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Functorial and algebraic properties of Browns P functor

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Luis-Javier Hernandez-Paricio

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.

http://www.tac.mta.ca/tac/volumes/1995/n2/v1n2.dvi

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