De rham cochain complex

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold without boundary. Its real de Rham cochain complex is $({\mathrm{\Omega }}^{\bullet }(M),d)$, where ${\mathrm{\Omega }}^k(M)$ is the space of smooth $k$-forms for $0\le k\le \dim M$ and is $0$ otherwise; $d$ has degree $+1$.

These are the sections in A smooth differential $k$-form. The identity $d^{k+1}{d}^k=0$ in The exterior derivative squares to zero makes this an instance of Cochain complex in an abelian category. On the empty manifold each section space is the zero vector space. Whenever a product with $[0,1]$ is used, forms mean smooth forms up to the endpoints, locally extendible across them.

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 17, pp.441–443, Proposition 17.2 and Corollary 17.3; local quotient calculations below.