A Frölicher space is one flavour of a generalized smooth space.
Frölicher smooth spaces are determined by a rule for
how to map the real line smoothly into it,
and how to map out of the space smoothly to the real line.
In the general context of space and quantity, Frölicher spaces take an intermediate symmetric position: they are both presheaves as well as copresheaves on their test domain (which here is the full subcategory of manifolds on the real line) and both of these structures determine each other.
The general abstract idea behind this is described at Isbell envelope.
A Frölicher Space is a triple where
- is a set;
- is a set of curves in ; that is, ;
- is a set of functionals on ; that is, ;
subject to the following saturation conditions
- if is a set map with the property that for all then , and
- if is a set map with the property that for all then .
A morphism of Frölicher spaces, say is a set map satisfying the following (equivalent) conditions:
- for all ,
- for all ,
- for all and .
Frölicher spaces and their morphisms form a category with an obvious faithful functor to the category of sets. The properties of this category are as follows.
To its eternal shame, the category of Frölicher spaces is not locally cartesian closed.
The notion goes back to Alfred Frölicher.
A detailed discussion of the category of Frölicher spaces and their relation to other notions of generalized smooth spaces is given in
- Andrew Stacey, Comparative Smootheology Theory and Applications of Categories, Vol. 25, 2011, No. 4, pp 64-117. (tac)
This also lists all the relevant further references.
A discussion of Lie algebras on Frölicher groups (group objects internal to the category of Frölicher spaces) is in
- Martin Laubinger, A Lie algebra for Frölicher groups (arXiv)
See also the unpublished thesis of Andreas Cap:
Discussion in the context of applications to continuum mechanics is in