Spahn reflective subcategory of a topos (Rev #3)

The following lemma improves on the following 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 left adjoint of 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.

  • EE is a topos if LL is left exact.

    • LL is left exact iff La HR !L\circ a_H\circ R_! is right exact. (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.)

Revision on December 8, 2012 at 04:55:11 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.