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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 18th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI edited [[adjoint functor theorem]] a bit: gave it an Idea-section and a References-section and, believe it or not, a toc. Then I opened an Examples-section and filled in what I think is an instructive simple example: the right adjoint for a colimit preserving functor on a category of presheaves.

   I edited adjoint functor theorem a bit: gave it an Idea-section and a References-section and, believe it or not, a toc.

   Then I opened an Examples-section and filled in what I think is an instructive simple example: the right adjoint for a colimit preserving functor on a category of presheaves.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 22nd 2010
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   Author: Urs
   Format: MarkdownItexinserted sketch of a proof of the [[adjoint functor theorem]] for the solution-set-condition-assumption (following Jaap's notes)

   inserted sketch of a proof of the adjoint functor theorem for the solution-set-condition-assumption (following Jaap’s notes)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 22nd 2010
   • (edited Oct 22nd 2010)
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   Author: Todd_Trimble
   Format: MarkdownItexI am tempted to rewrite the proof a little. I think it's a little easier than it's often made to seem. (1) Let $Y$ be a category. Call a small family of objects $F$ *weakly initial* if for every object $y$ of $Y$ there exists $x \in F$ and a morphism $f: x \to y$. (2) Suppose $Y$ has small products. If $F$ is a weakly initial family, then $\prod_{x \in F} x$ is a weakly initial object. (3) Suppose $Y$ is locally small and small complete. If $x$ is a weakly initial object, then the domain $e$ of the joint equalizer $i: e \to x$ of all arrows $x \to x$ is an initial object. Proof: clearly $e$ is weakly initial. Suppose given an object $y$ and arrows $f, g: e \to y$; we must show $f = g$. Let $j: d \to e$ be the equalizer of $f$ and $g$. There exists an arrow $k: x \to d$. The arrow $i: e \to x$ equalizes $1_x$ and $i j k: x \to x$, so $i j k i = i$. Since $i$ is monic, $j (k i) = 1_e$. Since $j (k i) j = j$ and $j$ is monic, $(k i) j = 1_d$. Hence $j$ is an iso, and therefore $f = g$. (4) If $D$ is locally small and small-complete and $S: D \to C$ preserves limits, then $c \downarrow S$ is locally small and small complete for every object $c$ of $C$. (5) If in addition each $c \downarrow S$ has a weakly initial family (solution set condition), then by (2) and (3) each $c \downarrow S$ has an initial object. This restates the condition that $S$ has a left adjoint.

   I am tempted to rewrite the proof a little. I think it’s a little easier than it’s often made to seem.

   (1) Let $Y$ be a category. Call a small family of objects $F$ weakly initial if for every object $y$ of $Y$ there exists $x \in F$ and a morphism $f: x \to y$.

   (2) Suppose $Y$ has small products. If $F$ is a weakly initial family, then $\prod_{x \in F} x$ is a weakly initial object.

   (3) Suppose $Y$ is locally small and small complete. If $x$ is a weakly initial object, then the domain $e$ of the joint equalizer $i: e \to x$ of all arrows $x \to x$ is an initial object. Proof: clearly $e$ is weakly initial. Suppose given an object $y$ and arrows $f, g: e \to y$; we must show $f = g$. Let $j: d \to e$ be the equalizer of $f$ and $g$. There exists an arrow $k: x \to d$. The arrow $i: e \to x$ equalizes $1_x$ and $i j k: x \to x$, so $i j k i = i$. Since $i$ is monic, $j (k i) = 1_e$. Since $j (k i) j = j$ and $j$ is monic, $(k i) j = 1_d$. Hence $j$ is an iso, and therefore $f = g$.

   (4) If $D$ is locally small and small-complete and $S: D \to C$ preserves limits, then $c \downarrow S$ is locally small and small complete for every object $c$ of $C$.

   (5) If in addition each $c \downarrow S$ has a weakly initial family (solution set condition), then by (2) and (3) each $c \downarrow S$ has an initial object. This restates the condition that $S$ has a left adjoint.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 22nd 2010
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   Author: Urs
   Format: MarkdownItex> I am tempted to rewrite the proof a little. Please do!

   I am tempted to rewrite the proof a little.

   Please do!

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 23rd 2010
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   Author: Todd_Trimble
   Format: MarkdownItexOkay, I did, and corrected a few errors while I was at it.

   Okay, I did, and corrected a few errors while I was at it.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 23rd 2010
   • (edited Oct 23rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Okay, I did, Thanks, Todd. That's very nice. > and corrected a few errors while I was at it. If that includes any serious technical errors le us know which ones, so that we learn something. From looking at "See changes" I see you edited the statement about the adjoint functor theorem for preorders. Is that what you are referring to?

   Okay, I did,

   Thanks, Todd. That’s very nice.

   and corrected a few errors while I was at it.

   If that includes any serious technical errors le us know which ones, so that we learn something. From looking at “See changes” I see you edited the statement about the adjoint functor theorem for preorders. Is that what you are referring to?

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 23rd 2010
   • (edited Oct 23rd 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNothing too serious. The word 'join' appeared where it should have been 'meet', and the precise statement of Freyd's theorem about small complete preorders was slightly off (referring to the size of the family of objects instead of the family of morphisms). By the way, I stuck in a little proof of the theorem, but there are some interesting subtleties involved. For example, there is a fairly extensive literature where you *can* have a small complete non-preorder inside a topos (the internal category of modest sets in the effective topos is an example). More material for the nLab one day.

   Nothing too serious. The word ’join’ appeared where it should have been ’meet’, and the precise statement of Freyd’s theorem about small complete preorders was slightly off (referring to the size of the family of objects instead of the family of morphisms).

   By the way, I stuck in a little proof of the theorem, but there are some interesting subtleties involved. For example, there is a fairly extensive literature where you can have a small complete non-preorder inside a topos (the internal category of modest sets in the effective topos is an example). More material for the nLab one day.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 23rd 2010
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   Author: Mike Shulman
   Format: MarkdownItexPeople often introduce the small complete category in the effective topos by saying "Freyd's argument carries over to any Grothendieck topos," but I've never understood how. The logic of a Grothendieck topos can be just as intuitionistic as that of the effective topos, no?

   People often introduce the small complete category in the effective topos by saying “Freyd’s argument carries over to any Grothendieck topos,” but I’ve never understood how. The logic of a Grothendieck topos can be just as intuitionistic as that of the effective topos, no?

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 23rd 2010
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   Author: Todd_Trimble
   Format: MarkdownItexMight it have something to do with restrictions on a morphism in a Grothendieck topos to be orthogonal to $2 = 1 + 1$? I seem to have some vague memories that this orthogonal category played an important role in the effective topos. It's to my dismay that I don't know this stuff.

   Might it have something to do with restrictions on a morphism in a Grothendieck topos to be orthogonal to $2 = 1 + 1$? I seem to have some vague memories that this orthogonal category played an important role in the effective topos. It’s to my dismay that I don’t know this stuff.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 24th 2010
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    Author: Mike Shulman
    Format: MarkdownItexAt least one of the small complete categories in the effective topos does, I believe, consist of the objects orthogonal to something, possibly 2 (or maybe $\Omega$, or $\nabla 2$?). I have no trouble believing that the particular construction of a small complete category that works in the effective topos couldn't possibly work in any Grothendieck topos, but how do we know that there could never be *any* small complete category in a Grothendieck topos?

    At least one of the small complete categories in the effective topos does, I believe, consist of the objects orthogonal to something, possibly 2 (or maybe $\Omega$, or $\nabla 2$?). I have no trouble believing that the particular construction of a small complete category that works in the effective topos couldn’t possibly work in any Grothendieck topos, but how do we know that there could never be any small complete category in a Grothendieck topos?

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeOct 24th 2010
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    Author: Mike Shulman
    Format: MarkdownItexI asked on [MO](http://mathoverflow.net/questions/43433/small-complete-categories-in-a-grothendieck-topos).

    I asked on MO.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2011
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    Author: Urs
    Format: MarkdownItexI had added to [[adjoint functor theorem]] its statement for locally presentable categories (in the main section) as well as a note on the situation for Grothendieck toposes (in the Examples-section)

    I had added to adjoint functor theorem its statement for locally presentable categories (in the main section) as well as a note on the situation for Grothendieck toposes (in the Examples-section)