In mathematics, a diffeology on a set generalizes the concept of a smooth atlas of a differentiable manifold, by declaring only what constitutes the "smooth parametrizations" into the set. A diffeological space is a set equipped with a diffeology. Many of the standard tools of differential geometry extend to diffeological spaces, which beyond manifolds include arbitrary quotients of manifolds, arbitrary subsets of manifolds, and spaces of mappings between manifolds.

The differential calculus on ${\displaystyle \mathbb {R} ^{n}}$, or, more generally, on finite dimensional vector spaces, is one of the most impactful successes of modern mathematics. Fundamental to its basic definitions and theorems is the linear structure of the underlying space.

The field of differential geometry establishes and studies the extension of the classical differential calculus to non-linear spaces. This extension is made possible by the definition of a smooth manifold, which is also the starting point for diffeological spaces.

A smooth ${\displaystyle n}$-dimensional manifold is a set ${\displaystyle M}$ equipped with a maximal smooth atlas, which consists of injective functions, called charts, of the form ${\displaystyle \phi :U\to M}$, where ${\displaystyle U}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$, satisfying some mutual-compatibility relations. The charts of a manifold perform two distinct functions, which are often syncretized:

A diffeology generalizes the structure of a smooth manifold by abandoning the first requirement for an atlas, namely that the charts give a local model of the space, while retaining the ability to discuss smooth maps into the space.

A diffeological space is a set ${\displaystyle X}$ equipped with a diffeology: a collection of maps$${\displaystyle \{p:U\to X\mid U{\text{ is an open subset of }}\mathbb {R} ^{n},{\text{ and }}n\geq 0\},}$$whose members are called plots, that satisfies some axioms. The plots are not required to be injective, and can (indeed, must) have as domains the open subsets of arbitrary Euclidean spaces.

A smooth manifold can be viewed as a diffeological space which is locally diffeomorphic to ${\displaystyle \mathbb {R} ^{n}}$. In general, while not giving local models for the space, the axioms of a diffeology still ensure that the plots induce a coherent notion of smooth functions, smooth curves, smooth homotopies, etc. Diffeology is therefore suitable to treat objects more general than manifolds.

Let ${\displaystyle M}$ and ${\displaystyle N}$ be smooth manifolds. A smooth homotopy of maps ${\displaystyle M\to N}$ is a smooth map ${\displaystyle H:\mathbb {R} \times M\to N}$. For each ${\displaystyle t\in \mathbb {R} }$, the map ${\displaystyle H_{t}:=H(t,\cdot ):M\to N}$ is smooth, and the intuition behind a smooth homotopy is that it is a smooth curve into the space of smooth functions ${\displaystyle {\mathcal {C}}^{\infty }(M,N)}$ connecting, say, ${\displaystyle H_{0}}$ and ${\displaystyle H_{1}}$. But ${\displaystyle {\mathcal {C}}^{\infty }(M,N)}$ is not a finite-dimensional smooth manifold, so formally we cannot yet speak of smooth curves into it.