Spahn reflective subcategory of a topos (Rev #5, changes)

Showing changes from revision #4 to #5: Added | Removed | Changed

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
  • 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.

  • Yoneda extension always commutes with small colimits.

  • We have a ELa HR !a_E\simeq L\circ a_H\circ R_! sends colimits into limits

    • EE is a topos.

    • LL is left exact.

    • La HR !L\circ a_H\circ R_! sends colimits to limits. (Since a HR !a_H\circ R_! is right exact.)

    • LL presrves limits in the image of aR !a\circ R_!.

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

  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.

  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 5. that LL preserves pullbacks in the image of a HR !a_H\circ R_!.

Revision on December 8, 2012 at 19:52:05 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.