The following lemma improves on the statement * *A reflective subcategory of a topos is a topos if the reflector is left exact.* +-- {: .num_lemma} ###### 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} ###### Proof 1. The Yoneda embeddings of $E$ and $H$ both posess left adjoints: $H$ and $E$ are [[total category|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. 1. We have $a_E\simeq L\circ a_H\circ R_!$ sends colimits into limits, since $a_E$ is a left adjoint. 1. $R_!$ preserves colimits since Yoneda extension always commutes with small colimits. And $R_!$ restricts to $R$ on representables. 1. $a_H$ sends colimits to limits 1. Hence $L$ needs to send limits in the image of $a_H\circ R_!$ to limits. 1. 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 $a_H\circ R_!$. 1. Let $j$ denote the canonical [[nLab: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 $a_H$. 1. (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 [[nLab:reflective subcategory]] as Bashir Velebil). In particular $E$ is a topos.) =--