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cohesive infinity-toposes - contents

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

Structures in a cohesive $(\mathrm{\infty },1)$-topos

structures in a cohesive (∞,1)-topos

• . concrete objects

• . cohesive ∞-groups

• . cohomology and principal ∞-bundles

• . concordance

• . geometric homotopy and Galois theory

• . van Kampen theorem

• . paths and geometric Postnikov towers

• . universal coverings and geometric Whitehead towers

• . flat ∞-connections and local systems

• . de Rham cohomology

• . ∞-Lie algebras

• . Maurer-Cartan forms and curvature characteristic forms

• . differential cohomology

• . Chern-Weil homomorphism and ∞-connections

• . higher holonomy and Chern-Simons functional

Structures with infinitesimal cohesion

infinitesimal cohesion

• . de Rham space

• formal cohesive ∞-groupoid

• . de Rham theorem

Models

• discrete ∞-groupoid

  • discrete space, discrete group, discrete groupoid

• Euclidean-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

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