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Exit paths and constructible stacks

  • David Treumann (a1)
Abstract
Abstract

For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP≤2(X,S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2-category to the 2-category of small categories.

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[2] T. Braden , Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002).

[3] T. Braden and M. Grinberg , Perverse sheaves on rank stratifications, Duke Math. J. 96 (1999).

[4] S. Gelfand , R. MacPherson and K. Vilonen , Perverse sheaves and quivers, Duke Math. J. 83 (1996).

[5] M. Goresky and R. MacPherson , Intersection homology II, Invent. Math. 72 (1983).

[7] R. MacPherson and K. Vilonen , Elementary construction of perverse sheaves, Invent. Math. 84 (1986).

[9] J. Milnor , Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) (1961), 575590.

[10] P. Polesello and I. Waschkies , Higher monodromy, Homology Homotopy Appl. 7(1) (2005), 109150.

[11] F. Quinn , Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988).

[12] L. Siebenmann , Deformations of homeomorphisms on stratified sets, Comm. Math. Helv. 47 (1972).

[13] R. Thom , Ensembles et morphismes stratifiées, Bull. Amer. Math. Soc. 75 (1969).

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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