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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

   with the special property that 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

   and if that is full and faithful, then one says that is local ,

   But I am wondering about 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 -topos is a locally contractible (infinity,1)-topos such that is a full and faithful (infinity,1)-functor.

   Here is one consistency check that this makes sense: in the case that is full and faithful we have that

   • the unit is an equivalence .

   This implies that the shape of in the sense of shape of an (infinity,1)-topos is that of the point. Which clearly matches the idea of 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 “-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 -connected object in an -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 “-connected” -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 -balls. An -ball is just a fat point, topologically. But crucially here the fact that maps are smooth maps remembers the smooth structure. So an -groupoid modeled on the smooth -balls, i.e. an oo-stack on 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 differs from just a little bit, but by a crucial bit. This is I think what the abstract formalism is seeing: that is “locally -connected and -connected” is a reflection of the fact that all the objects of 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 -category of -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 -category of -toposes we have a exhibited the terminal object as a retract of our -topos.

    I’ll think about it. But not tonight. I need to call it quits now.