Frölicher space

In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.

Definition

A Frölicher space consists of a non-empty set "X" together with a subset "C" of Hom(R, "X") called the set of smooth curves, and a subset "F" of Hom("X", R) called the set of smooth real functions, such that for each real function

:"f" : "X" → R

in "F" and each curve

:"c" : R → "X"

in "C", the following axioms are satisfied:

# "f" in "F" if and only if for each "γ" in "C", "f" . "γ" in C^∞(R, R)
# "c" in "C" if and only if for each "φ" in "F", "φ" . "c" in C^∞(R, R)

Let "A" and "B" be two Frölicher spaces. A map

:"m" : "A" → "B"

is called "smooth" if for each smooth curve "c" in "C"_"A", "m"."c" is in "C"_"B". Furthermore the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on ":"C^∞("A", "B")

are the images of:$S\; :\; F\_B\; imes\; C\_A\; imes\; mathrm\{C\}^\{infty\}(mathbf\{R\},\; mathbf\{R\})\text{'}\; o\; mathrm\{Mor\}(mathrm\{C\}^\{infty\}(A,\; B),\; mathbf\{R\})\; :\; (f,\; c,\; lambda)\; mapsto\; S(f,\; c,\; lambda),\; quad\; S(f,\; c,\; lambda)(m)\; :=\; lambda(f\; circ\; m\; circ\; c)$

References