In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:

Ω poly ∗ ( [ n ] ) = Q [ t 0 , . . . , t n , d t 0 , . . . , d t n ] / ( ∑ t i − 1 , ∑ d t i ) . {\displaystyle \Omega _{\text{poly}}^{*}([n])=\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum t_{i}-1,\sum dt_{i}).}

Varying n, it determines the simplicial commutative dg algebra:

Ω poly ∗ {\displaystyle \Omega _{\text{poly}}^{*}}

(each u : [ n ] → [ m ] {\displaystyle u:[n]\to [m]} induces the map Ω poly ∗ ( [ m ] ) → Ω poly ∗ ( [ n ] ) , t i ↦ ∑ u ( j ) = i t j {\displaystyle \Omega _{\text{poly}}^{*}([m])\to \Omega _{\text{poly}}^{*}([n]),t_{i}\mapsto \sum _{u(j)=i}t_{j}} ).

• Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and Rational Homotopy Type, Memoirs of the A. M. S., vol. 179, 1976.
• Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.