nLab integral cohomology

Idea

Integral cohomology or “ordinary cohomology” (see there) is the ordinary version of Whitehead-generalized cohomology, the one that is represented by the Eilenberg-MacLane spectrum $H \mathbb{Z}$ with coefficients in the integers.

Integral cohomology is best known maybe in its incarnation as singular cohomology or Čech cohomology with coefficients in the integers.

Geometric models

• integral cohomology in degree 2 classifies complex line bundles;

• integral cohomology in degree 3 classifies complex line bundle gerbes / line 2-bundles;

• integral cohomology in degree $n+1$ classifies line circle $n$-bundles.

Related concepts

• rational cohomology, real cohomology, complex cohomology

References

Discussion in homotopy type theory:

• Guillaume Brunerie, Axel Ljungström, Anders Mörtberg, Synthetic Integral Cohomology in Cubical Agda, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) 216 (2022) $[$doi:10.4230/LIPIcs.CSL.2022.11$]$

  (in cubical Agda)