Global analysis, analysis on manifolds Definition

Frölicher space

Builds on Smooth function Smooth manifold Diffeology Cartesian closed category

Definition

Let $X$ be a set, let $\mathcal{C}$ be a set of maps $c \colon \R \to X$ and let $\mathcal{F}$ be a set of maps $f \colon X \to \R$ . The triple $(X, \mathcal{C}, \mathcal{F})$ is a Frölicher space if both conditions hold.

A map $f \colon X \to \R$ lies in $\mathcal{F}$ if and only if $f \circ c$ is a smooth function $\R \to \R$ for every $c \in \mathcal{C}$ .
A map $c \colon \R \to X$ lies in $\mathcal{C}$ if and only if $f \circ c$ is a smooth function $\R \to \R$ for every $f \in \mathcal{F}$ .

The elements of $\mathcal{C}$ are the smooth curves and those of $\mathcal{F}$ the smooth functions. A map $\Phi \colon X \to Y$ of Frölicher spaces is smooth if $\Phi \circ c \in \mathcal{C}_Y$ for every $c \in \mathcal{C}_X$ ; equivalently, $f \circ \Phi \in \mathcal{F}_X$ for every $f \in \mathcal{F}_Y$ .

Remarks

Example: $X = \R^n$ with $\mathcal{C}$ the smooth curves and $\mathcal{F}$ the smooth functions. Condition 2 holds, since a curve is smooth when its coordinate functions are. Condition 1 is a theorem of Boman: a function on $\R^n$ is smooth if it is smooth along every smooth curve. The same works for every finite-dimensional smooth manifold.

Non-example: $X = \R$ , $\mathcal{C}$ the smooth curves and $\mathcal{F}$ the polynomial functions. Condition 1 fails: $\sin \circ c$ is smooth for every smooth curve $c$ , but $\sin$ is not a polynomial.

Every Frölicher space carries a diffeology: its plots are the maps $p$ from open subsets of Euclidean spaces with $f \circ p$ smooth for every $f \in \mathcal{F}$ . Frölicher spaces and smooth maps form a cartesian closed category.

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Tag	`8ZSFY9BWME`
URI	`m2://8ZSFY9BWME`
Kind	definition
Writing	W0 stub
MSC	58A40
Defines	Frölicher space
Uses	Smooth function, Smooth manifold, Diffeology, Cartesian closed category

Frölicher space

Definition

Remarks

Reading path

Further reading

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