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.

JacobiatorIdentity

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defined for a Lie algebra g. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ (g*) ⊗ S(g*) of its dual g*.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

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