In higher category theory in mathematics, the opposite simplicial set (or dual simplicial set) is an operation extending the opposite category (or dual category). It generalizes the concept of inverting arrows from 1-categories to ∞-categories. Similar to the opposite category defining an involution on the category of small categories, the opposite simplicial sets defines an involution on the category of simplicial sets. Both correspond to each other under the nerve construction.
Definition.
On the simplex category formula_1, there is an automorphism formula_2, which for a map formula_3 is given by formula_4. It fulfills formula_5 and is the only automorphism on the simplex category formula_1. By precomposition, it defines a functor formula_7 on the category of simplicial sets formula_8. For a simplicial set formula_9, the simplicial set formula_10is its "opposite simplicial set".