nLab
Toposes of laws of motion

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

This page collects links related to the lecture note

  • William Lawvere,

    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

  • Open problems in topos theory, April 2009 (pdf)

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.

category: reference

Revised on November 29, 2013 13:49:24 by Urs Schreiber (217.195.240.90)