Conjuncted: Microlinearity

A Frölicher space X is called microlinear providing that any finite limit diagram D in W yields a limit diagram X ⊗ D in FS, where X ⊗ D is obtained from D by putting X ⊗ to the left of every object and every morphism in D.

The following result should be obvious.

Proposition: Convenient vector spaces are microlinear.
Corollary: C^∞-manifolds are microlinear.
Proposition: If X is a Weil exponentiable and microlinear Frölicher space, then so is X ⊗ W for any Weil algebra W.

Proposition: If X and Y are microlinear Frölicher spaces, then so is X × Y. Proof. This follows simply from the familiar fact that the functor · × · : FS × FS → FS preserves limits.

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

Theorem: Weil exponentiable and microlinear Frölicher spaces, together with smooth mappings among them, form a cartesian closed subcategory FS_WE,ML of the category FS.

We note in passing that microlinearity is closed under transversal limits.

Theorem: If the limit of a diagram F of microlinear Frölicher spaces is transversal, then it is microlinear.

Proof:

Let X be the limit of the diagram F, i.e., X = Lim F

Let W be the limit of an arbitrarily given finite diagram D of Weil algebras, i.e.,

W = Lim D

We denote by F ⊗ D the diagram obtained from the diagrams F and D by the application of the bifunctor ⊗ : FS × W → FS. By recalling that double limits in a complete category commute, we have

Lim (X ⊗ D)

= Lim ((Lim F) ⊗ D)

= Lim_D Lim_F (F ⊗ D)

[since Lim F is the transversal limit] = Lim_F Lim_D (F ⊗ D)

[since double limits commute] = Lim (F ⊗ (Lim D))

[since every object in F is microlinear] = Lim (F ⊗ W ) 

= (Lim F) ⊗ W

[since Lim F is the transversal limit] = X ⊗ W

Therefore we have the desired result.

Proposition: If a Weil exponentiable Frölicher space X is microlinear, then any finite limit diagram D in W yields a transversal limit diagram X ⊗ D in FS.

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