#
Generalized covering space theories

##
Jeremy Brazas

In this paper, we unify various approaches to generalized covering space
theory by introducing a categorical framework in which coverings are
defined purely in terms of unique lifting properties. For each category C
of path-connected spaces having the unit disk as an object, we construct a
category of C-coverings over a given space X that embeds in the category
of $\pi_1(X,x_0)$-sets via the usual monodromy action on fibers. When C is
extended to its coreflective hull H(C), the resulting category of based
H(C)-coverings is complete, has an initial object, and often characterizes
more of the subgroup lattice of $\pi_1(X,x_0)$ than traditional covering
spaces.
We apply our results to three special coreflective subcategories: (1) The
category of $\Delta$-coverings employs the convenient category of
$\Delta$-generated spaces and is universal in the sense that it contains
every other generalized covering category as a subcategory. (2) In the
locally path-connected category, we preserve notion of generalized
covering due to Fischer and Zastrow and characterize the topology of such
coverings using the standard whisker topology. (3) By employing the
coreflective hull **Fan** of the category of all contractible spaces,
we characterize the notion of continuous lifting of paths and identify the
topology of **Fan**-coverings as the natural quotient topology
inherited from the path space.

Keywords:
Fundamental group, generalized covering map, coreflective hull, unique
path lifting property

2010 MSC:
55R65, 57M10, 55Q52

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 35, pp 1132-1162.

Published 2015-08-23.

http://www.tac.mta.ca/tac/volumes/30/35/30-35.pdf

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