Spahn reflective subcategory of a topos (Rev #12)

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 colimits into limits, since $a_E$ is a left adjoint.

3. $R_!$ preserves colimits since Yoneda extension always commutes with small colimits. And $R_!$ restricts to $R$ on representables.

4. $a_H$ sends colimits to limits

5. Hence $L$ needs to 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 Lawvere-Tierney 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.)