Applied Categorical Structures

, Volume 3, Issue 4, pp 321–356

Trunks and classifying spaces

Authors

  • Roger Fenn
    • School of Mathematical SciencesUniversity of Sussex
    • Mathematics InstituteUniversity of Warwick
  • Colin Rourke
    • School of Mathematical SciencesUniversity of Sussex
    • Mathematics InstituteUniversity of Warwick
  • Brian Sanderson
    • School of Mathematical SciencesUniversity of Sussex
    • Mathematics InstituteUniversity of Warwick
Article

DOI: 10.1007/BF00872903

Cite this article as:
Fenn, R., Rourke, C. & Sanderson, B. Appl Categor Struct (1995) 3: 321. doi:10.1007/BF00872903

Abstract

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack spaceBX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBXBAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].

The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

Mathematics Subject Classifications (1991)

57M57Q18C18F55R55U55P

Key words

trunkclassifying spacerackcubical setnervecrossed modulecrossed complexknotlinkcodimension 2

Copyright information

© Kluwer Academic Publishers 1995