nLab Global Homotopy Theory and Cohesion

representation theory

geometric representation theory

Ingredients

• linear algebra, algebra, higher algebra

Definitions

representation, 2-representation, ∞-representation

• group, ∞-group

• group algebra, algebraic group, Lie algebra

• vector space, n-vector space

• affine space, symplectic vector space

• action, ∞-action

• module, equivariant object

• bimodule, Morita equivalence

• induced representation, Frobenius reciprocity

• Hilbert space, Banach space, Fourier transform, functional analysis

• orbit, coadjoint orbit, Killing form

• unitary representation

• geometric quantization, coherent state

• socle, quiver

• module algebra, comodule algebra, Hopf action, measuring

Geometric representation theory

• D-module, perverse sheaf,

• Grothendieck group, lambda-ring, symmetric function, formal group

• principal bundle, torsor, vector bundle, Atiyah Lie algebroid

• geometric function theory, groupoidification

• Eilenberg-Moore category, algebra over an operad, actegory, crossed module

• reconstruction theorems

Theorems

• Borel-Weil-Bott theorem

• Be?linson-Bernstein localization?

• Kazhdan-Lusztig theory

• BBDG decomposition theorem

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

• (∞,1)-topos theory

• higher geometry

• motivation for cohesive toposes

Definition

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

Presentation over a site

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site

Models

• discrete ∞-groupoid

  • discrete space, discrete group, discrete groupoid

• D-topological ∞-groupoid

  • topological space, topological groupoid, topological group

• smooth ∞-groupoid

  • smooth space, diffeological space, smooth manifold, Lie group, Lie groupoid, differentiable stack, Lie 2-group, Lie 2-groupoid, smooth ∞-groupoid

• synthetic differential ∞-groupoid

  • Lie algebra, Lie algebroid, ∞-Lie algebra, ∞-Lie algebroid

• super ∞-groupoid

  • superalgebra, super Lie algebra, super L-∞ algebra

• smooth super ∞-groupoid

  • superpoint, supergeometry, supermanifold, super Lie group, super ∞-groupoid, synthetic differential super ∞-groupoid

• cohesion of global- over G-equivariant homotopy theory