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% Colimits for the Pro category of towers of simplicial sets
% David Blanc
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% January 18, 1995
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The Pro category of towers of spaces (and of other categories) has been
studied in several contexts, and used for a variety of applications in
homotopy theory, shape theory, geometric topology, and algebraic
geometry - as well as in the study of v_n-periodicity in unstable
homotopy theory.
One problem in the usual version of the Pro category of towers is
that, while finite limits and colimits exist, and may be
constructed in a straightforward (levelwise) manner, the same does not hold for
infinite colimits; and these were needed for the application to
v_n-periodicity.
The construction presented here embeds a suitable subcategory of the
Pro category Tow of towers of simplicial sets in a certain category
Net of strict Ind-towers, in which we have explicit constructions
for all colimits, as well as finite limits. This category Net
can thus be thought of as a cocompletion of the Pro category of towers
of spaces.
There are other cocomplete categories in which Tow may be embedded -
for example, the category of all pro-simplicial sets, or the full
category of all inductive systems of towers. One advantage of
the approach described here is that one obtains a smaller, and more mangeable,
cocompletion, in this special case, and the construction of the
colimits may be made quite explicitly. A side effect of our approach
is the elimination of certain ``phantom phenomena'' from the
Pro category of towers.