# nLab Toposes of laws of motion

Related entries

topos theory

## Theorems

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

• 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

Last revised on June 11, 2018 at 13:24:26. See the history of this page for a list of all contributions to it.