topology

algebraic topology

Basic concepts

• space

• locale

• topological space

  • continuous map

  • homeomorphism

• Top

  • nice topological space

  • nice category of spaces

  • convenient category of topological spaces

• homotopy theory

  • homotopy group

  • covering space

Theorems

• Whitehead's theorem

• Freudenthal suspension theorem

• nerve theorem

Extra stuff, structure, properties

• CW-complex, Hausdorff space, second-countable space, sober space

• compact space, paracompact space

• connected space, locally connected space, contractible space, locally contractible space

• topological vector space, Banach space, Hilbert space

• manifold

Examples

• point, real line, plane

• sphere, ball, annulus

• polytope, polyhedron

• loop space, path space

• Cantor space, Sierpinski space

• long line, Warsaw circle

differential geometry

synthetic differential geometry

Axiomatics

• smooth topos

  • infinitesimal space

    • amazing right adjoint

  • Kock-Lawvere axiom

  • integration axiom

  • microlinear space

• synthetic differential supergeometry

  • super smooth topos

• . infinitesimal cohesion

  • de Rham space

  • formally smooth morphism, formally unramified morphism, formally étale morphism

  • jet bundle

Models

• Models for Smooth Infinitesimal Analysis

  • Fermat theory

  • smooth algebra ($C^{\mathrm{\infty }}$-ring)

  • smooth locus

  • smooth manifold, formal smooth manifold, derived smooth manifold

  • smooth space, diffeological space, Frölicher space

  • smooth natural numbers

• Cahiers topos

• smooth ∞-groupoid

• synthetic differential ∞-groupoid

Concepts

• tangent bundle,

  • vector field, tangent Lie algebroid;

  • differentiation, chain rule

• differential forms

• differential equation, variational calculus

  • Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space

• connection on a bundle, connection on an ∞-bundle

• Riemannian manifold

  • isometry, Killing vector field

Theorems

• Hadamard lemma

• Borel's theorem

• Boman's theorem

• Steenrod-Wockel approximation theorem

• Poincare lemma

• Stokes theorem

• de Rham theorem

• Chern-Weil theory

Applications

• Lie theory, ∞-Lie theory

• Chern-Weil theory, ∞-Chern-Weil theory

  • gauge theory

  • ∞-Chern-Simons theory

• Klein geometry, Klein 2-geometry, higher Klein geometry

• Euclidean geometry, Cartan geometry, higher Cartan geometry

• Riemannian geometry

  • gravity, supergravity

mapping space

General abstract

• hom-set, hom-object, internal hom, exponential object, derived hom-space

• loop space object, free loop space object, derived loop space

Topology

• topology of mapping spaces

  • compact-open topology

• evaluation fibration of mapping spaces

• loop space, free loop space

Differential topology

• differential topology of mapping spaces

  • C-k topology

• manifold structure of mapping spaces

  • tangent spaces of mapping spaces

• smooth loop space

Stable homotopy theory

• mapping spectrum