# 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.

# Smooth Manifolds: Frölicher space as Weil exponentiable

Thus spoke André Weil,

Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory in another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when the illusion dissolves; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time.

The notion of Weil algebra is ordinarily defined for a Lie algebra g. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ (g*) ⊗ S(g*) of its dual g*.

A Frölicher space is one flavour of a generalized smooth space. Frölicher smooth spaces are determined by a rule for
• how to map the real line smoothly into it,
• and how to map out of the space smoothly to the real line.

In the general context of space and quantity, Frölicher spaces take an intermediate symmetric position: they are both presheaves as well as copresheaves on their test domain (which here is the full subcategory of manifolds on the real line) and both of these structures determine each other.

After assigning, to each pair (X, W ) of a Frölicher space X and a Weil algebra W , another Frölicher space X ⊗ W , called the Weil prolongation of X with respect to W, which naturally extends to a bifunctor FS × W → FS, where FS is the category of Frölicher spaces and smooth mappings, and W is the category of Weil algebras. We also know

The functor · ⊗ W : FS → FS is product-preserving for any Weil algebra W.

Weil Exponentiability

A Frölicher space X is called Weil exponentiable if (X ⊗ (W1 W2))Y = (X ⊗ W1)Y ⊗ W2 —– (1)

holds naturally for any Frölicher space Y and any Weil algebras W1 and W2. If Y = 1, then (1) degenerates into

X ⊗ (W1 W2) = (X ⊗ W1) ⊗ W2 —– (2)

If W1 = R, then (1) degenerates into

(X ⊗ W2)Y = XY ⊗ W2 —– (3)

Proposition: Convenient vector spaces are Weil exponentiable.
Corollary: C-manifolds are Weil exponentiable.

Proposition: If X is a Weil exponentiable Frölicher space, then so is X ⊗ W for any Weil algebra W.

Proposition: If X and Y are Weil exponentiable Frölicher spaces, then so is X × Y.

Proposition: If X is a Weil exponentiable Frölicher space, then so is XY for any Frölicher space Y .

Theorem: Weil exponentiable Frölicher spaces, together with smooth mappings among them, form a Cartesian closed subcategory FSWE of the category FS.

Generally speaking, limits in the category FS are bamboozling. The notion of limit in FS should be elaborated geometrically.

A finite cone D in FS is called a transversal limit diagram providing that D ⊗ W is a limit diagram in FS for any Weil algebra W , where the diagram D ⊗ W is obtained from D by putting ⊗ W to the right of every object and every morphism in D. By taking W = R, we see that a transversal limit diagram is always a limit diagram. The limit of a finite diagram of Frölicher spaces is said to be transversal providing that its limit diagram is a transversal limit diagram.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then DX is also a transversal limit diagram for any Frölicher space X, where DX is obtained from D by putting X as the exponential over every object and every morphism in D.

Proof: Since the functor ·X : FS → FS preserves limits, we have DX ⊗ W = (D ⊗ W)X

for any Weil algebra W , so that we have the desired result.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then D ⊗ W is also a transversal limit diagram for any Weil algebra W.

Proof: Since the functor W ⊗ · : W → W preserves finite limits, we have (D ⊗ W) ⊗ W′ = D ⊗ (W ⊗ W′)

for any Weil algebra W′, so that we have the desired result.