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cohomology group

Recall that cohomology in an (∞,1)-topos H\mathbf{H} on an object XX with coefficients in an object AA is the hom-set in the homotopy category of an (∞,1)-category

H(X,A)=π 0H(X,A). H(X,A) = \pi_0 \mathbf{H}(X,A) \,.

This is the cohomology set .

It is a pointed set if AA is a pointed object.

In the case that AA moreover carries the structure of a group object, the set H(X,A)H(X,A) inherits naturally itself the structure of a group. In this case one speaks of the cohomology group of XX with coefficients in AA.

Dually, is this why n-spheres are good for homotopy as they are cogroups? —David

Examples

generalized abelian cohomology

In all of what is called generalized cohomology – which is really generalized abelian cohomology, compare nonabelian cohomology – the coefficient object is taken to be not just a group object but a “maximally abelian” group object called a stable object in general and called a spectrum in the case that H\mathbf{H} = Top.

In that case all the deloopings B nA\mathbf{B}^n A of AA exists and are still stably abelian group objects.

So in that case not only is the cohomology set H(X,A)H(X,A) naturally an abelian group, but there is an infinite sequence of such cohomology groups, one for each delooping B nA\mathbf{B}^n A. This yields the traditional notation for graded cohomology groups by setting

H n(X,A):=H(X,B nA). H^n(X,A) := H(X, \mathbf{B}^n A) \,.

“ordinary” (integral/Eilenberg-Mac Lane-) cohomology

The standard example are the “ordinary” cohomology groups that come from taking H=\mathbf{H} = Top or = ∞Grpd (see homotopy hypothesis) and choosing the coefficient object to be the Eilenberg-Mac Lane spectrum

A:=B. A := \mathbf{B} \mathbb{Z} \,.

The for XHX \in \mathbf{H} any object (a topological space or an ∞-groupoid) the “ordinary” cohimology of XX in degree nn is

H n(X):=H n(X,):=H(X,B n)=π 0H(X,B)=:[X,K(n,)]. H^n(X) := H^n(X,\mathbb{Z}) := H(X, \mathbf{B}^n \mathbb{Z}) = \pi_0 \mathbf{H}(X, \mathbf{B}\mathbb{Z}) =: [X, K(n, \mathbb{Z})] \,.

Here on the left we have the standard notation for the ordinary cohomology groups, and on the right their expression in terms of homotopy classes of maps into an Eilenberg-Mac Lane space.

Cohomology groups in nonabelian cohomology

The standard counter-example to keep in mind for a nonabelian cohomology set that does not carry a group structure is “nonabelian cohomology in degree 1” that classifies GG-principal bundles, for GG some nonabelian group.

This cohomology set

H 1(X,G):=H(X,BG)=:[X,BG]GBund(X)/ H^1(X,G) := H(X, \mathbf{B}G) =: [X, \mathbf{B} G] \simeq G Bund(X)/_\sim

clearly has no natural group structure on it, unless GG is in fact abelian (in which case BG\mathbf{B}G is indeed a group object, namely a 2-group).

But when we pass from group-principal bundles to groupoid-principal bundles, then there may be cohomology sets with group structure even in nonabelian cohomology.

Let for instance G (2)G_{(2)} be a 2-group, i.e. a groupoid with group structure, such as the automrophism 2-group G (2):=AUT(H):=Aut Grpd(BH)G_{(2)} := AUT(H) := Aut_{Grpd}(\mathbf{B}H) of an ordinary group HH, then there is the nonabelian cohomology set

H 1(X,G (2)):=H(X,G (2))G (2)GrpdBund(X)/ . H^1(X, G_{(2)}) := H(X, G_{(2)}) \simeq G_{(2)} GrpdBund(X)/_\sim \,.

and this does carry a nonabelian (in general) group structure.

This is to be distinguished from the cohomology set

H 2(X,G (2)):=H(X,BG (2))G (2)Bund(X)/ H^2(X, G_{(2)}) := H(X, \mathbf{B} G_{(2)}) \simeq G_{(2)} Bund(X)/_\sim

that classifies G (2)G_{(2)} principal 2-bundles as opposed to groupoid principal 1-bundles and which is not in general a group (unless G (2)G_{(2)} in turn is sufficiently abelian).

For G (2)=AUT(H)G_{(2)} = AUT(H) both these cohomology sets play a role in the description of gerbes (see gerbe (as a stack) and gerbe (in nonabelian cohomology)).

Revised on July 11, 2009 15:11:44 by Eric Forgy (67.49.17.5)