Use this tag for questions about natural transformations from a functor defining a cohomology theory to itself. Common examples include Steenrod squares in mod 2 cohomology and Adams operations in K-theory.
The cohomology operation concept became central to algebraic topology, particularly homotopy theory, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Formally, a cohomology operation θ of type (n, q, π, G) is a natural transformation of functors θ : H$^n$ (−, π) $\rightarrow$ H$^q$(−, G) defined on CW complexes.
The ideas behind cohomology operations are that—
- they can be studied by combinatorial means, and
- their effect is to yield an interesting bicommutant theory.
The origin of those studies is the work of Pontryagin, Postnikov, and Steenrod who defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology in the case of modulo 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map at cochain level. The general theory of the Steenrod algebra of operations is closely related to that of the symmetric group.
In the Adams spectral sequence, the bicommutant aspect is implicit in the use of Ext functors, derived functors of Hom-functors; if there is a bicommutant aspect taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, for which information is hard to come by. That connection established deep interest in cohomology operations for homotopy theory. An extraordinary cohomology theory has its own cohomology operations which may exhibit a richer set on constraints.
