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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.
inserted sketch of a proof of the adjoint functor theorem for the solution-set-condition-assumption (following Jaap’s notes)
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.
I am tempted to rewrite the proof a little.
Please do!
Okay, I did, and corrected a few errors while I was at it.
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?
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.
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?
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.
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?
I asked on MO.
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)
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