In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:[1]
Varying n, it determines the simplicial commutative dg algebra:
(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)) ).