# Trunks and classifying spaces

- Received:
- Accepted:

DOI: 10.1007/BF00872903

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

- 57 Citations
- 117 Views

## 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 rack*X* gives rise to a trunk*T* (*X*) which has a single vertex and the set*X* as set of edges. The**rack space***BX* of*X* is the realisation of the nerve*NT* (*X*) of*T*(*X*). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural map*BX* ↦*BAs*(*X*) where*BAs*(*X*) is the classifying space of the associated group of*X*. 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.