Emancipating Microlinearity from within a Well-adapted Model of Synthetic Differential Geometry towards an Adequately Restricted Cartesian Closed Category of Frölicher Spaces. Thought of the Day 15.0

Differential geometry of finite-dimensional smooth manifolds has been generalized by many authors to the infinite-dimensional case by replacing finite-dimensional vector spaces by Hilbert spaces, Banach spaces, Fréchet spaces or, more generally, convenient vector spaces as the local prototype. We know well that the category of smooth manifolds of any kind, whether finite-dimensional or infinite-dimensional, is not cartesian closed, while Frölicher spaces, introduced by Frölicher, do form a cartesian closed category. It seems that Frölicher and his followers do not know what a kind of Frölicher space, besides convenient vector spaces, should become the basic object of research for infinite-dimensional differential geometry. The category of Frölicher spaces and smooth mappings should be restricted adequately to a cartesian closed subcategory.

Synthetic differential geometry is differential geometry with a cornucopia of nilpotent infinitesimals. Roughly speaking, a space of nilpotent infinitesimals of some kind, which exists only within an imaginary world, corresponds to a Weil algebra, which is an entity of the real world. The central object of study in synthetic differential geometry is microlinear spaces. Although the notion of a manifold (=a pasting of copies of a certain linear space) is defined on the local level, the notion of microlinearity is defined absolutely on the genuinely infinitesimal level. What we should do so as to get an adequately restricted cartesian closed category of Frölicher spaces is to emancipate microlinearity from within a well-adapted model of synthetic differential geometry.

Although nilpotent infinitesimals exist only within a well-adapted model of synthetic differential geometry, the notion of Weil functor was formulated for finite-dimensional manifolds and for infinite-dimensional manifolds. This is the first step towards microlinearity for Frölicher spaces. Therein all Frölicher spaces which believe in fantasy that all Weil functors are really exponentiations by some adequate infinitesimal objects in imagination form a cartesian closed category. This is the second step towards microlinearity for Frölicher spaces. Introducing the notion of “transversal limit diagram of Frölicher spaces” after the manner of that of “transversal pullback” is the third and final step towards microlinearity for Frölicher spaces. Just as microlinearity is closed under arbitrary limits within a well-adapted model of synthetic differential geometry, microlinearity for Frölicher spaces is closed under arbitrary transversal limits.

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