physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of

Regge calculus In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Available ...

. These objects combine attributes of a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...

with those of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

. There is no standard usage of this term in mathematics, and so the concept can refer to a triangulation in topology, or a

piecewise linear manifold In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise line ...

, or one of several different

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

s from either the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...

or the category of

simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...

s to the category of

A manifold made out of simplices

A simplicial manifold is a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

for which the geometric realization is homeomorphic to a

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout ma ...

. This is essentially the concept of a triangulation in topology. This can mean simply that a neighborhood of each vertex (i.e. the set of

simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

that contain that point as a vertex) is homeomorphic to a ''n''-dimensional ball.

A simplicial object built from manifolds

A simplicial manifold is also a

simplicial object In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...

in the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

s. This is a special case of a

simplicial space In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces In mathematics, the category of t ...

in which, for each ''n'', the space of ''n''-simplices is a manifold. For example, if ''G'' is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

, then the simplicial nerve of ''G'' has the manifold

G^n

as its space of ''n''-simplices. More generally, ''G'' can be a

Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are sm ...

. Structures on manifolds Simplicial sets {{Geometry-stub