# nLab Frölicher space

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Idea

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.

###### Goal

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 $n$-Lab to encourage contributions.

# Basics

###### Definition

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

1. $X$ is a set;
2. $C_X$ is a set of curves in $X$; that is, $C_X \subseteq Set(\mathbb{R},X)$;
3. $F_X$ is a set of functionals on $X$; that is, $F_X \subseteq Set(X, \mathbb{R})$;

subject to the following saturation conditions

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

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

1. $g c \in C_Y$ for all $c \in C_X$,
2. $f g \in F_X$ for all $f \in F_Y$,
3. $f g c \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $f \in F_Y$ and $c \in C_X$.

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.

###### Theorem

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

To its eternal shame, the category of Frölicher spaces is not locally cartesian closed.

# References

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)