Spahn reflective subcategory of a topos

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

1. 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.

2. We have $a_E\simeq L\circ a_H\circ R_!$. If this composite is left exact it exhibits $E$ as a left exact localization of a category of presheaves and hence in this case $E$ is a topos.

3. $a_H\circ R_!$ sends colimits to limits, since $R_!$ (as every Yoneda extension) commutes with colimits and $a_H$ as a left adjoint sends colimits to limits.

4. Hence $a_E\simeq L\circ a_H\circ R_!$ is left exact iff $L$ preserves limits in the image of $a_H\circ R_!$.

5. Since a reflector always preserves terminal objects (and all finite limits can be constructed from pullbacks and the terminal object), it is sufficient to check if $L$ preserves pullbacks in the image of $a_H\circ R_!$.

(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. Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at reflective subcategory as Bashir Velebil). In particular $E$ is a topos.)