nLab Frölicher space


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)


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.


The intention for these pages is to develop the basic tools of differential topology for Frölicher spaces. This means taking the basic pieces of “ordinary” differential topology and considering what they might look like for Frölicher spaces; including what looks the same and what looks different.

This project will both record existing structure and develop new ideas. It is intentionally in the main area of the nn-Lab to encourage contributions.



A Frölicher Space is a triple (X,C X,F X)(X,C_X,F_X) where

  1. XX is a set;
  2. C XC_X is a set of curves in XX; that is, C XSet(,X)C_X \subseteq Set(\mathbb{R},X);
  3. F XF_X is a set of functionals on XX; that is, F XSet(X,)F_X \subseteq Set(X, \mathbb{R});

subject to the following saturation conditions

  1. if cC Xc\in C_X and fF Xf\in F_X, then fcC (,)f c \in C^\infty(\mathbb{R}, \mathbb{R}),
  2. if c:Xc : \mathbb{R} \to X is a set map with the property that fcC (,)f c \in C^\infty(\mathbb{R}, \mathbb{R}) for all fF Xf \in F_X then cC Xc \in C_X, and
  3. if f:Xf : X \to \mathbb{R} is a set map with the property that fcC (,)f c \in C^\infty(\mathbb{R}, \mathbb{R}) for all cC Xc \in C_X then fF Xf \in F_X.

A morphism of Frölicher spaces, say (X,C X,F X)(Y,C Y,F Y)(X,C_X,F_X) \to (Y,C_Y,F_Y) is a set map g:XYg : X \to Y satisfying the following (equivalent) conditions:

  1. gcC Yg c \in C_Y for all cC Xc \in C_X,
  2. fgF Xf g \in F_X for all fF Yf \in F_Y,
  3. fgcC (,)f g c \in C^\infty(\mathbb{R}, \mathbb{R}) for all fF Yf \in F_Y and cC Xc \in C_X.

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.


The category of Frölicher spaces is complete, cocomplete, and cartesian closed. It is topological over SetSet. It is an amnestic, transportable construct.

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

Last revised on June 1, 2022 at 09:48:47. See the history of this page for a list of all contributions to it.