The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every  and {0,1,2,3,...}" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline2.gif" style="height:15px; width:103px" /> there are natural transformations of functors

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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,

(2)

is a degree  transformation of cohomology theories.

5. (Cartan relation)

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6. (Adem relations) For ,

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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 {Sq^i:i in {0,1,2,3,...}})/R" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline15.gif" style="height:21px; width:210px" />, where  is the free module functor that takes any set and sends it to the free  module over that set. We think of {Sq^i:i in {0,1,2,...}}" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline18.gif" style="height:21px; width:156px" /> as being a graded  module, where the th gradation is given by . This makes the tensor algebra {Sq^i:i in {0,1,2,3,...}})" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline22.gif" style="height:21px; width:191px" /> 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 .