nLab logical topology

Contents

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

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

The notion of Penon open is closely related to that of Zariski open. Consider the affine line 𝔸:fpRingSet\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 𝔸\mathbb{A} is a local ring. A ring is local if for all yy, either yy or 1y1- y is invertible; equivalently, we may ask that either yy or xyx - y is invertible for a given invertible element xx, by homogeneity. In this case, the locality of 𝔸\mathbb{A} is equivalent to the subset 𝔾 m𝔸\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.