nLab differential module

Roman Sikorski defined the term “differential module” in 1971/72. Mark Mostow further developed the theory in 1979. The following definition of a differentiable space (a kind of generalized smooth space) comes from Mostow’s paper where it is attributed to Sikorski.

Definition

A differentiable space is a topological space $X$ together with, for each open $U$ in $X$, a collection $C^\infty(U)$ of continuous real-valued functions on $U$, satisfying the closure conditions:

1. The rule $U \to C^\infty(U)$ defines a sheaf on $X$ (denoted $C^\infty X$).
2. For any $n$, if $f_1, \dots, f_n \in C^\infty(U)$ and $g \in C^\infty(\mathbb{R}^n)$ (with the usual meaning), then $g(f_1, \dots, f_n) \in C^\infty(U)$.

The elements of $C^\infty(X)$ are called smooth functions on $X$.

References

• Differential modules 1971/72
• The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations 1979.