A sequence of sets indexed by the nonnegative integers together with face and degeneracy maps satisfying the simplicial identities; a combinatorial object modelling simplices and their gluings used to represent homotopy types.

Principle

Organize combinatorial data by functorially encoding n-simplices, their faces and degeneracies so that algebraic relations between maps mirror geometric relations among simplices.

Demonstration

The nerve of a small category is a simplicial set: its n-simplices are composable n-tuples of morphisms. The standard representable simplicial set Δ[n] has one nondegenerate simplex in degree n and models the combinatorial n-simplex.

Misapplication

Treating an arbitrary sequence of sets with some maps as a simplicial set without checking all simplicial identities or ignoring degeneracy maps, which leads to incorrect calculations of homotopy invariants.

Consequence

When used correctly, simplicial sets admit geometric realization and model spaces up to homotopy; they support constructions such as Kan fibrations, homotopy colimits, and simplicial homotopy theory.

Reversal

A cosimplicial object reverses variance: instead of sets indexed contravariantly, a cosimplicial object has covariant structure maps and models cochain-like constructions rather than combinatorial simplices.

Boundary

Does not require an embedding in Euclidean space and is distinct from a simplicial complex; issues of finiteness, local finiteness, or manifold structure are outside the term unless specified.

Semantic Tension

Tension arises with 'simplicial complex' (geometric, nondegenerate gluings) and with 'Δ-set/Delta-set' (which may omit degeneracy maps): these nearby notions share simplices but differ in allowed degeneracy and combinatorial structure.

Synthesis

A simplicial set is the categorical encoding of simplices and their combinatorial relations via face and degeneracy maps satisfying simplicial identities, providing a flexible, purely combinatorial model for homotopy-theoretic constructions.