Beyond the Regnant Philosophy of Manifolds

TL;DR

This paper extends the theory of Frolicher spaces by establishing that the subcategory of Weil exponentiable and microlinear Frolicher spaces is cartesian closed, and shows that their embedding into the Cahiers topos preserves microlinearity.

Contribution

It introduces the notion of microlinearity for Frolicher spaces and proves that the subcategory of Weil exponentiable and microlinear Frolicher spaces is cartesian closed.

Findings

Weil exponentiable and microlinear Frolicher spaces form a cartesian closed subcategory.

The embedding into the Cahiers topos preserves microlinearity.

The paper generalizes synthetic differential geometry concepts to Frolicher spaces.

Abstract

Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model of synthetic differential geometry to Frolicher spaces, we get the notion of microlinearity for Frolicher spaces. It is shown in this paper that its full subcategory of Weil exponentiable and microlinear Frolicher spaces is cartesian closed. The canonical embedding of Weil exponentiable Frolicher spaces into the Cahiers topos is shown to preserve microlinearity.