Spahn reflective subcategory of a topos (Rev #3)

The following lemma improves on the following statement

A reflective subcategory of a topos is a topos if the reflector is left exact.

Let $(L\dashv R):E\stackrel{\R}{\hookrightarrow} H$ be a reflective subcategory of a topos.

Then $E$ is a topos if $L$ preserves pullbacks in the image of $a_H\circ R_!$ where

$(a_H\dashv Y_H):H\to Psh(H)$ is the left adjoint of the Yoneda embedding of $H$ .
$R_!:=Lan_{Y_E} Y_H\circ R$ is the left Kan extension of $Y_H\circ R$ along the left adjoint of the Yoneda embedding of $E$ .

\array{ Psh(E)&\stackrel{a_E\dashv Y_E}{\to}&E \\ \downarrow^{R_!}&&\downarrow^{L\dashv R} \\ Psh(H)&\stackrel{a_H\dashv Y_H}{\to}&H }

The Yoneda embeddings of $E$ and $H$ both posess left adjoints: $H$ and $E$ are total: Since $H$ is a topos, $H$ is total, since $E$ is a reflective subcategory of a total category $E$ is total. By the adjoint functor theorem for total categories this implies that the Yoneda embeddings of $E$ and $H$ both posess left adjoints.
Yoneda extension always commutes with small colimits.
$E$ is a topos if $L$ is left exact.
- $L$ is left exact iff $L\circ a_H\circ R_!$ is right exact. (Since $a_H\circ R_!$ is right exact.)
- $L$ presrves limits in the image of $a\circ R_!$ .
- $L$ preserves pullbacks in the image of $a\circ R_!$ . (Since a reflector always preserves the terminal object and all finite limits can be constructed from pullbacks and the terminal object.)

Revision on December 8, 2012 at 04:55:11 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.