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started essential geometric morphism
added a simple proposition and proof to essential geometric morphism in a new section "Properties".
Also replied to Mike's query box comment.
Re-replied and created locally connected geometric morphism.
Thanks, Mike.
I did the following at essential geometric morphism:
turned the query box discussion into genuine text in the Idea-section;
added a section Definition-Refinements with the locally connected case and two other refinements of "essential";
added references for these cases.
A question:
is there an established special name for the situation where we have an essential geometric morphism
$(f_! \dashv f^* \dashv f_*) : E \stackrel{\to}{\stackrel{\leftarrow}{\to}} T$with the special property that $f^*$ is full and faithful ?
I know (from Johnstone and Lawvere’s article referenced at essential geometric morphism) that in the case that we have yet one more right adjoint $f^!$
$(f_! \dashv f^* \dashv f_* \dashv f^!)$and if that $f^!$ is full and faithful, then one says that $f$ is local ,
But I am wondering about $f^*$ being full and faithful. (Because, unless I am mixed up, this is the case for the terminal geometric morphisms out of the objects in the class of locally contractible (oo,1)-toposes that I know how to build).
Probably this is a stupid question with an evident answer, but right now it escapes me.
Yes! It’s called a connected topos.
Yes! It’s called a connected topos.
Ah, nice. Okay, so this proves publically that I stil haven’t read the Elephant the way I should, but at least it’s the perfect answer for my purpose. :-)
Mike, I now want to make the following definition, but please give me a sanity check:
of course I want to say now that a contractible $(\infty,1)$ -topos is a locally contractible (infinity,1)-topos $\mathbf{H}$ such that $LConst : \infty Grpd \to \mathbf{H}$ is a full and faithful (infinity,1)-functor.
Here is one consistency check that this makes sense: in the case that $LConst$ is full and faithful we have that
This implies that the shape of $\mathbf{H}$ in the sense of shape of an (infinity,1)-topos is that of the point. Which clearly matches the idea of $\mathbf{H}$ being “contractible” .
I am inclined to make that into an nLab page connected (infinity,1)-topos.
at locally n-connected (infinity,1)-topos I
expanded the part on locally connected versus connected
added an analogous part locally contractible versus contractible
Would you have any objection to having a separate page locally connected topos about the 1-dimensional version?
I’m not sure I like calling that notion “contractible” – wouldn’t something like “$\infty$-connected” be closer to the mark? “Contractible” to me means “equivalent to a point,” which such a topos evidently is not always (or so I gather, otherwise you wouldn’t be interested in it). Is this at all similar to the notion of $\infty$-connected object in an $(\infty,1)$-topos (which also need not be trivial)?
Would you have any objection to having a separate page locally connected topos about the 1-dimensional version?
Right, I was thinking about that, too. We should do that. But I won’t do anything else tonight, need to catch some sleep.
I’m not sure I like calling that notion “contractible” – wouldn’t something like “∞-connected” be closer to the mark?
Maybe you are right.
“Contractible” to me means “equivalent to a point,” which such a topos evidently is not always (or so I gather, otherwise you wouldn’t be interested in it).
Right, so this touches on the crucial interesting point here: these “$\infty$-connected” $\infty$-toposes are in a way fat points with structure. For instance with smooth structure.
Consider the underlying site: the objects of CartSp are all contractible spaces. It is helpful to think of them (up to diffeomorphism) as the open $n$-balls. An $n$-ball is just a fat point, topologically. But crucially here the fact that maps are smooth maps remembers the smooth structure. So an $\infty$-groupoid modeled on the smooth $n$-balls, i.e. an oo-stack on $CartSp$ is much like a bare Kan complex, the only difference being that for around every point in the k-cells of the Kan complex, I have the information of what the ways are to extend that point smoothly to an open-ball-shaped family of points in its neightbourhood.
So $Sh_{(\infty,1)}(CartSp)$ differs from $Sh_{(\infty,1)}(pt)$ just a little bit, but by a crucial bit. This is I think what the abstract formalism is seeing: that $Sh_{(\infty,1)}(CartSp)$ is “locally $\infty$-connected and $\infty$-connected” is a reflection of the fact that all the objects of $CartSp$ are.
Is this at all similar to the notion of ∞-connected object in an (∞,1)-topos (which also need not be trivial)?
I was wondering about that today. One should look at the $(\infty,1)$-category of $(\infty,1)$-toposes and see if these conditions on the terminal morphism induce corresponding properties as connected objects. It’s probably an easy argument along the lines: if we have an essential geometric morphism to the terminal oo-topos with the inverse image full and faithful, then this means in the $(\infty,1)$-category of $(\infty,1)$-toposes we have a exhibited the terminal object as a retract of our $(\infty,1)$-topos.
I’ll think about it. But not tonight. I need to call it quits now.
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