cohomology

# Contents

## Idea

A concordance between cocycles in cohomology is a relation similar to but different from a plain coboundary, it is a “coboundary after geometric realization”.

A concordance is a left homotopy in an (∞,1)-topos with respect to a topological interval object, not with respect to the categorical interval .

For instance for $S = Diff$ the site of smooth manifolds, there is

• the “topological interval” $I \in \mathbf{H}_{diff}$ which is the smooth ∞-stack on $Diff$ represented by the manifold $I = [0,1]$;

• the “categorical interval” $Ex^\infty \Delta^1 \in \mathbf{H}_{Diff}$ is the smooth ∞-stack that is constant on the free groupoid on a single morphism.

## Definition

For $\mathbf{H}$ and (∞,1)-topos with a fixed notion of topological interval object $I$, for $A \in \mathbf{A}$ any coefficient object and $X \in \mathbf{H}$ any other object, a concordance between two objects

$c,d \in \mathbf{H}(X,A)$

(two cocycles in $A$-cohomology on $X$)

is an object $\eta \in A(X \times I)$ such that

$\begin{matrix} X&&\\ \downarrow&\searrow^{c}&\\ X \times I&\stackrel{\eta}{\to}& A\\ \uparrow& \nearrow_{d}&\\ X&& \end{matrix} \,.$

## Examples

### For topological principal bundles

###### Proposition

(concordant topological principal bundles are isomorphic)
With $kTopSp$ denoting the category of compactly generated weakly Hausdorff spaces, for $X \,\in\, kTopSp$ a k-topological space and $G \,\in\, Grp(kTopSp)$ a $k$-topological group, consider a concordance between a pair of $G$-principal bundles over $X$,

If

or

(e.g. if $X$ admits the structure of a smooth manifold)

then there exists already an isomorphism of principal bundles

(e.g. Roberts & Stevenson 2016, Cor. 15)
###### Proof

Observe that isomorphisms $f \,\colon\, P \xrightarrow{\;} P'$ between principal bundles over $X$ are equivalently global sections of the fiber bundle $(P \times_X P')/G$:

Here, from left to right, the dashed section follows by the universal property of the quotient space $X = P/G$. From right to left, the top morphism follows by pullback along the dashed section, using that

1. the bundle projections are effective epimorphisms by local triviality,

2. their kernel pairs are as shown, by principality,

3. in a regular category pullback preserves effective epimorphisms (this Prop.) and, of course, their kernel pairs.

In particular, for every $P$ the identity morphism on it corresponds to the canonical section of $(P \times_X P)/G$.

In the given situation, this means that we have a canonical local section $\sigma_0$ making the following solid diagram commute, exhibiting that the restriction of the bundle $P_0 \times [0,1]$ to $\{0\} \subset [0,1]$ is isomorphic to $P_0$, by construction:

Now

or

In either case, this implies that a lift exists, as shown by the dashed arrow above.

The resulting commutativity of the bottom right triangle says that this lift is a global section which hence exhibits an isomorphism of principal bundles (over $X \times [0,1]$) of this form:

$P \;\; \simeq_X \;\; P_0 \times [0,1] \,.$

The restriction of this isomorphism to $\{1\} \subset [0,1]$ is hence an isomorphism of the form $P_1 \,\simeq_X\, P_0$, as required.

### For topological vector bundles

For topological vector bundles over paracompact Hausdorff spaces, concordance classes coincide with plain isomorphism classes:

###### Proposition

(concordance of topological vector bundles)

Let $X$ be a paracompact Hausdorff space. If $E \to X \times [0,1]$ is a topological vector bundle over the product space of $X$ with the closed interval (hence a concordance of topological vector bundles on $X$), then the two endpoint-restrictions

$E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}}$

are isomorphic topological vector bundles over $X$.

For proof see at topological vector bundle this Prop..

### More examples

• For $A = VectrBund(-)$ the difference between concordance of vectorial bundles and isomorphism of vectorial bundles plays a crucial rule in the construction of K-theory from this model.

• The notions of coboundary and concordance exist in every cohesive (∞,1)-topos.

## References

Discussion of concordance of topological principal bundles (in fact for simplicial principal bundles parameterized over some base space):

Discussion of concordance in terms of the shape modality in the cohesive (∞,1)-topos of smooth ∞-groupoids (see at shape via cohesive path ∞-groupoid for more):