group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A cohomology operation is a family of morphism between cohomology groups, which is natural with respect to the base space.
Equivalently, if the cohomology theory has a classifying space (as it does for all usual notions of cohomology, in particular for all generalized (Eilenberg-Steenrod) cohomology theories) then, by the Yoneda lemma, cohomology operations are in natural bijection with homotopy-classes of morphisms between classifying spaces.
(This statement is made fully explicit for instance below def. 12.3.22 in (Aguilar-Gitler-Prieto).
every universal characteristic class is a cohomology operation.
Steenrod squares are the stable cohomology endo-operations on ordinary cohomology (mod 2)
Adams operations are the endo-cohomology operations on K-theory
Steenrod’s original colloquium lectures were published as:
Textbook accounts include the following.
Robert Mosher, Martin Tangora, p. 38 of Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968) (pdf)
Stanley Kochmann, section 2.5 and 3.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)
Peter May, chapter 22, section 5 of A concise course in algebraic topology (pdf)
A treatment of the differential refinement of cohomology operations is in