Spahn reflective subcategory of a topos (Rev #5, changes)

Showing changes from revision #4 to #5: Added | ~~Removed~~ | ~~Chan~~ged

The following lemma improves on the statement

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

Lemma

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 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 }

Proof

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.

We have $a_E\simeq L\circ a_H\circ R_!$ sends colimits into limits

$E$ is a topos.

$L$ is left exact.

$L\circ a_H\circ R_!$ sends colimits to limits. (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.)

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.
We have $a_E\simeq L\circ a_H\circ R_!$ sends colimits into limits, since $a_E$ is a left adjoint.
$R_!$ preserves colimits since Yoneda extension always commutes with small colimits.
$A_H$ sends colimits to limits
Hence $L$ needs to send limits in the image of $a_H\circ R_!$ to limits.
Since a reflector always preserves the terminal object (and all finite limits can be constructed from pullbacks and the terminal object), it is sufficient for 5. that $L$ preserves pullbacks in the image of $a_H\circ R_!$ .

Revision on December 8, 2012 at 19:52:05 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.