Abstract

For all integers $n$ and all homotopy modules $M_{\ast }$, we define certain operations ${\underset{¯}{K<!--FUNCTION\; APPLICATION-->}}_n^{\mathrm{MW}<!--FUNCTION\; APPLICATION-->}\to M_{\ast }$ and show that these generate the $M_{\ast }(k)$-module of all (in general nonadditive) operations ${\underset{¯}{K<!--FUNCTION\; APPLICATION-->}}_n^{\mathrm{MW}<!--FUNCTION\; APPLICATION-->}\to M_{\ast }$ in a suitable sense, if $M_{\ast }$ is $\mathbb{N}$-graded and has a ring structure. This also allows us to explicitly compute the abelian group $\mathrm{Hom}<!--FUNCTION\; APPLICATION-->({\underset{¯}{K<!--FUNCTION\; APPLICATION-->}}_n^{\mathrm{MW}<!--FUNCTION\; APPLICATION-->},{\underset{¯}{K<!--FUNCTION\; APPLICATION-->}}_m^{\mathrm{MW}<!--FUNCTION\; APPLICATION-->})$ and all operations between related theories such as Milnor, Witt and Milnor–Witt K-theory.

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