nLab
pullback in cohomology

Contents

Contents

Idea

Given any kind of cohomology theory, it is contravariant functor on some category 𝒞\mathcal{C} of spaces of sorts. For example, for ordinary cohomology or Whitehead-generalized cohomology theories this functor goes from pointed CW-complexes XX (or more generally: CW-pairs) to graded abelian groups E (X)E^\bullet(X).

The value of this functor on any morphism f:XYf \;\colon\; X \to Y is called the pullback in EE-cohomology f *:E (Y)E (Y)f^\ast \;\colon\; E^\bullet(Y) \to E^\bullet(Y).

Notice that, a priori, this is not related to the notion of pullback in the sense of a cospan-shaped limit in some category, though for good enough “geometric cycles” for EE-cohomology the notions may actually agree. For example, pullback in GG-non-abelian cohomology is given by forming the pullback bundles of the GG-principal bundles which are classified by the given cohomology classes.

Notions of pullback:

Last revised on January 17, 2021 at 01:28:20. See the history of this page for a list of all contributions to it.