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arxiv: 2606.07037 · v1 · pith:4G46VAZZ · submitted 2026-06-05 · math.CT · math.DG

Ehresmann connections in tangent categories

Geoffrey Cruttwell , Marcello Lanfranchi This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-27 20:32 UTCgrok-4.3pith:4G46VAZZrecord.jsonopen to challenge →

classification math.CT math.DG
keywords Ehresmann connectionstangent categoriesKoszul connectionsparallel transportcurvatureBianchi identitystructural equationhorizontal distribution
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The pith

Ehresmann connections extend to tangent categories through splittings of the tangent bundle that yield parallel transport and curvature satisfying the Bianchi identity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to formulate Ehresmann connections inside tangent categories by defining them as splittings of the tangent bundle into vertical and horizontal parts. This setup produces equivalent versions called full connections and abstract connections. The same definitions recover Koszul connections as a special case. Parallel transport along paths and a curvature operator are then introduced, with proofs that the curvature obeys both the structural equation and the Bianchi identity.

Core claim

In a tangent category an Ehresmann connection is a horizontal distribution complementary to the vertical sub-bundle of the tangent bundle of a submersion. This notion is equivalent to both a full connection and an abstract connection. It specializes to the classical Koszul connection when the tangent category is that of smooth manifolds. Parallel transport is defined by lifting paths horizontally, and curvature is defined by the failure of horizontal lifts to commute. The structural equation and Bianchi identity are proved for this curvature.

What carries the argument

Horizontal distribution complementary to the vertical sub-bundle of the tangent bundle in a tangent category.

If this is right

  • Parallel transport along paths becomes available for any submersion equipped with such a splitting.
  • Curvature is well-defined and satisfies the structural equation relating it to the Lie bracket of horizontal vector fields.
  • The Bianchi identity holds identically for the curvature in every tangent category.
  • Koszul connections on manifolds arise exactly as the special case when the tangent category is the usual one of smooth manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same horizontal-lift construction could be applied directly to tangent categories arising in algebraic geometry without passing through a manifold model.
  • Curvature identities might be used to define characteristic classes in any tangent category that supports a suitable notion of de Rham cohomology.
  • The equivalence between full, abstract, and Ehresmann connections suggests that any one of these notions could serve as the primitive definition in future axiomatizations.

Load-bearing premise

The tangent category must admit a splitting of its tangent bundles into vertical and horizontal parts that is compatible with the remaining tangent structure so that parallel transport and curvature are well-defined.

What would settle it

A concrete tangent category together with a submersion and a proposed horizontal distribution for which the induced curvature operator violates the Bianchi identity.

read the original abstract

The theory of connections is at the very core of differential geometry. Discovered by Levi-Civita and Christoffel and later studied by Cartan, Koszul, and others, connections appear in their most general form under the name of Ehresmann connections. An Ehresmann connection consists of a splitting of the tangent bundle of a submersion into the vertical sub-bundle and a given horizontal distribution. In this paper, we generalize Ehresmann connection to a categorical setting called tangent categories. Initially introduced by Rosick\'y in 1984 and later generalized by Cockett and the first author in 2014, tangent categories provide a categorical framework to study geometry that extends well beyond smooth manifolds, including algebraic geometry and non-commutative geometry. In this paper we introduce and study Ehresmann connections in the context of tangent categories. We give various equivalent formulations in term of full and abstract connections and prove that they generalize Koszul connections. We also define parallel transport and curvature for such connections, and prove the structural equation and the Bianchi identity for the curvature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper introduces Ehresmann connections in tangent categories by defining them as splittings of the tangent bundle into vertical and horizontal sub-bundles. It provides equivalent formulations in terms of full and abstract connections, proves that these generalize Koszul connections, and defines parallel transport and curvature, establishing the structural equation and Bianchi identity directly from the Cockett-Cruttwell tangent category axioms.

Significance. If the derivations hold, the work extends core concepts from differential geometry to a broad categorical setting applicable to algebraic geometry and non-commutative geometry. The direct use of existing tangent category axioms without extra parameters or ad-hoc assumptions, together with the explicit generalization to Koszul connections and the derivation of the Bianchi identity, constitutes a solid contribution to categorical differential geometry.

minor comments (1)
  1. The notation for the horizontal and vertical projections could be introduced with an explicit diagram in §2 to aid readability for readers less familiar with tangent categories.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending Ehresmann connections and related concepts to tangent categories, and recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines Ehresmann connections, full/abstract connections, parallel transport, and curvature as new structures on top of the Cockett-Cruttwell 2014 tangent category axioms. It proves equivalent formulations, generalization to Koszul connections, the structural equation, and Bianchi identity by direct construction from those axioms. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation chain; the 2014 reference supplies the external axiomatic foundation rather than importing a uniqueness theorem or ansatz from the present authors' prior work. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard axioms of a tangent category (vertical lift, tangent functor, etc.) together with the existence of a splitting that defines the connection; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption A tangent category satisfies the axioms introduced by Cockett and Cruttwell (2014) that equip it with a tangent functor and vertical lift satisfying the usual coherence conditions.
    All definitions of connections, parallel transport, and curvature are built inside this structure.

pith-pipeline@v0.9.1-grok · 5711 in / 1130 out tokens · 20984 ms · 2026-06-27T20:32:16.909204+00:00 · methodology

discussion (0)

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Reference graph

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