Search results
Results from the WOW.Com Content Network
A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).
Statement of assets, liabilities, and net worth [citation needed] — An annual document that all de jure government workers in the Philippines, whether regular or temporary, must complete and submit attesting under oath to their total assets and liabilities, including businesses and financial interests, that make up their net worth.
In mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets.
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
Formally, a Δ-set is a sequence of sets {} = together with maps : + for each and =,, …, +, that satisfy + = + whenever <.Often, the superscript of is omitted for brevity.. This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the are the associated face maps, each mapping the -th face of a simplex in + to a simplex in .
Formally, consider a real-valued function on a simplicial complex : that is non-decreasing on increasing sequences of faces, so () whenever is a face of in .Then for every the sublevel set = ((,]) is a subcomplex of K, and the ordering of the values of on the simplices in (which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration
A simplicial set is called a Kan complex if the map from {}, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, { ∗ } {\displaystyle \{*\}} is the terminal object and so a Kan complex is exactly the same as a fibrant object .