What is called generalized Eilenberg-Steenrod cohomology is really the general fully abelian subcase of cohomology.
The archetypical example of this is , the stable (∞,1)-category of spectra and this is the context in which generalized Eilenberg-Steenrod cohomology is usually understood. So
One may conceptualize the axioms as ensuring that certain nice properties that hold in the category Top will be preserved by our cohomology functor.
Let and be topological spaces, such that is a subspace of . Notation: .
Note that if only one space is listed, the subspace is assumed to be the empty set .
The Eilenberg-Steenrod axioms are the following:
from the homotopy category of pairs of topological spaces to the category Ab of abelian groups, as well as a natural transformation . These functors and natural transformations satisfy and are characterized by the following axioms.
Ordinary cohomology theories require and additional axiom, the dimension axiom .
The Atiyah-Hirzebruch spectral sequence serves to express generalized cohomology in terms of ordinary cohomology with coefficients in .
|linear homotopy type theory||generalized cohomology theory||quantum theory|
|multiplicative conjunction||smash product of spectra||composite system|
|dependent linear type||module spectrum bundle|
|Frobenius reciprocity||six operation yoga in Wirthmüller context|
|dual type (linear negation)||Spanier-Whitehead duality|
|invertible type||twist||prequantum line bundle|
|dependent sum||generalized homology spectrum||space of quantum states (“bra”)|
|dual of dependent sum||generalized cohomology spectrum||space of quantum states (“ket”)|
|linear implication||bivariant cohomology||quantum operators|
|exponential modality||Fock space|
|dependent sum over finite homotopy type (of twist)||suspension spectrum (Thom spectrum)|
|dualizable dependent sum over finite homotopy type||Atiyah duality between Thom spectrum and suspension spectrum|
|(twisted) self-dual type||Poincaré duality||inner product|
|dependent sum coinciding with dependent product||ambidexterity, semiadditivity|
|dependent sum coinciding with dependent product up to invertible type||Wirthmüller isomorphism|
|-counit||pushforward in generalized homology|
|(twisted-)self-duality-induced dagger of this counit||(twisted-)Umkehr map/fiber integration|
|linear polynomial functor||correspondence||space of trajectories|
|linear polynomial functor with linear implication||integral kernel (pure motive)||prequantized Lagrangian correspondence/action functional|
|composite of this linear implication with daggered-counit followed by unit||integral transform||motivic/cohomological path integral|
|trace||Euler characteristic||partition function|
remark Originally Eilenberg and Steenrod had written down axioms that characterized the behaviour of ordinary integral cohomology, what is now understood to be cohomology with coefficients in the Eilenberg-MacLane spectrum. Generalized Eilenberg-Steenrod cohomology is originally defined as anything that satisfies this list of axioms except the first one. Later it was proven, by the Brown representability theorem, that all the models for these axioms arise in terms of homotopy classes of maps into a spectrum. In our revisionist perspective above, we take this historically secondary point of view as the conceptually primary one.
A pedagogical introduction to spectra and generalized (Eilenberg-Steenrod) cohomology is in
A comprehensive account is in
More references relating to the nPOV on cohomology include: