry, they are defined as locally trivial projection map of smooth manifolds with an atlas whose transition maps are given by group multiplication. In Topology they are $G$-equivariantly trivial $G$-spaces. In Algebraic Geometry, they are Étale locally isotrivial geometric quotients of $G$-varieties. The goal of this work is to have a categorical notion that recovers all of them. While they are different structures, they are all locally isomorphic to the Cartesian product of a base space with a group. There are a variety of other results on group objects and their tangent bundle. In particular we show that the tangent bundle is the product of the tangent space and the group object and that the tangent space has an external Lie algebra structure, generalizing the correspondence between Lie groups and Lie algebras. In order to give a purely categorical definition of a principal bundle, we formulate this notion in the language of join restriction categories. Restriction categories were developed by Cockett and Lack to generalize partial maps (maps defined only on a subset of the domain) and have since then found applications in mathematics and computer science. Join restriction categories, as described by Guo are restriction categories where local restrictions can be joined to obtain a global map. Together with a manifold construction due to Grandis, that allows us to glue together objects, we can describe principal bundles entirely in the language of join-restriction categories.