# 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

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

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.

The concept was introduced in

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