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# Homology, Homotopy and Applications

## Volume 17 (2015)

### Number 2

### A topological fibrewise fundamental groupoid

Pages: 37 – 51

DOI: https://dx.doi.org/10.4310/HHA.2015.v17.n2.a4

#### Author

#### Abstract

It is well known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological groupoids to groupoids. This article shows that for a map $Y \to X$ with certain relative local connectivity assumptions, the *fibrewise* fundamental groupoid can also be lifted to a topological groupoid over the space $X$. This allows the construction of a simply connected covering space in the setting of fibrewise topology, assuming a local analogue of the definition of an ex-space. When applied to maps which are up-to-homotopy locally trivial fibrations, the result is a categorified version of a covering space. The fibrewise fundamental groupoid can also be used to define a topological fundamental bigroupoid of a (suitably locally connected) topological space.

This is the published version of arXiv:1411.5779.

#### Keywords

fundamental groupoid, parameterised homotopy theory, topological groupoid

#### 2010 Mathematics Subject Classification

Primary 55R70. Secondary 18B40, 22A22.

Published 3 December 2015