Spahn reflective subcategory of a topos (Rev #13, 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 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 $\mathrm{<span><del\; class="diffmod">\; j</del><ins\; class="diffmod">\; J</ins></span>}$ j J denote the canonical topology onLawvere-Tierney topology$H$ on st.${\mathrm{Sh}}_J(H)\simeq {Psh}_J(H)\simeq H$ Sh_J(H)\simeq \Psh_J(H)\simeq H st. and${\mathrm{<span><del\; class="diffmod">\; Sh</del><ins\; class="diffmod">\; a</ins></span>}}_{\mathrm{<span><del\; class="diffmod">\; j</del><ins\; class="diffmod">\; H</ins></span>}}(H)\simeq {Psh}_{\mathrm{Sh}}(H-)\simeq H$ Sh_j(H)\simeq a_H\simeq \Psh_j(H)\simeq Sh_J(-) H . and$a_H\mathrm{Psh}\simeq {\mathrm{Sh}}_j(-E)$ a_H\simeq Psh(E) Sh_j(-) . is cocomplete and hence every diagram$D:I\to \mathrm{Psh}(E)$ D:I\to Psh(E) has a colimit which is cocomplete preserved and by hence every diagram$D{R}_{!}:I\to \mathrm{Psh}(E)$ D:I\to R_! Psh(E) has and a colimit which is preserved by${\mathrm{<span><del\; class="diffmod">\; R</del><ins\; class="diffmod">\; a</ins></span>}}_{!}$ R_! a_H 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.)