**algebraic topology** – application of higher algebra and higher category theory to the study of (stable) homotopy theory

Given a topological space $X$, its *Čech homology* is the limit of simplicial homology-groups of the Čech nerves of its open covers, under refinement of open covers.

Compare to the singular homology of $X$, which is the simplicial homology of its singular simplicial complex. For well behaved topological spaces the two notions agree and are jointly known as computing the *ordinary homology* of $X$.

Given a topological space $X$ with an open cover,

$\mathcal{U}
\;=\;
\Big\{
U_i \xhookrightarrow{open} X
\Big\}_{i \in I}$

we write

(1)$C(X,\mathcal{U})
\;\coloneqq\;
\pi_0
\big(
U^{\times^\bullet_X}
\big)
\;\in\;
sSet$

for the simplicial set which is its Čech+nerve: whose $n$-simplices are the inhabited $(n + 1)$-fold intersections of the open subsets $U_i$ in $\mathcal{U}$.

For sufficiently well-behaved topological spaces $X$ (paracompact spaces) and *good* open covers, the Čech nerve (1) is simplicially homotopy equivalent to the singular simplicial complex of $X$ – this is the statement of the *nerve theorem*.

However, in general $C(X,\mathcal{U})$ may differ from the weak homotopy type of $X$ (even in the limit below) in which case Čech homology may *differ* from the singular homology of $X$. (See for instance the discussion at *well group*.)

If $\mathcal{U}$ is a refinement open cover, i.e. such that for each $U' \in \mathcal{U}$, there is a $U \in \mathcal{U}$ with $U' \subseteq U$, then these inclusions induce a morphism of simplicial sets

$C(X,\mathcal{U}) \longrightarrow C(X,\mathcal{U}')$

This yields an inverse system of simplicial sets.

The $n$th **Čech homology group** of the space $X$ is the limit

$\check{H}_n(X)
\;\coloneqq\;
\underset{
\underset{\mathcal{U}}{\longleftarrow}
}{lim}
H_n(X,\,\mathcal{U})$

over the inverse system of open covers $\alpha$, of the simplicial homology groups of the Čech nerve $C(X,\alpha)$:

$H_n(X,\,\mathcal{U})
\;\coloneqq\;
H_n\big(C(X,\,\mathcal{U})\big)
\,.$

It is to be noted that these groups do not constitute a homology theory in the sense of the Eilenberg-Steenrod axioms as the *exactness axiom* fails in general. There is a “corrected” theory known under the name strong homology.

Exposition:

- David H. Fremlin, Section 2 of:
*Singular homology for amateurs*(2016) $[$pdf, pdf$]$

On closed covers in Čech homology:

- Edwin E. Floyd,
*Closed coverings in Čech homology theory*, Trans. Amer. Math. Soc.**84**(1957) 319-337 [doi:10.1090/S0002-9947-1957-0087100-2, pdf]

Last revised on October 17, 2023 at 22:17:06. See the history of this page for a list of all contributions to it.