ON THE CONSTRUCTION OF THE GROTHENDIECK FUNDAMENTAL GROUP OF A TOPOS BY PATHS

The purpose of this paper is to compare the construction of the Grothe ndieck, fundamental group of a topos using locally constant sheaves, w ith the construction using paths given by Moerdijk and Wraith. Our dis cussion focuses on the Grothendieck fundamental group in the general c ase of an unpointed (possibly pointless) topos, as constructed by Bung e. Corresponding results for topoi with a chosen base-point are then e asily derived. The main result states that the basic comparison map fr om the paths fundamental group to the (unpointed version of the) Groth endieck fundamental group is an equivalence, under assumptions of the ''locally paths simply connected'' sort, as for topological spaces. (C ) 1997 Elsevier Science B.V.