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Combinatorial notions relating to principal fibre bundles. (English) Zbl 0575.18005

This work represents a continuation of the approach begun by the author [Mathematical applications of category theory, Contemp. Math. 30, 132-144 (1984; Zbl 0542.18007)], which uses purely combinatorial notions to clarify ideas and constructions in differential geometry. This combinatorial approach includes the classical case, when the results are interpreted in the context of a well-adapted model for synthetic differential geometry. In particular, this article examines the relationships between principal fibre bundles and groupoids and connections in a bundle and the associated connection forms.
The central construction is that of a pregroupoid over a base B. This consists of a total space E with a map \(E\to^{\pi}B\) together with a partially defined ternary operation \(\lambda\). If \(\pi (x)=\pi (z)\), \(\lambda\) (x,y,z) is defined and \(\lambda\) ”behaves like” the group theoretic operation \(yx^{-1}z\). This ternary operation seems to play an important role in differential geometry, as analogous operations appear in the recent work of J. Pradines [Cah. Topologie Géom. Différ. Catégoriques 26, 339-380 (1985)] under the same ”rule of three”.
Given a pregroupoid \(E\to^{\pi}B\), one can canonically associate a groupoid \(E^*\) and a group \(E_*\), which act on E on the left and right respectively, making E into a principal \(E_*\)-bundle. Assuming B has a reflexive, symmetric relation \(\sim\) defined on it (in SDG, this would be the ”first neighbourhood of the diagonal” relation), one can proceed to discuss connections on \(E^*\) and \(E_*\)-valued forms on E and their relationship.
The paper goes on to consider curvature-free connections and integrability, path-lifting along a connection, as well as holonomy. The results obtained serve as a further testament to the elegance and clarity provided by synthetic reasoning in differential geometry.
Reviewer: K.I.Rosenthal

MSC:

18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
53C05 Connections (general theory)
51K10 Synthetic differential geometry

Citations:

Zbl 0542.18007
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References:

[1] Ambrose, W.; Singer, I. M., A theorem on holonomy, Trans. Amer. Math. Soc., 75, 428-443 (1953) · Zbl 0052.18002
[2] Cartan, E., Les groupes d’holonomie des espaces généralisés, Acta Math., 48, 1-23 (1926) · JFM 52.0723.01
[3] Dubuc, E.; Kock, A., On 1-form classifiers, Comm. Algebra, 12, 1471-1531 (1984) · Zbl 1254.51005
[4] Ehresmann, C., Les connexions infinitesimales dans un espace fibré différentiable, (Coll. de Topo.. Coll. de Topo., Bruxelles (1950), C.B. R.M.), 29-55 · Zbl 0054.07201
[5] Ehresmann, C., Sur les connexions d’ordre superieur, (Atti V. Cong. Un. Math. Italiana. Atti V. Cong. Un. Math. Italiana, Pavia-Torino (1956)), 326-328
[6] Kock, A., Formal manifolds and synthetic theory of jet bundles, Cahiers Topologie Géom. Différentielle, 21, 227-246 (1980) · Zbl 0434.18012
[7] Kock, A., Synthetic Differential Geometry (1981), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0466.51008
[8] Kock, A., Diffential forms with values in groups, Bull. Austr. Math. Soc., 25, 357-386 (1982) · Zbl 0484.58005
[9] Kock, A., The algebraic theory of moving frames, Cahiers Topologie Géom. Différentielle, 23, 347-362 (1982) · Zbl 0518.18011
[10] Kock, A., Ehresmann and the fundamental structures of differential geometry seen from a synthetic viewpoint, (Commentary in C. Ehresmann, Oeuvres, Vol. 1 (1985), Amiens)
[11] Malgrange, B., Equations de Lie, I, J. Differential Geom., 6, 503-522 (1972)
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