On the construction of the Grothendieck fundamental group of a topos by paths
Since the original work of \textit{A. Grothendieck} in SGA 4 [Lect. Notes Math. 269 (1972; Zbl 0234.00007)] on constructing the fundamental group of a topos, there have been several different approaches to understanding it better, including work by both authors [\textit{M. Bunge}, ``Classifying toposes and fundamental localic groupoids'', in: Category theory '91, CMS Conf. Proc. 13, 75-96 (1992; Zbl 0787.18005)], as well as \textit{I. Moerdijk} and \textit{G. C. Wraith} [``Connected locally connected toposes are path-connected'', Trans. Am. Math. Soc. 295, 849-859 (1986; Zbl 0592.18003)]. Others who have worked on the fundamental group include Barr, Diaconescu and Kennison. The paper under review compares the Grothendieck construction using locally constant sheaves with the Moerdijk-Wraith construction using paths. The basic setting involves looking at a connected, locally connected (unpointed) topos \({\mathcal E}\). Several results are derived showing that the comparison map from the path construction fundamental group to the Grothendieck fundamental group is an equivalence, provided some additional assumptions, such as \({\mathcal E}\) is locally path-simply connected, are made. These results also readily follow in the case of toposes with a chosen base point.
scientific article; zbMATH DE number 1004393
scientific article