Orderable groups and bundles

TL;DR

This paper introduces a categorical framework for strict total orders, applies it to ordered bundles and G-sets, and relates these to fundamental group orderings and embeddings of universal covers.

Contribution

It defines a categorical notion of strict total order and connects it to orderings of fundamental groups and bundle embeddings, generalizing Farrell's theorem.

Findings

Established a categorical definition of strict total order.

Connected orderings of (X) to embeddings of universal covers.

Generalized Farrell's theorem to bi-orderings and path space embeddings.

Abstract

We define what is meant by a strict total order in a category having subobjects, products and fibre products. This allows us to define the notions of an ordered bundle X and an ordered G-set; when G=\pi_1(X) we relate these structures to orderings of \pi_1(X). We apply this to prove a theorem of Farrell relating right-orderings of \pi_1(X) to embeddings of the universal cover into line bundles over X, and generalize it by relating bi-orderings of \pi_1(X) to embeddings of the path space into line bundles over X \times X.