physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
This page collects links related to the lecture note
Toposes of laws of motion ,
transcript of a talk in Montreal, Sept. 1997
(pdf)
on the formulation of differential equations/continuum mechanics in synthetic differential geometry and the notion of toposes of laws of motion.
Another early text in this direction is Lawvere’s Categorical dynamics. Related texts on the foundations of physics in topos theory include the collection Categories in Continuum Physics.
An open question concerning the characterization of “Toposes of laws of motion” is raised as question 7 “The algebra of time” in
Entries with related discussion include geometry of physics and higher category theory and physics. Refinement to higher topos theory is discussed at Higher toposes of laws of motion.