On the construction of the Grothendieck fundamental group of a topos by paths Original Research Article

The purpose of this paper is to compare the construction of the Grothendieck fundamental group of a topos using locally constant sheaves, with the construction using paths given by Moerdijk and Wraith. Our discussion focuses on the Grothendieck fundamental group in the general case of an unpointed (possibly pointless) topos, as constructed by Bunge. Corresponding results for topoi with a chosen base-point are then easily derived. The main result states that the basic comparison map from the paths fundamental group to the (unpointed version of the) Grothendieck fundamental group is an equivalence, under assumptions of the “locally paths simply connected” sort, as for topological spaces.