Conjuncted: Microlinearity

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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 XY for any Frölicher space Y .

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

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