 in mathematics, specifically algebraic topology, Äech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. it is named for the mathematician eduard Äech. ==motivation== let x be a topological space, and let \mathcal{u} be an open cover of x. define a simplicial complex n(\mathcal{u}), called the nerve of the covering, as follows: * there is one vertex for each element of \mathcal{u}. * there is one edge for each pair u_1,u_2\in\mathcal{u} such that u_1 \cap u_2 \ne \emptyset. * in general, there is one k-simplex for each k+1-element subset \{u_0,\ldots,u_k\}\,\! of \mathcal{u} for which u_0\cap\cdots\cap u_k\ne\emptyset\,\!. geometrically, the nerve n(\mathcal{u}) is essentially a "dual complex" (in the sense of a dual graph, or poincarÃ© duality) for the covering \mathcal{u}. the idea of Äech cohomology is that, if we choose a "nice" cover \mathcal{u} consisting of sufficiently small open sets, the resulting simplicial complex n(\mathcal{u}) should be a good combinatorial model for the space x. for such a cover, the Äech cohomology of x is defined to be the simplicial cohomology of the nerve. this idea can be formalized by the notion of a good cover, for which every open set and every finite intersection of open sets is contractible. however, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of x, ordered by refinement. this is the approach adopted below. ==construction== let x be a topological space, and let \mathcal{f} be a presheaf of abelian groups on x. let \mathcal{u} be an open cover of x. ===simplex=== a q-simplex \sigma of \mathcal{u} is an ordered collection of q+1 sets chosen from \mathcal{u}, such that the intersection of all these sets is non-empty. this intersection is called the support of \sigma and is denoted |\sigma|. now let \sigma = (u_i)_{i \in \{ 0 , \ldots , q \}} be such a q-simplex. th
Cohomology_theories Homology_theory