topos theory

# Contents

## Definition

A sheaf topos $ℰ$ is called strongly connected if it is a locally connected topos

$\left({\Pi }_{0}⊣\Delta ⊣\Gamma \right):ℰ\stackrel{\stackrel{{\Pi }_{0}}{\to }}{\stackrel{\stackrel{\Delta }{←}}{\underset{\Gamma }{\to }}}\mathrm{Set}$(\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} Set

such that the extra left adjoint ${\Pi }_{0}$ in addition preserves finite products (the terminal object and binary products).

This means it is in particular also a connected topos.

If ${\Pi }_{0}$ preserves even all finite limits then $ℰ$ is called a totally connected topos.

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

## Terminology

The “strong” in “strongly connected” may be read as referring to ${f}_{!}⊣{f}^{*}$ being a strong adjunction in that we have a natural isomorphism for the internal homs in the sense that

$\left[{f}_{!}X,A\right]\simeq {f}_{*}\left[X,{f}^{*}A\right]\phantom{\rule{thinmathspace}{0ex}}.$[f_! X, A] \simeq f_* [X, f^* A] \,.

This follows already for $f$ connected and essential if ${f}_{!}$ preserves products, because this already implies the equivalent Frobenius reciprocity isomorphism. See here for more.

and

Revised on December 7, 2011 19:47:10 by Urs Schreiber (131.174.40.86)