# nLab cohesive

Contents

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## 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

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

cohesion

tangent cohesion

differential cohesion

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