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
 |
(1)
|
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)
 |
(3)
|
6. (Adem relations) For 
,
 |
(4)
|
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 
.
