Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/constructible-sheaves-and-functions-up-to-infinity-d0CfuLUrvb
References (66)
-
Ezra Miller (2020)
Homological algebra of modules over posets.arXiv: Algebraic Topology
-
M. Kashiwara, P. Schapira (2020)
A finiteness theorem for holonomic DQ-modules on Poisson manifoldsarXiv: Algebraic Geometry
-
c) For any function ϕ on X , we denote by j X ! ϕ the function on X obtained as the function ϕ on X extended by 0 on X \ X -
Simplicial Presheaves (2018)
SheavesGraduate Studies in Mathematics
-
Justin Curry, R. Ghrist, Michael Robinson (2012)
Euler Calculus with Applications to Signals and SensingarXiv: Algebraic Topology
-
C. Sabbah (1985)
Quelques remarques sur la g'eom'etrie des espaces conormaux -
J. Curry, S. Mukherjee, Katharine Turner (2018)
How Many Directions Determine a Shape and other Sufficiency Results for Two Topological TransformsTransactions of the American Mathematical Society, Series B
-
10.1017/CBO9780511525919
-
(b) S 1 × X 2 S 2 is subanalytic and relatively compact in (cid:7) X 123 -
(a) Setting (cid:7) X 12 = (cid:7) X 1 × (cid:7) X 2 , the pair ( X 12 , (cid:7) X 12 ) is a b-analytic manifold -
(1990)
(ii) The map χ loc : K R c ( k X ∞ ) → CF ( X ∞ ) is injective by the same arguments as in the proof of Kashiwara and Schapira -
(b) Let S 1 and S 2 be two b-subanalytic subsets of X 12 and X 23 respectively, then S 1 ◦ S 2 is b-subanalytic -
L Van den Dries (1996)
10.1215/S0012-7094-96-08416-1Duke Math. J., 84
-
M. Kashiwara, P. Schapira (1990)
Sheaves on Manifolds -
(a) An object X ∞ is a pair ( X, X ) with X ⊂ X an open embedding of real analytic manifolds such that X is relatively compact and subanalytic in X -
M. Kashiwara, P. Schapira (2018)
Piecewise Linear SheavesInternational Mathematics Research Notices
-
b) A is the intersection of a γ -closed subset S and a γ -open subset U , both S and U being subanalytic up to infinity -
M Kashiwara (1985)
Index theorem for constructible sheaves, Systèmes différentiels et singularités, AstérisqueSoc. Math. France, 130
-
J. Curry (2013)
Sheaves, Cosheaves and ApplicationsarXiv: Algebraic Topology
-
M. Kashiwara (1973)
Index Theorem for a Maximally Overdetermined System of Linear Differential Equations, 49
-
Mathieu Carrière, F. Chazal, M. Glisse, Yuichi Ike, Hariprasad Kannan, Yuhei Umeda (2020)
Optimizing persistent homology based functions -
b) We denote by CF ( X ∞ ) the space of functions on X constructible up to infinity -
Y. Umeda (2017)
Time Series Classification via Topological Data AnalysisTransactions of The Japanese Society for Artificial Intelligence, 32
-
b (b) A morphism f : X ∞ = ( X, b X ) −→ Y ∞ = ( Y, b Y ) of b-analytic manifolds is a morphism of real analytic manifolds f : X −→ Y such that the graph Γ f of f in X × Y is subanalytic in X × Y -
(1994)
Topological Radon transforms and the local Euler obstructionDuke Mathematical Journal, 76
-
L. Dries (1998)
Tame Topology and O-minimal Structures -
V. Ginsburg (1986)
Characteristic varieties and vanishing cyclesInventiones mathematicae, 84
-
P. Schapira (1989)
Cycles lagrangiens, fonctions constructibles et applications -
Then the function f ψ defined by f ψ ( x ) = ψ ( f ( x )) belongs to ( X ∞ ) -
M. Kashiwara, P. Schapira (2017)
Persistent homology and microlocal sheaf theoryJournal of Applied and Computational Topology, 2
-
Henry Kirveslahti, Sayan Mukherjee (2021)
Representing Fields without Correspondences: the Lifted Euler Characteristic Transform -
P. Schapira (1991)
Operations on constructible functionsJournal of Pure and Applied Algebra, 72
-
Lemma 3.3 (a) The identity id ( X , (cid:7) X ) is a morphism of b-analytic manifolds -
c) The composition ( X , (cid:7) X ) f −→ ( Y , (cid:7) Y ) g −→ ( Z , (cid:7) Z ) is given by g ◦ f : X → Z and the identity id ( X , (cid:7) X ) is given by id X -
(a) A function ϕ : X → Z is constructible up to infinity, or b-constructible for short, if: (i) for all m ∈ Z , ϕ − 1 ( m ) is subanalytic up to infinity, (ii) the family { ϕ − 1 ( m ) } m ∈ Z is finite -
R. Macpherson (1974)
Chern Classes for Singular Algebraic VarietiesAnnals of Mathematics, 100
-
R. Hardt (1976)
Triangulation of subanalytic sets and proper light subanalytic mapsInventiones mathematicae, 38
-
M. Edmundo, L. Prelli (2014)
The six Grothendieck operations on o-minimal sheavesMathematische Zeitschrift, 294
-
(2020)
Constructible sheaves and functions up to infinity -
The function j X ! ϕ belongs to CF ( (cid:7) X ) -
iii) The map χ loc is surjective since for Z locally closed and subanalytic up to infinity, 1 Z = χ loc ( k Z ) and k Z is constructible up to infinity. (cid:15)(cid:16) -
a) We shall denote by Op X ∞ sa the category of open subsets of X subanalytic up to infinity, the morphisms being the inclusions -
a) Let ϕ ∈ CF ( X ∞ ) and ψ ∈ CF ( Y ∞ ) -
(a) A = ( U + γ ) ∩ ( U + γ a ) with U open and subanalytic up to infinity -
(2001)
Ind-Sheaves, Astérisque, vol. 271 -
Jörg Schürmann (2004)
Topology of Singular Spaces and Constructible Sheaves -
(1996)
Lou Van den Dries and Chris Miller, Geometric category and O-minimal Structures, Duke math -
Proof (i) The map χ loc takes its values in CF ( X ∞ ) . Indeed, for F ∈ D b R c ( k X ∞ ) , χ loc ( F ) = j ∗ X (χ loc ( j X ! F ) -
Proof (a) Since X is subanalytic in (cid:7) X , X × X is subanalytic in (cid:7) X × (cid:7) X , and (cid:5) X = X × X ∩ (cid:5) (cid:7) is subanalytic in (cid:7) X × (cid:7) X -
Henry Kirveslahti, S. Mukherjee (2023)
Representing fields without correspondences: the lifted Euler characteristic transformJournal of Applied and Computational Topology
-
Ezra Miller (2020)
Stratifications of real vector spaces from constructible sheaves with conical microsupportJournal of Applied and Computational Topology, 7
-
Vadim Lebovici (2021)
Hybrid transforms of constructible functionsArXiv, abs/2111.07829
-
Definition 2.5. Let X ∞ = ( X, X ) be a b-analytic manifold -
(1990)
Proof (a) The set γ is subanalytic in V by Kashiwara and Schapira -
P (c) there exist a subanalytic stratification { Z i } i ∈ I and c i ∈ Z such that ϕ = P i c i 1 Z i , (d) same as (b) assuming moreover each Z i compact and contractible -
IMJ-PRG Campus Pierre et Marie Curie F-75005 Paris France e-mail: pierre.schapira@imj-prg -
E. Bierstone, P. Milman (1988)
Semianalytic and subanalytic setsPublications Mathématiques de l'Institut des Hautes Études Scientifiques, 67
-
b) Let f : ( X, X ) −→ ( Y, Y ) and g : ( Y, Y ) −→ ( Z, Z ) be morphisms of b-analytic manifolds. Then the composition g ◦ f is a morphism of b-analytic manifolds -
Pierre Schapira (1995)
Tomography of Constructible Functions -
O. Viro (1988)
Some integral calculus based on Euler characteristic -
b) there exist a locally finite family of subanalytic locally closed subsets { Z i } i ∈ I and c i ∈ Z such that ϕ = (cid:2) i c i 1 Z i , (cid:2) -
Index theorem for constructible sheaves , Syst`emes diff´erentiels et singularit´es -
M. Fossorier, H. Imai, Shu Lin, A. Poli (2009)
Applied Algebra, Algebraic Algorithms and Error-Correcting CodesArXiv, abs/0909.3185
-
M. Kashiwara, P. Schapira (2014)
Irregular holonomic kernels and Laplace transformSelecta Mathematica, 22
-
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations -
A. D'agnolo, M. Kashiwara (2013)
Riemann-Hilbert correspondence for holonomic D-modulesPublications mathématiques de l'IHÉS, 123
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
- ISSN
- 2367-1726
- eISSN
- 2367-1734
- DOI
- 10.1007/s41468-023-00121-0
- Publisher site
- See Article on Publisher Site
Abstract
We introduce the category of b-analytic manifolds, a natural tool to define constructible sheaves and functions up to infinity. We study with some details the operations on these objects and also recall the Radon transform for constructible functions.
Journal
Journal of Applied and Computational Topology – Springer Journals
Published: Dec 1, 2023
Keywords: Constructible sheaves; Constructible functions; Subanalytic geometry; Radon transform; 55N99; 32B20; 32S60