The algebra of oriented simplexes. (English) Zbl 0661.18005

Street, Ross

J. Pure Appl. Algebra 49, 283-335 (1987).

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.

Reviewer: Georges Hoff (Paris)

MSC:

18G30	Simplicial sets; simplicial objects in a category (MSC2010)
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18D05	Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55N99	Homology and cohomology theories in algebraic topology

Keywords:

nerve of a category; simplicial set; n-categories; cohomology; oriented simplexes; nerve functor

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