nLab cohesive

In solid state physics

In physics and chemistry (solid state physics), cohesion refers to the tendency of certain types of matter/substance to hold together.

In physics, the intermolecular attractive force acting between two adjacent portions of a substance, particularly of a solid or liquid. It is this force that holds a piece of matter together. (Enc. Britannica).

• J. S. Rowlinson, Cohesion A Scientific History of Intermolecular Forces (web)

Analogy with a quality of space

In natural philosophy

In Georg Hegel‘s Encyclopedia of the Philosophical Sciences there is discussion of the cohesion of some substance.

In categorical logic / topos theory

William Lawvere argued that the “objective logic” of this discussion is to be formalized via categorical logic by the axiomatics of cohesive toposes, i.e. by modal type theory equipped with shape modality and flat modality.

Entries discussing aspects of cohesion in this sense include the following

• motivation for cohesion, geometry of physics

• cohesive topos

  • cohesive site

• cohesive (∞,1)-topos, cohesive homotopy type theory,

  • ∞-cohesive site

• differential cohesion

Hegel goes on to speak of cohesion being refined to elasticity:

PN§297Zusatz Elasticity is the whole of cohesion.

Moreover, according to PN§298 this elasticity is related to the unity of opposites that constitute Zeno's paradox of motion, hence to the modern concept of differentiation via a limit of a sequence. In terms of categorical logic this is precisely what is encoded in the infinitesimal shape modality and infinitesimal flat modality of

• differential cohesion

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

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

Examples

• discrete cohesion

• stable cohesion

• global equivariant homotopy theory

• continuous cohesions

• smooth cohesion

• synthetic differential cohesion

• supergeometric cohesion

Applications

• differential calculus in cohesion

• differential cohomology/differential cohomology diagram

• ∞-Chern-Weil theory introduction

• local prequantum field theory

• motivic quantization

Related concepts

• elasticity, solidity

category: adjective