Simplicial

Simplicial is an adjective used in geometry, topology, and related fields to describe constructions built from simplices, the simplest possible polyhedra. A k-simplex is the convex hull of k+1 affinely independent points in some Euclidean space. A simplicial complex is a collection of simplices of various dimensions that intersect only along common faces and that is closed under taking faces. Triangulations of manifolds and surfaces are standard examples of simplicial complexes; the geometric realization of such a complex is a topological space assembled from simplices glued along shared faces.

A simplicial map between simplicial complexes sends vertices to vertices in a way that preserves incidences and face relations; such maps induce continuous maps between the corresponding geometric realizations. Simplicial homology uses a chain complex generated by oriented simplices of a complex, with boundary maps that record how lower-dimensional faces fit together; its homology groups detect holes in different dimensions. Along with cohomology theories, these ideas form foundational tools in algebraic topology. Computationally, simplicial methods underpin mesh generation, computer graphics, and topological data analysis.

In category theory, a simplicial object in a category C is a functor from the opposite of the simplex category Delta to C. A simplicial set is a simplicial object in the category of sets, encoding a combinatorial model of space. The geometric realization of a simplicial set yields a topological space, and constructions such as the nerve of a category provide canonical simplicial descriptions of categorical structure.

Simplicial concepts are widely used to model and analyze shapes, spaces, and data in both theoretical and applied contexts.