Spahn reflective subcategory of a topos (Rev #4, changes)

Proof

• 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.

• Yoneda extension always commutes with small colimits.

• $E$We have is a topos if $a_E\simeq L\circ a_H\circ R_!$$L$ sends colimits into limits is left exact.

  • $\mathrm{<span><del\; class="diffmod">\; L</del><ins\; class="diffmod">\; E</ins></span>}$ L E is left a exact topos. iff$L\circ a_H\circ R_!$ is right exact. (Since $a_H\circ R_!$ is right exact.)

  • $L$ presrves is limits left in exact. the image of$a\circ R_!$.

  • $L\circ a_H\circ R_{!}$ L L\circ a_H\circ R_! preserves sends pullbacks colimits in to the limits. image (Since of$a{a}_H\circ R_{!}$ a\circ a_H\circ R_! . (Since is a right reflector exact.) always preserves the terminal object and all finite limits can be constructed from pullbacks and the terminal object.)

  • $L$ presrves limits in the image of $a\circ R_!$.

  • $L$ preserves pullbacks in the image of $a\circ R_!$. (Since a reflector always preserves the terminal object and all finite limits can be constructed from pullbacks and the terminal object.)