Weil Prolongations of Banach Manifolds in an Analytic Model of SDG

EDUARDO J. DUBUC, JORGE ZILBER

CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE CATEGORIQUES

Año: 2005

0008-0004

Weil prolongations were introduced for paracompact real C^oo manifolds as a generalization of Ehresmann's Jet-bundles, and they play a central role in SDG (Synthetic Differential Geometry). We define and develop Weil prolongations for open sets of complex Banach spaces (we do so in a way that automatically yields the version of Weil prolongations for any Banach manifold). We introduce a definition different than the classical one. It has the desired properties in this infinite dimensional and analytic case, and it coincides with the classical one in the real finite dimensional case. We give an explicit construction of the Weil bundle B[W] for an open subset B of a Banach space and a Weil algebra W. Given an holomorphic function between open subsets of complex Banach spaces, we give an explicit formula in terms of higher derivatives for the induced map between the respective Weil bundles. We show that the embedding of the category of open subsets of complex Banach spaces into the analytic model of SDG developed in a previous paper is compatible with the differential calculus. That is, we show that under this embedding the usual differential calculus corresponds with the intrinsic differential calculus of the topos. This means that the Weil bundle B[W] is the exponential in the topos of B with the infinitesimal manifold associated to W.