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Abstract velocity functors

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Abstract velocity functors CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES R. A. BOWSHELL Abstract velocity functors Cahiers de topologie et géométrie différentielle catégoriques, tome 12, no 1 (1971), p. 57-91. <http://www.numdam.org/item?id=CTGDC_1971__12_1_57_0> © Andrée C. Ehresmann et les auteurs, 1971, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 57 ABSTRACT VELOCITY FUNCTORS by R. A. Bowshell CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE Vol.XII, 1 It is still uncertain which category differential geometry studies.One general definition of a structured manifold is given by the notion of a higher order G-structure, but the definition of admissible maps is elusive. Since the theory of connections on groupoids plays a central role, it would be desirable that a category of structured manifolds admit many groupoids - a groupoid can, of course, be defined in any category with pull- backs. This requirement indicates that the proper domain for geometry is a category of groupoids. In any case all the familiar constructions on a mani- fold B can be considered as constructions on the trivial groupoid TT (B) = B X B , and as such have immediate generalizations to any Lie groupoid. Lacking any clear conception of what is required from a category of structured manifolds, especially because to date differential geometry has focussed on first order differential equations: existence of flows and the Frobenius theorem, it seems premature to attempt a definition of admissible

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