# nLab microlocal sheaf theory

Contents

topos theory

## Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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

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

• Makoto Sakai, 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

• Vivek Shende, Microsheaves from Hitchin fibers via Floer theory, arXiv:2108.13571