The algebra of oriented simplexes
type: Item
instance of: scholarly article
title: The algebra of oriented simplexes
zbMATH Open document ID: 0661.18005
DOI: 10.1016/0022-4049(87)90137-X
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: 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.
reviewed by: Georges Hoff
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 55N99
zbMATH DE Number: 4079716
zbMATH Keywords: nerve of a category
zbMATH Keywords: simplicial set
zbMATH Keywords: n-categories
zbMATH Keywords: cohomology
zbMATH Keywords: oriented simplexes
zbMATH Keywords: nerve functor
Wikidata QID: Q56687256
author: Ross H. Street
MaRDI profile type: Publication
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full work available at URL: 0022-4049(87)90137-x
OpenAlex ID: W2076580453
description: scientific article; zbMATH DE number 4079716