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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 12th 2010
   • PermaLink
   Author: Urs
   Format: Markdownstarted [[essential geometric morphism]]

   started essential geometric morphism

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2010
   • PermaLink
   Author: Urs
   Format: Markdownadded a simple proposition and proof to [[essential geometric morphism]] in a new section "Properties". Also replied to Mike's query box comment.

   added a simple proposition and proof to essential geometric morphism in a new section "Properties".

   Also replied to Mike's query box comment.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMar 10th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownRe-replied and created [[locally connected geometric morphism]].

   Re-replied and created locally connected geometric morphism.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2010
   • (edited Mar 10th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownThanks, 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2010
   • (edited May 6th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexA 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes! It's called a [[connected topos]].

   Yes! It’s called a connected topos.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2010
   • (edited May 6th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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 * the unit $Id_{\infty Grpd} \to \Gamma LConst $ is an equivalence . 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]].

   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

   • the unit $Id_{\infty Grpd} \to \Gamma LConst$ is an equivalence .

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexat [[locally n-connected (infinity,1)-topos]] I * expanded the part on <a href="http://ncatlab.org/nlab/show/locally+n-connected+(infinity%2C1)-topos#Connected">locally connected versus connected</a> * added an analogous part <a href="http://ncatlab.org/nlab/show/locally+n-connected+(infinity%2C1)-topos#Contractible">locally contractible versus contractible</a>

   at locally n-connected (infinity,1)-topos I

   • expanded the part on locally connected versus connected

   • added an analogous part locally contractible versus contractible

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWould 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?

   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)?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2010
    • (edited May 7th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> 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.

    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.