nLab stably locally connected topos

Redirected from "strongly connected geometric morphism".
Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Geometry

Contents

Definition

A sheaf topos \mathcal{E} is called stably locally connected if it is a locally connected topos

(Π 0ΔΓ):ΓΔΠ 0Set (\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set

such that the extra left adjoint Π 0\Pi_0 in addition preserves finite products (the terminal object and binary products).

This means it is in particular also a connected topos.

If Π 0\Pi_0 preserves even all finite limits then \mathcal{E} is called a totally connected topos.

If a stably locally connected topos is also a local topos, then it is a cohesive topos.

and

References

  • Peter Johnstone. Remarks on punctual local connectedness. Theory and Application of Categories 25.3 (2011): 51-63. (TAC)

Last revised on December 18, 2022 at 16:59:08. See the history of this page for a list of all contributions to it.