Spahn reflective subcategory of a topos (Rev #8)

The following lemma improves on the statement

  • A reflective subcategory of a topos is a topos if the reflector is left exact.
Lemma

Let (LR):ERH(L\dashv R):E\stackrel{\R}{\hookrightarrow} H be a reflective subcategory of a topos.

Then EE is a topos if LL preserves pullbacks in the image of a HR !a_H\circ R_! where

  • (a HY H):HPsh(H)(a_H\dashv Y_H):H\to Psh(H) is the left adjoint of the Yoneda embedding of HH.

  • R !:=Lan Y EY HRR_!:=Lan_{Y_E} Y_H\circ R is the left Kan extension of Y HRY_H\circ R along the Yoneda embedding of EE.

Psh(E) a EY E E R ! LR Psh(H) a HY H H\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
  1. The Yoneda embeddings of EE and HH both posess left adjoints: HH and EE are total: Since HH is a topos, HH is total, since EE is a reflective subcategory of a total category EE is total. By the adjoint functor theorem for total categories this implies that the Yoneda embeddings of EE and HH both posess left adjoints.

  2. We have a ELa HR !a_E\simeq L\circ a_H\circ R_! sends colimits into limits, since a Ea_E is a left adjoint.

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

  4. a Ha_H sends colimits to limits

  5. Hence LL needs to send limits in the image of a HR !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 LL preserves pullbacks in the image of a HR !a_H\circ R_!.

  7. Now let JJ be the canonical Grothendieck topology on HH st. Sh(H)Psh(H)HSh(H)\simeq \Psh(H)\simeq H. Psh(E)Psh(E) is cocomplete and hence every diagram D:IPsh(E)D:I\to Psh(E) has a colimit which is preserved by R !R_! and a Ha_H. This means that to give a diagram on HH is equivalent to give a diagram on Psh(E)Psh(E)

  • (If \circ denotes a duality and lrl\dashv r then r l r^\circ\dashv l^\circ. Hence if R !R_! has a left adjoint, then RR has a right adjoint.)

Revision on December 9, 2012 at 03:53:22 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.