Free and based path groupoids

TL;DR

This paper provides explicit descriptions of free and based path and loop groupoids within the Morita bicategory, linking them to topological spaces and establishing homotopy and fibration concepts.

Contribution

It introduces explicit models for free and based path and loop groupoids in the context of translation topological groupoids, connecting them to topological spaces.

Findings

Free path groupoid of a discrete group acting on X is a translation groupoid.

Based path and loop groupoids are equivalent to topological spaces.

Homotopy and fibration notions are established in this framework.

Abstract

We give an explicit description of the free path and loop groupoids in the Morita bicategory of translation topological groupoids. We prove that the free path groupoid of a discrete group acting on a topological space $X$ is a translation groupoid given by the same group acting on the topological path space $X^I$. We give a detailed description of based path and loop groupoids and show that both are equivalent to topological spaces. We also establish the notion of homotopy and fibration in this context.