To handle complex sets of scientific data, various concepts are already available, e.g., the concept of fiber bundles . This concept enables the separation of data structural components from the data storage mechanisms. A further step towards a data structure model was investigated  where a possible identification of the discretized base space by a CW-cell complex of the fiber bundle was introduced. This section formally introduces the concept of fiber bundles and gradually refines this concept to cover the concepts of a CW-complex and chains within a common theoretical framework. A possible separation of combinatorial data structure and traversal properties modeled by a CW-complex and data access attributes is also provided here. The first required concept is a projection map to deal with correlated spaces.
is called the total space, is called the base space, and is called the fiber space. Commutativity of the diagram means that carries each fiber homeomorphically onto . Thus the fibers are arranged locally as in the product , though not necessarily globally. An example related to the discussed homeomorphism is given in Figure 1.12.
The concept of an -dimensional CW-cell complex introduced in Section 1.3 can be seen as a hierarchical system of spaces . This system of spaces can then be used to describe discrete base spaces by the identification of each , with , as a separate base space, with fibers that describe the relationship to the remaining spaces with . The total space of a complete CW-complex is then modeled by intrinsic fiber bundles on the set of -skeletons.
An important fiber bundle related to the transition from a local fiber bundle to a global:
An important representative of a fiber bundle is a vector bundle, where operations from linear algebra can be utilized and the fiber space is fully described by its dimensionality, also called the multiplicity, as it defines the number of quantities per point. The original fiber bundle data model approach  was based on the concept of a vector bundle concept.
|Figure 1.13: Zero section of a vector bundle .|
To render the final transition to the approximation of physical semantics by a discrete setting for a space , as given in Section 1.3, the concept of a manifold is introduced. This enables a local as well as global abstraction from the Euclidean perspective, but maintains the identification of Euclidean structure locally. The already declared concepts are sufficient to establish these abstractions.
The neighborhood is called Euclidean. is called a chart and assigns a set of values from , commonly called coordinates, to the points in the neighborhood . This models the local Euclidean structure, where the elements are usually called points.
In non-empty intersections of two neighborhoods it is possible to define a transition from one chart to another in the following manner:
To be able to construct more complex algebraic structures, which allow an appropriate modeling of physical fields, additional requirements are imposed on the purely topological manifold to arrive at a differentiable manifold.
To use the well known concepts of integration and properties of functions, such as continuity and differentiability, the concept of a pullback is introduced.
A mapping of an interval describes a curve in the manifold . A curve is called smooth if the composition with an arbitrary chart is continuously differentiable with respect to at least one component of does not vanish.
Different charts lead to different representatives of the same tangent vector. This concept finally allows tangent spaces to be attached at each point.
Finally a connection between a differentiable manifold and the attached tangent spaces, and fiber bundles can be established by the concept of a tangent bundle , formally expressed in the following definition:
A fiber of a point is the tangent space with a special dimensional restriction of the tangent space. The dimension of the bundle is twice the dimension of the underlying manifold ; its elements are points in addition to tangent vectors.
Where this given dimensional restriction of a bundle is a very specific specialization of the fiber bundle concept, several other identification can be obtained in a more general way by the amount of information that is available on the data in the fiber space (Figure 1.14). The fiber space of a fiber bundle has a certain dimension and thus an element has the same dimensionality at each point. If the dimensionality is unknown or may vary at each point, then the generalization of fiber bundles to a sheaf with stalks is modeled [57,58]. If more information for an object with fixed dimension is available, e.g., some linearity relationship, then a vector bundle is specified. In the special case of the same dimensionality of the fiber space and the base space, a tangent bundle is obtained, and so it is a special case of the vector bundle that is built directly from the derivatives.
|Figure 1.14: A hierarchy of concepts with partial specialization. The most general form is represented by a sheaf concept. The concept of fiber bundles is obtained by using fibers with a certain dimension. If the fiber space satisfies linear vector space properties, the concept of a vector bundle is derived. Finally, by confining the dimension of the base and fiber space, a tangent bundle is obtained.|
A hierarchy for various abstractions to deal with scientific data is thereby available, but all are confined to one type of attached space. The following concepts now introduce spaces where additional attributes can be specified in a separate space. The basic properties of combinatorial elements and attached physical quantities are thereby possible. First, a non-degenerate mapping between two spaces is required.
The scalar product is noted by and enables the concept of a dual vector space.
The elements of a dual vector space act as linear mappings1.5, or linear forms, on elements of . In the same fashion that the vector space has a dual vector space , the dual vector space is also associated with a dual vector space that again contains the linear forms on the elements of . These linear forms can be obtained by applying the scalar product to the linear forms for fixed
Figure 1.15 graphically illustrates the correlation of the concepts, whereas the linear forms on are called covectors or 1-forms and are an important step to describe arbitrary abstract quantities. The linear forms on are called (contravariant) vectors or multivectors [59,36] and can be used to enable a discrete combinatorial setting of a manifold. An important step of this translation is the identification of the given subdomains of the manifold by the concept of so-called multivectors [31,34].
As already mentioned, the basic approach to describe scientific data by the concept of fiber bundles started by using the vector bundle concepts, Figure 1.14. To model cospaces, e.g., a cotangent space, the dimensionality alone is not sufficient to describe a fiber space, but further requirements have to be met to identify covariant and contravariant elements, thus requiring more structure. By using vector bundles, e.g., a tangent vector cannot be distinguished from a normal vector. To address this lack of structure an extension of the original vector bundle approach was given  to provide the fiber bundle with additional semantic meta-information.