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This work was motivated in part by the following question of Soulé: given a simplicial presheaf X on a site C, how does one produce a map of simplicial presheaves X → L HZ X in such a way that each of the maps in sections X(U ) → L HZ X(U ), U ∈ C, is an integral homology localization map in the sense of Bousfield? Secondly, if Y is a simplicial presheaf which is integrally homology local in a suitable sense, is it the case that the map X → L HZ X induces an isomorphismdoi:10.4153/cjm-1998-051-1 fatcat:pqi3bxtgqjgmndngr6lkdek4ke