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I typed at local topos in the section Local over-toposes statement and poof that sufficient for a slice topos $\mathcal{E}/X$ to be local is that $X$ is tiny .
What are necessary conditions? Is this already necessary?
started adding a little bit of content at local topos in the section Elementary axiomatization
started writing out details of the proof of Awodey-Birkedal’s Lemma 2.3 here but am again being interrupted now.
(Also I realize that I need to think about the next step…)
I changed the $#$s to $\sharp$s.
I changed the #s to ♯s.
All right, thanks.
I realize that I am suffering from puzzlement over axiom 2 for local toposes: every object is the subquotient of a discrete one.
In the example of sheaves on CartSp. What is the subquotient-of-a-discrete-object realization of a manifold?
That example isn’t localic, is it? So you need the version of the axiom that involves a bound as well.
Ah right, thanks.
Following a request of a reader who found the entry local topos intransparent I have
highlighted more explicitly the simple definition of a local topos over Set itself, here
(previously that was a bit hidden in the full generality of the definition of local geometric morphism);
added a bunch of basic details in the section Easy examples.
Urs, I’m looking over what you just wrote about a (Grothendieck) topos $C$ being a local topos over $Set$. Isn’t it enough just to say $\Gamma = C(1, -)$ has a right adjoint, i.e., that its right adjoint $codisc$ being fully faithful will come for free? Thus, $C$ is a local topos iff the terminal object $1$ is connected and projective.
My reasoning is that $codisc$ is fully faithful iff $disc$ is fully faithful, by an argument at adjoint triple. Now $disc: Set \to C$ takes a set $S$ to the $S$-indexed coproduct $S \cdot 1$ of copies of $1$ in $C$. The functor $disc$ is fully faithful iff the unit $S \to \hom(1, S \cdot 1)$ is an isomorphism. But since $1$ is connected, $\hom(1, -)$ preserves this $S$-indexed coproduct. Thus the unit, being a composite
$S \cong S \cdot \hom(1, 1) \to \hom(1, S \cdot 1)$of two isomorphisms, is an isomorphism.
That also follows from the first of the four equivalent additional conditions on the right adjoint in the relative case, since every functor is $Set$-indexed.
Thanks, Mike.
Also, Urs asked (back in January 2011, above) if $X$ being tiny in a Grothendieck topos $C$ follows from $C/X$ being local. It does. For, $C/X$ being local means the terminal $1_X$ is tiny in $C/X$. Then, for any colimit $colim_i Y_i$ in $C$, the functor $\hom(X, -)$ preserves this colimit iff $1_X$ preserves $colim_i X \times Y_i \cong X \times colim_i Y_i$ in $C/X$, which it does.
Hey Todd,
so I am not sure what you want me to say. I thought I did spell out that $\Gamma = Hom(*,-)$ and that this has a right adjoint and that this is full and faithful because Disc is.
Maybe you are saying that my argument looks lengthy? I wrote it intentionally in plenty of and supposedly elementary detail , because I was asked about the details by somebody who found the previous version of the entry too abstract.
This is in the Examples-section. So we can indulge a bit in spelling out details there, I think.
But if you say you’d rather spell out details differently, then please just add that, too.
Urs, I was focusing on the precise point you directed us to in #9, where you said “here”. I did not look through the whole article.
Definition. A sheaf topos $\mathcal{T}$ is a local topos if the global section geometric morphism $\mathcal{T} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Set$ has a further right adjoint functor $(LConst \vdash \Gamma \vdash CoDisc)$
$CoDisc \colon Set \hookrightarrow \mathcal{T}$which is furthermore a full and faithful functor.
Since you said that you wrote that definition as a favor to someone who found the article intransparent, I thought we could go one step further for that person. Maybe he or she didn’t read through the whole article either?
Edit: I’ve now looked through the article. Maybe I missed it, but the only place I saw mention of
and that this is full and faithful because Disc is.
was in the course of example 1. My point is that it holds more generally (not just in the presheaf example).
I should also say that my commenting on this was not criticism – it was purely in the spirit of trying to say something that might be helpful for someone.
Looking one more time, I see that you did say – in the sentence before the subsection on Local Topos that you pointed us to – that that condition was automatic. Sorry I didn’t see that earlier. I will now edit, again.
Todd, thanks and no problem. I didn’t regard it as a criticism. And I do welcome criticism. I just meant to say that it wasn’t clear to me what kind of action from my side your comment was calling for, if any.
Please, if you feel like it, edit the article. All I did was to drop there some lines which I expected somebody I was talking to would find useful. I am not making any claims that the entry needs to have just the structure that it has now.
what kind of action from my side your comment was calling for, if any.
No action was being called for. I was just running something past you. Anyway, I am done editing for the time being.
Added to local topos an explicit description of the extra right adjoint in the case of a sheaf topos over a topological space containing a focal point.
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