nLab logical topology

Contents

Context

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

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

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

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

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

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a given intuitionistic logic framework, such as a type theory, the logical topology (Penon 81) on a given object (type) $X$ is that whose open subsets are the sets $U$ for which any $X - \{x\}$ and $U$ cover $X$. This implies that an open subset contains for every point $x$ also the collection of points that are indistinguishable from it in classical logic, hence that are not not equal to $x$.

The notion of Penon open is closely related to that of Zariski open. Consider the affine line $\mathbb{A} : \text{fpRing} \to \text{Set}$ as the forgetful functor from finitely presented rings to sets. The (category theorists’) big Zariski topos is the largest subtopos of this topos of functors in which $\mathbb{A}$ is a local ring. A ring is local if for all $y$, either $y$ or $1- y$ is invertible; equivalently, we may ask that either $y$ or $x - y$ is invertible for a given invertible element $x$, by homogeneity. In this case, the locality of $\mathbb{A}$ is equivalent to the subset $\mathbb{G}_m \hookrightarrow \mathbb{A}$ of invertible elements being logically open.

References

The concept was introduced in

Review in the context of cohesive toposes, modal type theory and cohesive homotopy type theory includes

Last revised on July 26, 2022 at 14:26:42. See the history of this page for a list of all contributions to it.