Pith. sign in

REVIEW 1 minor 22 references

Ehresmann connections in tangent categories

T0 review · 0 major / 1 minor · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Ehresmann connections extend to tangent categories through splittings of the tangent bundle that yield parallel transport and curvature satisfying the Bianchi identity.

desk verdict This paper cleanly moves Ehresmann connections, parallel transport, and curvature into tangent categories while recovering Koszul connections and deriving the expected identities from the existing axioms. read the letter →

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

Geoffrey Cruttwell , Marcello Lanfranchi This is my paper · ORCID
classification math.CTmath.DG
keywords EhresmannconnectionstangentcategoriesKoszulparalleltransportcurvatureBianchiidentitystructuralequationhorizontaldistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

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.

Watch

Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ehresmann connections in tangent categories." pith.science (2026). https://pith.science/paper/4G46VAZZ

@misc{pith2026260607037,
  author       = {Pith},
  title        = {Pith review of: Ehresmann connections in tangent categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4G46VAZZ}},
  note         = {Machine review of arXiv:2606.07037}
}
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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages

  1. [1]

    J.R.B.CockettandG.S.H.Cruttwell, DifferentialStructure,TangentStructure,andSDG,Applied CategoricalStructures22(2014),no.2,331–417

    2014

  2. [2]

    J. R. B. Cockett and G. S. H. Cruttwell, The Jacobi identity for tangent categories,Cahiers de TopologieetGéométrieDifférentielleCatégoriques56(2015),301–316

    2015

  3. [3]

    J.R.B.CockettandG.S.H.Cruttwell, Connectionsintangentcategories,TheoryandApplications ofCategories32(2017),no.26,835–888

    2017

  4. [4]

    J.R.B.CockettandG.S.H.Cruttwell, DifferentialBundlesandFibrationsforTangentCategories, CahiersdeTopologieetGéométrieDifférentielleCatégoriquesLIX(2018),10–92

    2018

  5. [5]

    (Springer)

    J.R.B.Cockett,G.S.H.Cruttwell,andJ.-S.P.Lemay, Differentialequationsinatangentcategory i: Complete vector fields, flows, and exponentials,Applied Categorical Structures(2021), 1–53. (Springer)

    2021

  6. [6]

    ISSN1201- 561X.DOI:10.70930/tac/1kntlbz3

    G.S.H.CruttwellandJ.-S.P.Lemay, Differentialbundlesincommutativealgebraandalgebraic geometry,TheoryandApplicationsofCategories39(2023),PaperNo.36,1077–1120. ISSN1201- 561X.DOI:10.70930/tac/1kntlbz3

    work page doi:10.70930/tac/1kntlbz3 2023

  7. [7]

    G.S.H.CruttwellandM.Lanfranchi, Pullbacksintangentcategoriesandtangentdisplaymaps, preprint,arXiv:2502.20699(2025)

    work page arXiv 2025

  8. [8]

    M.J.D.Hamilton,MathematicalGaugeTheory, Universitext,SpringerInternationalPublishing,

Show all 22 references
  1. [9]

    DOI:10.1007/978-3-319-68439-0

    doi

  2. [10]

    Soc.23(1986),no.1,19–33

    G.B.ImandG.M.Kelly, Someremarksonconservativefunctorswithleftadjoints,J.KoreanMath. Soc.23(1986),no.1,19–33

    1986

  3. [11]

    S.Ikonicoff,M.Lanfranchi,andJ.-S.P.Lemay, TheRosickýtangentcategoriesofalgebrasoveran operad,High.Struct.8(2024),no.2,332–385

    2024

  4. [12]

    I.Kolář,P.W.Michor,andJ.Slovák,Naturaloperationsindifferentialgeometry, SpringerScience& BusinessMedia,2013

    2013

  5. [13]

    D.C.Kertesz,J.Szilasi,andR.L.Lovas,Connections,SpraysandFinslerStructures, WorldScientific Publishing,2013

    2013

  6. [14]

    J.M.Lee,IntroductiontoSmoothManifolds, GraduateTextsinMathematics218,2nded.,Springer, 2013

    2013

  7. [15]

    J.-S.P.LemayandV.Geoff, ImportantClassesofMorphismsandtheRelativeCotangentSequence inTangentCategories, preprint,arXiv:2506.07874(2025), https://arxiv.org/abs/2506.07874

    2025

  8. [16]

    R.B.B.Lucyshyn-Wright, Onthegeometricnotionofconnectionanditsexpressionintangent categories,TheoryandApplicationsofCategories33(2018),no.28,832–866. 88

    2018

  9. [17]

    B.MacAdam, Vectorbundlesanddifferentialbundlesinthecategoryofsmoothmanifolds,Applied CategoricalStructures29(2021),no.2,285–310

    2021

  10. [18]

    B.MacAdam,ThefunctorialsemanticsofLietheory, Ph.D.thesis,UniversityofCalgary,2022

    2022

  11. [19]

    Marle, The works of Charles Ehresmann on connections: from Cartan connections to connectionsonfibrebundles,BanachCenterPublications76(2005),65–86

    C.-M. Marle, The works of Charles Ehresmann on connections: from Cartan connections to connectionsonfibrebundles,BanachCenterPublications76(2005),65–86

    2005

  12. [20]

    J.Rosický, AbstractTangentFunctors,Diagrammes12(1984),JR1–JR11

    1984

  13. [21]

    R.Street, Theformaltheoryofmonads,JournalofPureandAppliedAlgebra2(1972),no.2,149–168

    1972

  14. [22]

    R.M.Wald,GeneralRelativity, TheUniversityofChicagoPress,1984. 89

    1984

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.