Glossary

Here we propose some definitions for the main concepts encounter in DEC and EC. Our objective is to provide intuitive and easy to grasp understanding of these concepts to complement the formal mathematical definitions provided in the main text.

We however strongly encourage interested readers to go beyond these first definitions by engaging in the reading of real mathematical literature on the topic.

k-blade: The exterior product of k independent vectors in an n-dimensional space (k ≤ n), encoding a higher-dimensional, oriented geometric object whose magnitude is the (hyper)volume of the parallelotope spanned by those vectors. If vectors can be seen as elementary 1D bricks to construct oriented lines, k-blades play a similar role for higher-dimensional geometrical objects such as oriented surfaces (2-blades) or volumes (3-blades).
k-simplex: Given k+1 linearly independent points within a n-dimensional space (k ≤ n), the k-simplex they form corresponds to the smallest convex space (called the convex hull) that contains them all. From a geometric perspective, k-simplices are a realization of a k-blade.
k-vector: The linear combination of k-blades.
k-chain: The linear combination (with integer coefficients) of k-simplices within a simplicial complex. One could see a k-chain as a realization of a k-vector within the context of a given simplicial complex.
k-form: (k-differential form) A smooth, k-linear and antisymmetric function that takes a k-vector as input and returns a real number. Its antisymmetry means that if two vectors constituting the k-vector are flipped, the sign of the result is inverted. A 0-form corresponds to a usual scalar field and a 1-form to a covector field. A 2-form would correspond to a twice-covariant antisymmetric tensor fields.
cochain: The discrete version of a k-form that takes as input k-chains and returns a real number.
Exterior product: (or wedge product) A bilinear, associative, and anti-symmetric operation in geometric algebra, noted $\wedge$ . It combines lower-degree k-blades into higher-degree ones.
Manifold: A topological space locally similar to a Euclidean space, i.e., where every point has a (at least small) neighborhood that behaves like a Euclidean space.
Field: A mathematical object that assigns a value (such as a number, vector, tensor or differential form) to every point on a differentiable smooth manifold. The calculus-based tools provided by the differentiable structure of the manifold enables their manipulation.
Homomorphism: A structure-preserving map between two algebraic structures of the same type. This means that the map respects the operations that define the structures. For instance a map between two groups is an homomorphism.
Homeomorphism: A continuous bijective mapping between two topological spaces (the inverse mapping must also be continuous). Homeomorphics spaces are topologically equivalent, geometrically they can be continuously deformed into one another without tearing or gluing (think about the notorious example of a cup of coffee and a donut).
Simplicial Complex: a combinatorial structure formed by gluing together simplices of various dimension along their boundaries.
(co)tangent bundle: The union of all the (co)tangent spaces at every point of a manifold. Such a structure forms a vector space over the supporting manifold. N.B.: A tangent space at a given point of a manifold forms a vector space centered on this point and populated with vectors and k-vectors; While a cotangent also forms a vector space but for the differential forms acting on these vectors. Because of this vector space structure, differential forms are sometimes called covectors.