Spahn reflective subcategory of a topos (Rev #2, changes)

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A The reflective following subcategory lemma of improves a on topos is a topos if the reflector following is statement left exact.

Let H≃Sh J(H)H\simeq Sh_J(H) be a topos for the canonical topology JJ. Let

  • A reflective subcategory of a topos is a topos if the reflector is left exact.
(L⊣R):E↪RH(L\dashv R):E\stackrel{R}{\hookrightarrow} H
Lemma

Let (L⊣R):E↪RH(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 H∘R !a_H\circ R_! where

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

  • R !:=Lan Y EY H∘RR_!:=Lan_{Y_E} Y_H\circ R is the left Kan extension of Y H∘RY_H\circ R along the left adjoint of the Yoneda embedding of EE.
Psh(E) →a E⊣Y E E ↓ R ! ↓ L⊣R Psh(H) →a H⊣Y 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 }

be the inclusion of a reflective subcategory into a topos. Let

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 L∘a H∘R !L\circ a_H\circ R_! is left exact. Since a H∘R !a_H\circ R_! is left exact.

    • LL presrves colimits in the image of a∘R !a\circ R_!.

    • LL preserves pullbacks in the image of a∘R !a\circ R_!

a E≃L∘a H∘R !a_E\simeq L\circ a_H\circ R_!
a:=Sh J(−):Psh(H)→Ha:=Sh_J(-):Psh(H)\to H

the sheafification functor.

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. This implies the the Yoneda embeddings of EE and HH both posess left adjoints
(a H⊣Y H):H↪Psh(H)(a_H\dashv Y_H):H\hookrightarrow Psh(H)
(a E⊣Y E):E↪Psh(E)(a_E\dashv Y_E):E\hookrightarrow Psh(E)

apparently a H=Sh J(H)a_H=Sh_J(H).

Let R !:=Lan Y EY HRR_!:=Lan_{Y_E} Y_H R then we have L∘a H∘R !L\circ a_H\circ R_! is the identity on representables and
a E∘Y E≃L∘a H∘R !a_E\circ Y_E\simeq L\circ a_H\circ R_!

is a natural equivalence.

Revision on December 8, 2012 at 04:01:53 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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