The "higher CI operators" complete the structure of the ordinary CI operators on (sometimes called "Eisenbud operators") on a resolution over a complete intersection in the same sense that the "higher homotopies" complete the structure of homotopies on with respect to a sequence of elements. Details will appear in a preprint in preparation by Burke, Eisenbud and Schreyer.
The higher CI operators are constructed by the routine higherCIOperators.
Just as a system of higher homotopies for a regular sequence f_1..f_c on a resolution over a ring S allow one to construct the Shamash resolution over R = S/(f_1..f_c), the higher CI operators are involved in a sort of dual construction: from a resolution F over R, lifted to a sequence of maps A over S, and lifted higher CI operators on A\otimes L, where L is the Koszul complex on f, one can construct a nonminimal resolution AL over S using the routine ciOperatorResolution.
This documentation describes version 0.5 of HigherCIOperators.
The source code from which this documentation is derived is in the file HigherCIOperators.m2.