nLab
microlocal sheaf theory

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          In microlocal sheaf theory one generalizes the concepts of microlocal analysis, such as the wave front set of a distributions, to sheaf theory, notably to a concept of microsupport? of a sheaf on a smooth manifold, as a subset of the cotangent bundle.

          This microlocal point of view was then extended to sheaf theory? by Kashiwara-Schapira 82, Kashiwara-Schapira 82) who introduced the notion of microsupport? of sheaves giving rise to microlocal sheaf theory (see Schapira 17). Beware that some authors still use the term “microlocal analysis” for this sheaf theoretic concept.

          References

          The generalization of the concepts of microlocal analysisto microlocal sheaf theory via the concept of microsupport? of sheaves was introduced in spring

          Review of this incldues

          See also:

          • V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84, 327–402 (1986), MR87j:32030, doi

          • Masaki Sato, Restriction, localization and microlocalization, (expository paper) pp. 195–205 in collection “Quadrature domains and their application” (Operator theory: advances and applications 156), doi

          • M. Kashiwara, P. Schapira, F. Ivorra, I. Waschkies, Microlocalization of ind-sheaves, in “Studies in Lie theory”, Progress in Math. 243, Birkhäuser 2006.

          • Dmitry Tamarkin, Microlocal condition for non-displaceablility, arxiv/0809.1584 (application of microlocal analysis to symplectic/Lagrangean geometry).

          • Goro Kato, Daniele Carlo Struppa, Fundamentals of algebraic microlocal analysis, M. Dekker 1999, googB

          • Masaki Kashiwara, Systems of microdifferential equations, Birkhäuser 1983, 87 pp.

          • Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of algebraic analysis, Transl. from Japanese by Goro Kato. Princeton Mathematical Series 37, 1986. xii+255 pp.

            MR87m:58156; Jean-Luc Brylinski, Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 104–108, doi

          • Masaki Kashiwara, Pierre Schapira, Hochschild homology and microlocal Euler classes, arxiv/1203.4869

          Last revised on September 1, 2017 at 09:29:38. See the history of this page for a list of all contributions to it.