A simplicial set is a functor X: Δᵒᵖ → Set, where Δ is the simplex category (objects are the finite ordinals [0], [1], [2], …, morphisms are order-preserving maps) and Set is the category of sets. Concretely, a simplicial set consists of a set Xₙ of n-simplices for each n ≥ 0, together with face maps dᵢ: Xₙ → Xₙ₋₁ (removing the i-th vertex) and degeneracy maps sᵢ: Xₙ → Xₙ₊₁ (repeating the i-th vertex), satisfying the simplicial identities.
Simplicial sets encode combinatorial topology: a 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is a triangle, and so on. Every topological space has a singular simplicial set (whose n-simplices are continuous maps from the standard n-simplex into the space), and every simplicial set has a geometric realization as a topological space. This adjunction connects combinatorial and topological approaches to homotopy theory.
Simplicial sets provide the combinatorial foundation for higher category theory. A quasi-category is a simplicial set satisfying a horn-filling condition, and this model of ∞-categories is the basis of Lurie’s framework for (∞,1)-categories and (∞,1)-topoi. Simplicial sets also serve as the base for simplicial enrichment of categories, replacing hom-sets with hom-simplicial-sets whose simplices witness homotopies between morphisms.