In 1975 E. M. Brown constructed a functor $\cal P$ which carries the tower of fundamental groups of the end of a (nice) space to the Brown-Grossman fundamental group. In this work, we study this functor and its extensions and analogues defined for pro-sets, pro-pointed sets, pro-groups and pro-abelian groups. The new versions of the $\cal P$ functor are provided with more algebraic structure. Examples given in the paper prove that in general the $\cal P$ functors are not faithful, however, one of our main results establishes that the restrictions of the corresponding $\cal P$ functors to the full subcategories of towers are faithful. We also prove that the restrictions of the $\cal P$ functors to the corresponding full subcategories of finitely generated towers are also full. Consequently, in these cases, the towers of objects in the categories of sets, pointed sets, groups and abelian groups, can be replaced by adequate algebraic models ($M$-sets, $M$-pointed sets, near-modules and modules.) The article also contains the construction of left adjoints for the $\cal P$ functors.
Keywords: Category of fractions, Procategory, Monoid, M set, Brown's P functor, Tower, Proobject, Nearring, Nearmodule, Generator, Proset, Progroup, Proabelian group.
AMS Classification (1990): 18B15,18E20, 18A40, 16Y30, 55N05, 55N07, 55Q52.
Theory and Applications of Categories, Vol. 1, 1995, No. 2, pp 10-35.
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