The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every and there are natural transformations of functors

satisfying:

1. for .

2. for all and all pairs .

3. .

4. The maps commute with the coboundary maps in the long exact sequence of a pair. In other words,

is a degree transformation of cohomology theories.

5. (Cartan relation)

6. (Adem relations) For ,

7. where is the cohomology suspension isomorphism.

The existence of these cohomology operations endows the cohomology ring with the structure of a module over the Steenrod algebra , defined to be , where is the free module functor that takes any set and sends it to the free module over that set. We think of as being a graded module, where the th gradation is given by . This makes the tensor algebra into a graded algebra over . is the ideal generated by the elements and for . This makes into a graded algebra.

By the definition of the Steenrod algebra, for any space , is a module over the Steenrod algebra , with multiplication induced by . With the above definitions, cohomology with coefficients in the ring , is a functor from the category of pairs of topological spaces to graded modules over .