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Inphysics, the termsimplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study ofRegge calculus. These objects combine attributes of asimplex with those of amanifold. There is no standard usage of this term inmathematics, and so the concept can refer to atriangulation in topology, or apiecewise linear manifold, or one of several differentfunctors from either thecategory of sets or the category ofsimplicial sets to the category ofmanifolds.

A simplicial manifold is asimplicial complex for which thegeometric realization ishomeomorphic to atopological manifold. This is essentially the concept of atriangulation in topology. This can mean simply that aneighborhood of each vertex (i.e. the set ofsimplices that contain that point as a vertex) ishomeomorphic to an-dimensionalball.

A simplicial manifold is also asimplicial object in thecategory ofmanifolds. This is a special case of asimplicial space in which, for eachn, the space ofn-simplices is a manifold.

For example, ifG is aLie group, then thesimplicial nerve ofG has the manifold${\displaystyle G^n}$ as its space ofn-simplices. More generally,G can be aLie groupoid.

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