A topological fibrewise fundamental groupoid
Date
2015
Authors
Roberts, D.
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Journal article
Citation
Homology, Homotopy and Applications, 2015; 17(2):37-51
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David Michael Roberts
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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→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.
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This is the published version of arXiv:1411.5779
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