In JP May's paper *A general algebraic approach to Steenrod operations*, Steenrod operations are constructed in wide generality. In this context, it is not necessarily true that negative Steenrod operations are zero. However, I could not find any examples (not in May's paper, not even elsewhere), where one actually proves in some specific situation that some negative Steenrod operation is non-zero. It appears that sheaves of differential graded algebras $A$ (not concentrated in degree $0$) on a topological space $X$ should provide a counterexample, but I would like to see a specific example of $X$ and $A$, possibly with a proof.



Examples of non-zero negative Steenrod operations
Notice the switch in grading from homology to cohomology on May's page 182: $P^s(x) = P\_{-s}(x)$. Operations that raise degree when graded homologically lower degree when graded cohomologically. A large part of the motivation for the paper was to give a common framework for the Dyer-Lashof operations in the homology of iterated loop spaces and the Steenrod operations in the cohomology of spaces. But if one grades these the same way, either homologically or cohomologically, one has positive operations and the other has negative operations. The grading May found most sensible for the Steenrod operations on the cohomology of cocommutative Hopf algebras (p. 226) also gives examples. There are a number of more recent papers that make serious use of the large Steenrod algebra (sometimes called the Kudo-Araki-May algebra) constructed using all operations.


