Connections in tangent categories. (English) Zbl 1374.18016
Authors’ abstract: “Connections are an important tool of differential geometry. This paper investigates their definition and structure in the abstract setting of tangent categories. At this level of abstraction we derive several classically important results about connections, including the Bianchi identities for curvature and torsion, almost complex structure, and parallel transport.”
A tangent category is a category equipped with an endofunctor with abstract properties modelling those of the tangent bundle functor on the category of smooth manifolds. Examples include many settings for differential geometry; for example, convenient manifolds,\(C^\infty\) -rings, and models of synthetic differential geometry all give rise to tangent categories. The tangent categories have been studied by several authors including: [J. Rosický, Diagrammes 12, JR 1-JR 11 (1984; Zbl 0561.18008)], [J. R. B. Cockett and G. S. H. Cruttwell, Appl. Categ. Struct. 22, No. 2, 331–417 (2014; Zbl 1304.18031)], [R. F. Blute et al., Math. Struct. Comput. Sci. 16, No. 6, 1049–1083 (2006; Zbl 1115.03092); Theory Appl. Categ. 22, 622–672 (2009; Zbl 1262.18004)], [T. Ehrhard and L. Regnier, Theor. Comput. Sci. 309, No. 1–3, 1–41 (2003; Zbl 1070.68020); ibid. 364, No. 2, 166–195 (2006; Zbl 1113.03054)], and others.
A tangent category is a category equipped with an endofunctor with abstract properties modelling those of the tangent bundle functor on the category of smooth manifolds. Examples include many settings for differential geometry; for example, convenient manifolds,\(C^\infty\) -rings, and models of synthetic differential geometry all give rise to tangent categories. The tangent categories have been studied by several authors including: [J. Rosický, Diagrammes 12, JR 1-JR 11 (1984; Zbl 0561.18008)], [J. R. B. Cockett and G. S. H. Cruttwell, Appl. Categ. Struct. 22, No. 2, 331–417 (2014; Zbl 1304.18031)], [R. F. Blute et al., Math. Struct. Comput. Sci. 16, No. 6, 1049–1083 (2006; Zbl 1115.03092); Theory Appl. Categ. 22, 622–672 (2009; Zbl 1262.18004)], [T. Ehrhard and L. Regnier, Theor. Comput. Sci. 309, No. 1–3, 1–41 (2003; Zbl 1070.68020); ibid. 364, No. 2, 166–195 (2006; Zbl 1113.03054)], and others.
Reviewer: Ioan Pop (Iaşi)
MSC:
| 18D99 | Categorical structures |
| 53B05 | Linear and affine connections |
| 53B15 | Other connections |
| 51K10 | Synthetic differential geometry |
Keywords:
tangent categories (TC); differential bundle in TC; bundle morphisms in TC; vertical connections; canonical(affine) verical connection; covariant derivative of a vertical connection; abstract curvature; flat vertical connection; curvature of a verical connection; curvature tensor; abstract torsion; torsion-free vertical connection; Finsler connection; horizontal connection; connection on a differential bundle; dynamical system; total curvature object; parallel transport.Citations:
Zbl 0561.18008; Zbl 1304.18031; Zbl 1115.03092; Zbl 1262.18004; Zbl 1070.68020; Zbl 1113.03054References:
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