Spahn reflective subcategory of a topos (Rev #8, changes)

Showing changes from revision #7 to #8: 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.

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 }

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. And $R_!$ restricts to $R$ on representables.
$a_H$ 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 the previous point 5. that $L$ preserves pullbacks in the image of $a_H\circ R_!$ .
Now let $J$ be the canonical Grothendieck topology on $H$ st. $Sh(H)\simeq \Psh(H)\simeq H$ . $Psh(E)$ is cocomplete and hence every diagram $D:I\to Psh(E)$ has a colimit which is preserved by $R_!$ and $a_H$ . This means that to give a diagram on $H$ is equivalent to give a diagram on $Psh(E)$

(If $\circ$ denotes a duality and $l\dashv r$ then $r^\circ\dashv l^\circ$ . Hence if $R_!$ has a left adjoint, then $R$ has a right adjoint.)

Revision on December 9, 2012 at 03:53:22 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.