Spahn reflective subcategory of a topos (Rev #14, changes)

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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 If colimits this into composite limits, is since left exact it exhibitsa EE a_E E is as a left adjoint. exact localization of a category of presheaves and hence in this caseEE is a topos.

  3. a HR ! a_H\circ R_! preserves sends colimits to limits, since Yoneda extension always commutes with small colimits. AndR !R_! restricts (as to every Yoneda extension) commutes with colimits andRa H R a_H on as representables. a left adjoint sends colimits to limits.

  4. a Ha_HHence sends colimits to limitsa ELa HR !a_E\simeq L\circ a_H\circ R_! is left exact iff LL preserves limits in the image of a HR !a_H\circ R_!.

  5. Hence Since a reflector always preserves terminal objects (and all finite limits can be constructed from pullbacks and the terminal object), it is sufficient to check ifLL needs preserves to pullbacks send limits in the image ofa 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. Let JJ denote the canonical topology on HH st. Sh J(H)Psh J(H)HSh_J(H)\simeq \Psh_J(H)\simeq H and a HSh J()a_H\simeq Sh_J(-). 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.

  8. (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. Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at reflective subcategory as Bashir Velebil). In particular EE is a topos.)

(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. Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at reflective subcategory as Bashir Velebil). In particular EE is a topos.)

Revision on December 10, 2012 at 17:27:14 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.