Diffeology

Definition

Let $X$ be a set. A parametrization of $X$ is a map $p \colon U \to X$, where $U$ is an open subset of $\R^n$ for some $n \ge 0$. A diffeology on $X$ is a set $\mathcal{D}$ of parametrizations of $X$, called plots, with these properties.

1. Covering: every constant parametrization $U \to X$ is a plot.
2. Locality: if a parametrization $p \colon U \to X$ is such that each $u \in U$ has an open neighbourhood $V \subseteq U$ with $p|_V \in \mathcal{D}$, then $p \in \mathcal{D}$.
3. Smooth compatibility: let $p \colon U \to X$ be a plot and let $V$ be an open subset of some $\R^m$. For every smooth map $f \colon V \to U$, the map $p \circ f$ is a plot.

A diffeological space is a set with a diffeology. A map $F \colon X \to Y$ between diffeological spaces is smooth if $F \circ p$ is a plot of $Y$ for every plot $p$ of $X$.