FRöLICHER SPACE

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

Contents

Definition

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'
and each curve
:''c'' : 'R' → ''X''
# ''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
:$S\; :\; F\_B\; imes\; C\_A\; imes\; mathrm\{C\}^\{infty\}(mathbf\{R\},\; mathbf\{R\})\text{'}\; o\; mathrm\{Mor\}(mathrm\{C\}^\{infty\}(A,\; B),\; mathbf\{R\})\; :\; (f,\; c,\; lambda)\; mapsto\; S(f,\; c,\; lambda),\; quad\; S(f,\; c,\; lambda)(m)\; :=\; lambda(f\; circ\; m\; circ\; c)$