5.1 Čech complex

The Čech complex is a powerful tool in sheaf theory, bridging covers of topological spaces with simplicial structures. It captures the overlap patterns of open sets, enabling the study of sheaf cohomology through combinatorial means.

By constructing simplices from intersecting open sets, the Čech complex provides a concrete way to compute cohomological invariants. This approach connects local information from covers to global properties of spaces and sheaves, making it invaluable in algebraic topology.

Definition of Čech complex

• The Čech complex is a simplicial complex constructed from a cover of a topological space
• It captures the combinatorial structure of the overlaps between open sets in the cover
• The Čech complex is a key tool in the study of sheaves and their cohomology

Čech complex for a cover

Top images from around the web for Čech complex for a cover

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Topological sound | Communications Physics View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Topological sound | Communications Physics View original

  Is this image relevant?

1 of 2

Top images from around the web for Čech complex for a cover

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Topological sound | Communications Physics View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Topological sound | Communications Physics View original

  Is this image relevant?

1 of 2

• Given a cover $\mathcal{U} = \{U_i\}_{i \in I}$ of a topological space $X$, the Čech complex $\check{C}(\mathcal{U})$ is defined
• An $n$-simplex in $\check{C}(\mathcal{U})$ is an $(n+1)$-tuple of open sets $(U_{i_0}, \ldots, U_{i_n})$ with non-empty intersection
• The face maps in $\check{C}(\mathcal{U})$ are given by omitting one of the open sets in the tuple

Čech complex vs simplicial complex

• The Čech complex is a special type of simplicial complex
• Unlike a general simplicial complex, the simplices in a Čech complex are determined by the intersections of open sets
• The Čech complex captures the topology of the space through the combinatorics of the cover

Properties of Čech complex

• The Čech complex is functorial with respect to refinements of covers
• If $\mathcal{U}$ refines $\mathcal{V}$, there is a natural simplicial map $\check{C}(\mathcal{U}) \to \check{C}(\mathcal{V})$
• The Čech complex is homotopy equivalent to the nerve of the cover, which is a simplicial complex encoding the intersection pattern of the open sets

Construction of Čech complex

• The construction of the Čech complex relies on the notion of the nerve of a cover
• It involves building simplices based on the intersections of open sets in the cover
• The Čech complex is a functorial construction, compatible with refinements of covers

Nerve of a cover

• The nerve $N(\mathcal{U})$ of a cover $\mathcal{U} = \{U_i\}_{i \in I}$ is a simplicial complex
• The vertices of $N(\mathcal{U})$ are the open sets $U_i$
• An $n$-simplex in $N(\mathcal{U})$ is an $(n+1)$-tuple of open sets with non-empty intersection

Simplices in Čech complex

• The simplices in the Čech complex $\check{C}(\mathcal{U})$ are determined by the nerve of the cover
• An $n$-simplex in $\check{C}(\mathcal{U})$ corresponds to an $(n+1)$-tuple of open sets with non-empty intersection
• The face maps in $\check{C}(\mathcal{U})$ are induced by the face maps in the nerve $N(\mathcal{U})$

Čech complex as a functor

• The Čech complex construction is functorial
• A refinement of covers $\mathcal{U} \to \mathcal{V}$ induces a simplicial map $\check{C}(\mathcal{U}) \to \check{C}(\mathcal{V})$
• This functoriality allows for the study of the Čech complex under refinements of covers

Cohomology of Čech complex

• The cohomology of the Čech complex is a powerful tool in the study of sheaves
• It relates the combinatorial structure of the cover to the cohomological information of the sheaf
• The Čech cohomology groups can be computed using the Čech complex and provide invariants of the space and the sheaf

Čech cohomology groups

• Given a sheaf $\mathcal{F}$ on a topological space $X$ and a cover $\mathcal{U}$, the Čech cohomology groups $\check{H}^n(\mathcal{U}, \mathcal{F})$ are defined
• The Čech cohomology groups are the cohomology groups of the cochain complex $C^*(\check{C}(\mathcal{U}), \mathcal{F})$
• The cochain complex is obtained by applying the sheaf $\mathcal{F}$ to the simplices of the Čech complex $\check{C}(\mathcal{U})$

Čech cohomology vs sheaf cohomology

• The Čech cohomology groups $\check{H}^n(\mathcal{U}, \mathcal{F})$ depend on the choice of the cover $\mathcal{U}$
• As the cover $\mathcal{U}$ becomes finer, the Čech cohomology groups approximate the sheaf cohomology groups $H^n(X, \mathcal{F})$
• Under certain conditions (acyclicity of the cover), the Čech cohomology groups coincide with the sheaf cohomology groups

Computation of Čech cohomology

• The Čech cohomology groups can be computed using the Čech complex and the sheaf
• The computation involves constructing the cochain complex $C^*(\check{C}(\mathcal{U}), \mathcal{F})$ and calculating its cohomology
• In some cases, the Čech cohomology can be computed using alternative methods (acyclic covers, Leray's theorem)

Applications of Čech complex

• The Čech complex has numerous applications in algebraic topology and sheaf theory
• It provides a bridge between the combinatorial structure of covers and the cohomological information of sheaves
• The Čech complex is a valuable tool for computing topological invariants and studying the properties of spaces and sheaves

Čech complex in algebraic topology

• The Čech complex is used in the study of the topology of spaces
• It provides a way to compute topological invariants (homology, cohomology) using covers
• The nerve theorem relates the homotopy type of a space to the homotopy type of the nerve of a cover, which is closely related to the Čech complex

Čech complex for computing invariants

• The Čech complex can be used to compute cohomological invariants of spaces and sheaves
• The Čech cohomology groups, obtained from the Čech complex, provide information about the global structure of the space and the sheaf
• In some cases, the Čech complex allows for the computation of invariants that are difficult to access directly ($\check{H}^1$ and line bundles, $\check{H}^2$ and gerbes)

Čech complex vs other complexes

• The Čech complex is one of several complexes used in algebraic topology and sheaf theory
• Other complexes include the simplicial complex, the CW complex, and the singular complex
• The Čech complex is particularly well-suited for studying the relationship between covers and sheaves, and for computing cohomological invariants

Refinements and generalizations

• The notion of Čech complex can be refined and generalized in various ways
• Refinements of covers lead to simplicial maps between Čech complexes, allowing for the study of the Čech complex under refinements
• Generalizations of the Čech complex extend its applicability to a wider range of settings and provide new insights into the structure of spaces and sheaves

Refinement of covers

• A refinement of a cover $\mathcal{U}$ is another cover $\mathcal{V}$ such that every open set in $\mathcal{V}$ is contained in some open set of $\mathcal{U}$
• Refinements of covers induce simplicial maps between the corresponding Čech complexes
• The study of refinements allows for the analysis of the behavior of the Čech complex and the associated cohomological invariants

Generalized Čech complexes

• The notion of Čech complex can be generalized to various settings
• Generalized Čech complexes can be defined for covers of sites, where the notion of open set is replaced by a suitable generalization (sieves)
• Generalized Čech complexes can also be defined for covers of $\infty$-topoi, which are generalizations of topological spaces and sheaves

Čech complex in abstract settings

• The Čech complex construction can be adapted to abstract settings, such as simplicial sets and $\infty$-categories
• In these settings, the notion of cover is replaced by a suitable generalization (hypercovers, $\infty$-covers)
• The abstract Čech complex provides a way to study the cohomological properties of objects in these general settings, extending the scope of the classical Čech complex