nLab Toposes of laws of motion

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physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Topos Theory

topos theory



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synthetic differential geometry


from point-set topology to differentiable manifolds

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smooth space


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infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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This page collects links related to the lecture note

  • William Lawvere,

    Toposes of laws of motion ,

    transcript of a talk in Montreal, Sept. 1997


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 17:24:26. See the history of this page for a list of all contributions to it.