The algebra of oriented simplexes
The well known notion of nerve of a category, the simplicial set whose \(p\)-simplexes are abutting \(p\)-tuples of morphisms, generalizes to \(n\)-categories where an element of dimension \(p\) is a \(p\)-simplex with an \(m\)-cell in each face of dimension \(m\). The obtained nerve of an \(n\)-category \(A\) is the basis for cohomology with coefficients in \(A\). This paper presents a \(p\)-category \(O_ p\) such that a \(p\)-functor \(O_ p\to A\) is precisely a \(p\)-simplex in the \(n\)-category \(A\). History and motivation begin this self-contained paper. In low dimensions the data which generate \(O_ p\) are visually given by many diagrams. Taking the notion of oriented simplexes one has \(O_ p\), called the \(p\)-th oriental, as a free \(p\)-category. The central idea is to describe the free \(\omega\)-category \(O_{\omega}\) (\(\omega\) being the first infinite ordinal) in which \(O_ p\) is a sub-\(p\)-category (a \(p\)-category being an \(\omega\)-category for which all elements are \(p\)-cells). The nerve functor from \(\omega\)-categories to simplicial sets is a right adjoint.
scientific article; zbMATH DE number 4079716
scientific article