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The fundamental category of a stratified space
#
The fundamental category of a stratified space

##
Jon Woolf

The fundamental groupoid of a locally $0$ and $1$-connected
spaceclassifiescovering spaces, or equivalently local systems. When the space
istopologicallystratified, Treumann, based on unpublished ideas of MacPherson,
constructed an`exit category' (in the terminology of this paper, the
`fundamentalcategory') which classifies constructible sheaves, equivalently
stratifiedetale covers.This paper generalises this construction to
homotopicallystratified sets, inaddition showing that the fundamental category
dually classifiesconstructiblecosheaves, equivalently stratified branched
covers.

The more general setting has several advantages. It allows us toremove
atechnical `tameness' condition which appears in Treumann's work; toshow
thatthe fundamental groupoid can be recovered by inverting allmorphisms and,
perhaps most importantly, to reduce computations to the two-stratumcase.
Thisprovides an approach to computing the fundamental category in termsof
homotopygroups of strata and homotopy links. We apply these techniques
tocompute thefundamental category of symmetric products of $\C$, stratified
bycollisions.

Two appendices explain the close relations respectively
betweenfiltered and pre-ordered spaces and between cosheaves and branchedcovers
(technically locally-connected uniquely-complete spreads).

Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 359-387