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\begin{document}

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\section*{geometryofphysicscontents}

\begin{itemize}%
\item About this page

\begin{itemize}%
\item Scope and perspective
\item Layers of exposition
\item The full story in a few formal words

\end{itemize}

\item Coordinate systems

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item The continuum real (world-)line
\item Cartesian spaces and smooth functions
\item The fundamental theorems about smooth functions

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item The algebraic theory of smooth algebras
\item The coverage of differentially good open covers
\item The slice category

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Judgments about types and terms
\item Natural deduction rules for product types
\item Natural deduction rules for dependent sum types
\item Dictionary: type theory / category theory

\end{itemize}


\end{itemize}

\item Smooth spaces

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Plots of smooth spaces and their gluing
\item Homomorphisms of smooth spaces
\item Products and fiber products of smooth spaces
\item Smooth mapping spaces and smooth moduli spaces
\item The smooth moduli space of smooth functions
\item Outlook

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Toposes


\item Subobjects


\item Slice toposes


\item Local, connected and cohesive toposes


\item The topos of smooth spaces

\begin{itemize}%
\item Connectedness
\item Locality
\item Cohesion

\end{itemize}


\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Natural deduction rules for dependent product types
\item Internal logic of a toposThe type of propositions
\item Cohesive modality I: Sharp types

\end{itemize}


\end{itemize}

\item Differential forms

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Differential forms on abstract coordinate systems
\item Differential forms on smooth spaces

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Concrete smooth spaces
\item Smooth space of differential forms on a smooth space

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Images

\end{itemize}


\end{itemize}

\item Differentiation

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Differentiation of smooth functions and differential forms

\begin{itemize}%
\item Differentiation on coordinate patches
\item Differentiation on smooth spaces
\item Example: The electromagnetic field strength

\end{itemize}

\item Variational calculus

\begin{itemize}%
\item Discrete points of a smooth space
\item Smooth functionals
\item Functional derivative / variational derivative
\item Euler-Lagrange equations

\end{itemize}

\item $\mathcal{D}$-geometry

\begin{itemize}%
\item Infinitesimal smooth loci
\item de Rham space
\item Jet bundles

\end{itemize}


\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Synthetic differential geometry
\item Tangent bundle
\item Differential equations
\item Differential cohesion of the topos of smooth spaces
\item Differential cohesion
\item de Rham space
\item Jet bundle
\item Formally \'e{}tale / formally unramified / formally smooth

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Differential homotopy type theory

\end{itemize}


\end{itemize}

\item Smooth homotopy types

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Gauge transformations in electromagnetism
\item Groupoids
\item Smooth groupoids
\item Differential 1-forms are smooth incremental path measures

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item ∞-Toposes
\item Slice ∞-toposes
\item Local, ∞-connected and cohesive ∞-toposes
\item The ∞-topos of smooth ∞-groupoids
\item Local ∞-Connectedness
\item Locality
\item Cohesion

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Identity types
\item Homotopy type theory
\item Cohesive modality II: flat types

\end{itemize}


\end{itemize}

\item Groups

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Groups
\item Dold-Kan correspondence

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item A ∞-types
\item ∞-Groups

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item Pointed connected types
\item Identity types of connected types

\end{itemize}


\end{itemize}

\item Principal bundles

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Connected groupoids
\item Universal principal bundle
\item Principal bundles
\item Weakly principal simplicial bundles

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Principal ∞-bundles

\end{itemize}

\item Syntactic Layer



\end{itemize}

\item Actions, representations and associated bundles

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Actions
\item Associated bundle
\item Representations up to coherent homotopy

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item ∞-Actions
\item Associated ∞-bundles

\end{itemize}

\item Syntactic Layer

\begin{itemize}%
\item The context of a pointed connected type: representation theory
\item Dependent product over a pointed connected type: invariants
\item Dependent sum over a pointed connected type: quotients

\end{itemize}


\end{itemize}

\item Manifolds

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Smooth manifolds
\item Tangent bundle

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Manifolds modeled on an object

\end{itemize}

\item Syntactic Layer



\end{itemize}

\item $G$-Structure

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Reduction of structure group
\item Vielbein, orthogonal structure
\item Almost complex structure
\item Almost Hermitean structure
\item Almost symplectic structure
\item Metaplectic structure
\item Metalinear structure
\item Generalized complex geometry
\item Type II geometry
\item Generalized Calabi-Yau structure
\item Exceptional generalized geometry
\item Spin structure, String structure, Fivebrane structure

\end{itemize}

\item Semantic Layer


\item Syntactic Layer



\end{itemize}

\item Riemannian geometry

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item Integration

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Integration
\item Integration of differential forms
\item Integration in ordinary differential cohomology
\item Lie integration
\item Transgression
\item Transgression of differential forms
\item Transgression of circle n-bundles with connection
\item Action functionals from transgression

\end{itemize}

\item Semantic Layer


\item Syntactic Layer



\end{itemize}

\item Flat connections

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item de Rham Coefficients

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Lie-algebra valued differential 1-forms
\item The de Rham complex

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item De Rham coefficient objects
\item Recovering smooth differential forms from cohesive de Rham coefficients

\end{itemize}

\item Syntactic Layer



\end{itemize}

\item Maurer-Cartan forms

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Maurer-Cartan form on a Lie group

\end{itemize}


\end{itemize}
\emph{Semantic Layer}

\begin{itemize}%
\item Maurer-Cartan form on a cohesive ∞-group


\item Maurer-Cartan forms on smooth ∞-groups


\item Cohesive differentiation


\item Universal curvature characteristic forms


\item Syntactic Layer



\end{itemize}

\item Circle-principal connections

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Circle-principal connections
\item Dirac charge quantization and the electromagnetic field
\item Deligne cohomology and Cheeger-Simons differential characters
\item Circle-principal 2-connection
\item Magnetic charge and the B-field
\item Circle-principal 3-connection
\item The 3d Chern-Simons action functional and the C-field

\end{itemize}

\item Semantic Layer

\begin{itemize}%
\item Smooth differential cohomology
\item Higher holonomy

\end{itemize}

\item Syntactic Layer



\end{itemize}

\item Characteristic classes

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item Circle-principal n-connection

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item Action functionals for Chern-Simons-type gauge theories

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item Abelian Chern-Simons theory

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item Principal connections

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Yang-Mills theory

\end{itemize}

\item Semantic Layer


\item Syntactic Layer



\end{itemize}

\item Associated bundles

\begin{itemize}%
\item Model Layer

\begin{itemize}%
\item Associated vector bundle
\item Spin geometry

\end{itemize}

\item Semantic Layer


\item Syntactic Layer



\end{itemize}

\item Covariant derivative


\item Einstein-Yang-Mills theory


\item Symplectic geometry


\item Geometric quantization


\item Supergeometric coordinate systems

\begin{itemize}%
\item Model Layer
\item Semantic Layer
\item Syntactic Layer

\end{itemize}

\item References

\begin{itemize}%
\item General
\item Differential forms and parallel transport
\item Mathematical quantum field theory
\item Further

\end{itemize}


\end{itemize}


\end{document}