nLab
generalized (Eilenberg-Steenrod) cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Homotopy theory

Contents

Idea

What is called generalized Eilenberg-Steenrod cohomology is really the general fully abelian subcase of cohomology.

This means that generalized Eilenberg-Steenrod cohomology is the cohomology in an (∞,1)-category H\mathbf{H} that happens to be a stable (∞,1)-category.

The archetypical example of this is H=Sp(Top)\mathbf{H} = Sp(Top), the stable (∞,1)-category of spectra and this is the context in which generalized Eilenberg-Steenrod cohomology is usually understood. So

Generalized Eilenberg-Steenrod cohomology is cohomology H(X,A)H(X,A) with coefficient object AA a spectrum.

The Eilenberg-Steenrod axioms

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 UU and XX be topological spaces, such that UU is a subspace of XX. Notation: (X,U):=UX(X,U) := U \hookrightarrow X.

Note that if only one space is listed, the subspace is assumed to be the empty set (X,)(X, \emptyset).

The Eilenberg-Steenrod axioms are the following:

Definition

A cohomology theory is a collection {A n} n\{A^n\}_{n \in \mathbb{Z}} of functors

A n:(Top ) opAb A^n : (Top^{\hookrightarrow})^{op} \to Ab

from the homotopy category Top Top^{\hookrightarrow} of pairs of topological spaces to the category Ab of abelian groups, as well as a natural transformation δ:A n(X,)A n+1(X,U)\delta: A^n(X, \emptyset) \to A^{n+1}(X, U). These functors and natural transformations satisfy and are characterized by the following axioms.

  1. Exactness: The following sequence is exact. Note that the inclusions UXU \hookrightarrow X and (X,)(X,U)(X, \emptyset) \hookrightarrow (X, U) induce the unlabeled arrows.

A n(X,U)A n(X,)A n(U,)δA n+1(X,U) \cdots \to A^n(X, U) \to A^n(X, \emptyset) \to A^n(U, \emptyset) \xrightarrow{\delta} A^{n+1}(X, U) \to \cdots

  1. Weak homotopy equivalence: if f:XYf : X \to Y is a weak homotopy equivalence then A n(f):A n(Y)A n(X)A^n(f) : A^n(Y) \to A^n(X) is an isomorphism

  2. Additivity: If (X,U)= i(X i,U i) (X, U) = \coprod_i (X_i, U_i), then A n(X,U)= iA n(X i,U i)A^n(X, U) = \coprod_i A^n(X_i, U_i).

  3. Excision: Let SS be a subspace of UU, the natural inclusion of the pair i:(XS,US)(X,U)i:(X-S, U-S) \hookrightarrow (X, U) induces an isomorphism A n(i):A n(XS,US)A n(X,U)A^n(i): A^n(X-S, U-S) \to A^n(X, U).

Ordinary cohomology theories require and additional axiom, the dimension axiom A n(pt)=0A^n(pt) = 0.

Examples

Properties

Expression by ordinary cohomology via Atiyah-Hirzebruch

The Atiyah-Hirzebruch spectral sequence serves to express generalized cohomology E E^\bullet in terms of ordinary cohomology with coefficients in E (*)E^\bullet(\ast).

twisted generalized cohomology theory is ∞-categorical semantics of linear homotopy type theory:

linear homotopy type theorygeneralized cohomology theoryquantum theory
linear type(module-)spectrum
multiplicative conjunctionsmash product of spectracomposite system
dependent linear typemodule spectrum bundle
Frobenius reciprocitysix operation yoga in Wirthmüller context
dual type (linear negation)Spanier-Whitehead duality
invertible typetwistprequantum line bundle
dependent sumgeneralized homology spectrumspace of quantum states (“bra”)
dual of dependent sumgeneralized cohomology spectrumspace of quantum states (“ket”)
linear implicationbivariant cohomologyquantum operators
exponential modalityFock space
dependent sum over finite homotopy type (of twist)suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy typeAtiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual typePoincaré dualityinner product
dependent sum coinciding with dependent productambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible typeWirthmüller isomorphism
( ff *)(\sum_f \dashv f^\ast)-counitpushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit(twisted-)Umkehr map/fiber integration
linear polynomial functorcorrespondencespace of trajectories
linear polynomial functor with linear implicationintegral kernel (pure motive)prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unitintegral transformmotivic/cohomological path integral
traceEuler characteristicpartition 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.

References

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:

Revised on May 25, 2015 09:51:51 by Anonymous Coward (77.9.20.9)