topos theory

Contents

Definition

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

$(\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set$

such that the extra left adjoint $\Pi_0$ 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 $\mathcal{E}$ 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 $f_! \dashv f^*$ being a strong adjunction in that we have a natural isomorphism for the internal homs in the sense that

$[f_! X, A] \simeq f_* [X, f^* A] \,.$

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

Last revised on July 23, 2014 at 06:04:14. See the history of this page for a list of all contributions to it.