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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 When the space istopologicallystratified, Treumann, based on unpublished ideas of MacPherson, constructed an`exit category' (in the terminology of this paper, the `fundamentalcategory') which classifies constructible sheaves,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