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 âÂÂ-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.
On the simplex category , there is an automorphism , which for a map is given by . It fulfills and is the only automorphism on the simplex category . By precomposition, it defines a functor on the category of simplicial sets . For a simplicial set , the simplicial set is its opposite simplicial set.