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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 31st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexwrote [[Isbell duality]].

   wrote Isbell duality.

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeAug 31st 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexNot sure which way the links should go, or where in the pages, but this should presumably link to [[Isbell envelope]].

   Not sure which way the links should go, or where in the pages, but this should presumably link to Isbell envelope.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 31st 2010
   • (edited Aug 31st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI further added sections * on the fact that the left adjoint $\mathcal{O}$ lands in limit-preserving copresheaves; * on the fact that if the site $C := T$ is a Lawvere theory, we may think of $\mathcal{O}$ as homming into the $T$-line object. All entirely tautological. The point being to make these tautologies manifest. > Not sure which way the links should go, or where in the pages, but this should presumably link to Isbell envelope. Right. I need to make a telephone call now. If after that you haven't done it, then I'll do it.

   I further added sections

   • on the fact that the left adjoint $\mathcal{O}$ lands in limit-preserving copresheaves;

   • on the fact that if the site $C := T$ is a Lawvere theory, we may think of $\mathcal{O}$ as homming into the $T$-line object.

   All entirely tautological. The point being to make these tautologies manifest.

   Not sure which way the links should go, or where in the pages, but this should presumably link to Isbell envelope.

   Right. I need to make a telephone call now. If after that you haven’t done it, then I’ll do it.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2010
   • (edited Sep 1st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFurther expanded the last subsection, which is supposed to isolate the purely 1-categorical adjunction that underlies the discussion at [[function algebras on infinity-stacks]]. Here is the story I am meaning to tell: For $C := T$ an algebraic theory, we may think of ordinary Isbell duality $$ (\mathcal{O} \dashv Spec) : T Alg^{op} = [C,Set]_\times^{op} \stackrel{\leftarrow}{\to} [C^{op}, Set] $$ equivalently as given by taking the left adjoint $\mathcal{O}$ to be homs into the **line object** $\mathbb{A} := Spec(F_T(1))$. But in applications, we actually want $C$ to sit in between the free $T$-algebras and general $T$-algebras $$ T \subset C \subset T Alg^{op} $$ such that we still have the adjunction $$ (\mathcal{O} \dashv Spec) : T Alg^{op} = [T,Set]_\times^{op} \stackrel{\leftarrow}{\to} [C^{op}, Set] $$ (notice now $T \neq C$!). For this case the original simple definition of the Isbell duality does not quite make sense anymore. But that in terms of homs into the line object does, and still gives an adjunction.

   Further expanded the last subsection, which is supposed to isolate the purely 1-categorical adjunction that underlies the discussion at function algebras on infinity-stacks.

   Here is the story I am meaning to tell:

   For $C := T$ an algebraic theory, we may think of ordinary Isbell duality

   $$(\mathcal{O} \dashv Spec) : T Alg^{op} = [C,Set]_\times^{op} \stackrel{\leftarrow}{\to} [C^{op}, Set]$$

   equivalently as given by taking the left adjoint $\mathcal{O}$ to be homs into the line object $\mathbb{A} := Spec(F_T(1))$.

   But in applications, we actually want $C$ to sit in between the free $T$-algebras and general $T$-algebras

   $$T \subset C \subset T Alg^{op}$$

   such that we still have the adjunction

   $$(\mathcal{O} \dashv Spec) : T Alg^{op} = [T,Set]_\times^{op} \stackrel{\leftarrow}{\to} [C^{op}, Set]$$

   (notice now $T \neq C$!).

   For this case the original simple definition of the Isbell duality does not quite make sense anymore. But that in terms of homs into the line object does, and still gives an adjunction.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2010
   • (edited Sep 1st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexwrote out a second proof -- <a href="http://ncatlab.org/nlab/show/Isbell+duality#ProofB">Proof B</a> -- of the Isbell adjunction, one that highlights maybe a different aspect, which is useful in generalizations and variations of the situation. I <a href="http://ncatlab.org/nlab/show/Isbell+duality#FuncCompDerStacks">point out</a> that this proof is effectively what Ben-Zvi and Nadler write out in the context of $\infty$-preshaves over duals of dg-algebras.

   wrote out a second proof – Proof B – of the Isbell adjunction, one that highlights maybe a different aspect, which is useful in generalizations and variations of the situation.

   I point out that this proof is effectively what Ben-Zvi and Nadler write out in the context of $\infty$-preshaves over duals of dg-algebras.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI finally noticed that I should rename the entry to [[Isbell conjugation]], which i now did. As soon as we get the cache cleared (I can't do it right now) I'll split off [[Isbell duality]] again with a proper discussion of that,

   I finally noticed that I should rename the entry to Isbell conjugation, which i now did.

   As soon as we get the cache cleared (I can’t do it right now) I’ll split off Isbell duality again with a proper discussion of that,

7. • CommentRowNumber7.
   • CommentAuthorAndrew Stacey
   • CommentTimeSep 1st 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexYou should be able to simply create a new "Isbell duality" even without clearing the cache. Simply go to <http://ncatlab.org/nlab/new/Isbell+duality>. By creating the new page, it'll clear out the old one properly. (That's not to say that it shouldn't have cleared it out anyway automatically.)

   You should be able to simply create a new “Isbell duality” even without clearing the cache. Simply go to http://ncatlab.org/nlab/new/Isbell+duality. By creating the new page, it’ll clear out the old one properly.

   (That’s not to say that it shouldn’t have cleared it out anyway automatically.)

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2010
   • (edited Sep 1st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> You should be able to simply create a new "Isbell duality" But I am not, unfortunately: when I go to [[Isbell duality]] I see the old page displayed there, and when I hit edit on that page I am being sent to the HomePage. That's the cache bug at work.

   You should be able to simply create a new “Isbell duality”

   But I am not, unfortunately: when I go to Isbell duality I see the old page displayed there, and when I hit edit on that page I am being sent to the HomePage.

   That’s the cache bug at work.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 1st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, sorry, I didn't read what you said properly. Got it now

   Oh, sorry, I didn’t read what you said properly. Got it now

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, wrote something at [[Isbell duality]]. But have to rush off now.

    Okay, wrote something at Isbell duality. But have to rush off now.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded a link to [[Isbell conjugation]] in the Application-section at [[Yoneda lemma]].

    added a link to Isbell conjugation in the Application-section at Yoneda lemma.

12. • CommentRowNumber12.
    • CommentAuthormaxsnew
    • CommentTimeJan 31st 2017
    • PermaLink
    Author: maxsnew
    Format: MarkdownItexAre Dedekind cuts the same thing as Isbell dual pairs when enriching over truth values? ($x \in L$ iff $\forall y \in U. x \le U$ means that L is the dual of U, right?) If so that's a really good intuition for Isbell duality. And assuming that's true, what is known about the analogue of the Dedekind completion? Is it complete, cocomplete, and has an embedding preserving limits and colimits of the original category? Sounds almost too good to be true!

    Are Dedekind cuts the same thing as Isbell dual pairs when enriching over truth values?

    ($x \in L$ iff $\forall y \in U. x \le U$ means that L is the dual of U, right?)

    If so that’s a really good intuition for Isbell duality. And assuming that’s true, what is known about the analogue of the Dedekind completion? Is it complete, cocomplete, and has an embedding preserving limits and colimits of the original category? Sounds almost too good to be true!

13. • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2017
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexYes, and compare [[MacNeille completion]]. However, don't expect (co)completeness, for I believe the Isbell completion of a small category is again small; for it to be (co)complete, it would have to be a preorder, and since the original category embeds fully faithfully into the Isbell completion, it too would have to be a preorder.

    Yes, and compare MacNeille completion. However, don’t expect (co)completeness, for I believe the Isbell completion of a small category is again small; for it to be (co)complete, it would have to be a preorder, and since the original category embeds fully faithfully into the Isbell completion, it too would have to be a preorder.

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 31st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexDid we ever bring over Simon Willerton's [material](https://golem.ph.utexas.edu/category/2013/09/classical_dualities_and_formal.html) on Galois connections and closures to nLab? The story continued to a fourth [post](https://golem.ph.utexas.edu/category/2014/02/galois_correspondences_and_enr.html). Dedekind cuts are certainly one case in the truth value-enriched case.

    Did we ever bring over Simon Willerton’s material on Galois connections and closures to nLab? The story continued to a fourth post. Dedekind cuts are certainly one case in the truth value-enriched case.

15. • CommentRowNumber15.
    • CommentAuthormaxsnew
    • CommentTimeJan 31st 2017
    • PermaLink
    Author: maxsnew
    Format: MarkdownItexOk, what about something more basic like finite (co)-completeness? Specifically how would you construct products? If you have $(L,U), (L', U')$ that satisfy Isbell duality, you have 2 choices for a product, $(L \times L', I(L \times L')$ or $(I(U + U'), U + U')$. I'm trying to work it out myself but the number of quantifiers makes it a bit hairy.

    Ok, what about something more basic like finite (co)-completeness? Specifically how would you construct products? If you have $(L,U), (L', U')$ that satisfy Isbell duality, you have 2 choices for a product, $(L \times L', I(L \times L')$ or $(I(U + U'), U + U')$.

    I’m trying to work it out myself but the number of quantifiers makes it a bit hairy.