Spahn reflective subcategory of a topos (Rev #14, changes)

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 $a_E\simeq L\circ a_H\circ R_!$ . sends If colimits this into composite limits, is since left exact it exhibits$a_{E}E$ a_E E is as a left adjoint. exact localization of a category of presheaves and hence in this case$E$ is a topos.

3. $a_H\circ R_{!}$ a_H\circ R_! preserves sends colimits to limits, since Yoneda extension always commutes with small colimits. And$R_!$ restricts (as to every Yoneda extension) commutes with colimits and$R{a}_H$ R a_H on as representables. a left adjoint sends colimits to limits.

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

5. Hence 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$L$ needs preserves to pullbacks send limits in the image of$a_H\circ R_!$ . to limits.

6. 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 $a_H\circ R_!$.

7. Let $J$ denote the canonical topology on $H$ st. $Sh_J(H)\simeq \Psh_J(H)\simeq H$ and $a_H\simeq Sh_J(-)$. $Psh(E)$ is cocomplete and hence every diagram $D:I\to Psh(E)$ has a colimit which is preserved by $R_!$ and $a_H$.

8. (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.)