Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJan 16th 2011
   • (edited Jan 16th 2011)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexMike reminds me that the traditional meaning of "$n$-connected" in topology (see [Wikipedia](http://secure.wikimedia.org/wikipedia/en/wiki/n-connected), for example) is such that a $1$-connected space is *simply* connected. I had earlier suggested that we might subtly change "$n$-connected" to "$n$-simply connected" to improve this; see [[k-simply connected n-category]], for example. However, this doesn't match the usage at [[locally n-connected (n,1)-topos]]. In particular, the list there jumps from a "locally connected space" to a "locally $2$-connected space" without catching the intervening "locally simply connected space". So I have renumbered things and moved the page to [[locally n-connected (n+1,1)-topos]]. Now here are some questions: * Do we go further and rename this [[locally n-simply connected (n+1,1)-topos]]? (There's a Forum bug that breaks this link; try [this](http://ncatlab.org/nlab/show/locally+n-simply+connected+%28n%2B1%2C1%29-topos).) * From a remark there, Lurie seems to have adopted "$n$-connective" to mean $(n-1)$-(simply) connected. Should we use this? (Should we even change "connected space" to "connective space"? I don't like that!) * Do the numbers have to match at all? If I can define a locally $k$-(simply) connected $n$-category in general (in a way that seems relevant to something else, [[k-tuply monoidal n-categories]], via the [[delooping hypothesis]] and so shouldn't be altogether wrong), can't we define a locally $k$-(simply) connected $(n,1)$-topos in general? (Note that the correspondence here is that an $n$-category is *an object of* an $(n+1)$-topos just as much as that an $n$-topos is a kind of $n$-category; see [[n-connected object]].) But I don't see how; I barely understand what Urs has written about these things.

   Mike reminds me that the traditional meaning of “$n$-connected” in topology (see Wikipedia, for example) is such that a $1$-connected space is simply connected. I had earlier suggested that we might subtly change “$n$-connected” to “$n$-simply connected” to improve this; see k-simply connected n-category, for example.

   However, this doesn’t match the usage at locally n-connected (n,1)-topos. In particular, the list there jumps from a “locally connected space” to a “locally $2$-connected space” without catching the intervening “locally simply connected space”. So I have renumbered things and moved the page to locally n-connected (n+1,1)-topos.

   Now here are some questions:

   • Do we go further and rename this locally n-simply connected (n+1,1)-topos? (There’s a Forum bug that breaks this link; try this.)
   • From a remark there, Lurie seems to have adopted “$n$-connective” to mean $(n-1)$-(simply) connected. Should we use this? (Should we even change “connected space” to “connective space”? I don’t like that!)
   • Do the numbers have to match at all? If I can define a locally $k$-(simply) connected $n$-category in general (in a way that seems relevant to something else, k-tuply monoidal n-categories, via the delooping hypothesis and so shouldn’t be altogether wrong), can’t we define a locally $k$-(simply) connected $(n,1)$-topos in general? (Note that the correspondence here is that an $n$-category is an object of an $(n+1)$-topos just as much as that an $n$-topos is a kind of $n$-category; see n-connected object.) But I don’t see how; I barely understand what Urs has written about these things.
2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeJan 16th 2011
   • PermaLink
   Author: Tim_Porter
   Format: HtmlJust a side comment on a related entry: I was looking [[Whitehead tower]] and it seemed to me that when killing homotopy groups the suffix was wrong. (These are notoriously difficult to get consistent!) I changed it as I think it needs, but please galnce and check that I have not slipped up.;-(

   Just a side comment on a related entry: I was looking Whitehead tower and it seemed to me that when killing homotopy groups the suffix was wrong. (These are notoriously difficult to get consistent!) I changed it as I think it needs, but please galnce and check that I have not slipped up.;-(
3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeJan 16th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI wrote: >Note that the correspondence here is that an $n$-category is *an object of* an $(n+1)$-topos just as much as that an $n$-topos is a kind of $n$-category Actually, the latter correspondence doesn't apply at all. We're looking at arbitrary functors of $n$-categories but only geometric morphisms of $n$-toposes (which also go the other way). For example, an $n$-category is $(-1)$-connected (or inhabited) iff it has at least one object, or iff (classically) it's not the empty $n$-category (which is initial under arbitrary functors), while an $n$-topos always has an object; however, an $n$-topos is $(-1)$-connected (or positive) iff (classically) it has at least two objects, or iff (classically) it's not the one-object $n$-topos (which is initial under geometric morphisms).

   I wrote:

   Note that the correspondence here is that an $n$-category is an object of an $(n+1)$-topos just as much as that an $n$-topos is a kind of $n$-category

   Actually, the latter correspondence doesn’t apply at all. We’re looking at arbitrary functors of $n$-categories but only geometric morphisms of $n$-toposes (which also go the other way). For example, an $n$-category is $(-1)$-connected (or inhabited) iff it has at least one object, or iff (classically) it’s not the empty $n$-category (which is initial under arbitrary functors), while an $n$-topos always has an object; however, an $n$-topos is $(-1)$-connected (or positive) iff (classically) it has at least two objects, or iff (classically) it’s not the one-object $n$-topos (which is initial under geometric morphisms).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexToby, not sure what I can say, there is a slight mess of terminology and its hard to impossible to enforce a standard on the $n$Lab. I think the best we can do is have all variants and pitfalls discussed nicely on the relevant pages. If I understand correctly, you are suggesting to say "$n$-simply connected" instead of "$n$-connected" to make the counting match the default for $n=1$ instead of $n=0$. Right? While I see the logic, that looks cumbersome to me. i'd rather say "1-connected" for "simply connected". The use of "$n$-connective" for "vanishing homotopy groups below degree $n$" has some advantages. One is that it is less standard and hence lends itself better to a systematic definition for all $n$, while the standard terminology with "connected" and"simply-connected" will always suffer from a counting ambiguity. Not sure what we should do.

   Toby,

   not sure what I can say, there is a slight mess of terminology and its hard to impossible to enforce a standard on the $n$Lab. I think the best we can do is have all variants and pitfalls discussed nicely on the relevant pages.

   If I understand correctly, you are suggesting to say “$n$-simply connected” instead of “$n$-connected” to make the counting match the default for $n=1$ instead of $n=0$. Right?

   While I see the logic, that looks cumbersome to me. i’d rather say “1-connected” for “simply connected”.

   The use of “$n$-connective” for “vanishing homotopy groups below degree $n$” has some advantages. One is that it is less standard and hence lends itself better to a systematic definition for all $n$, while the standard terminology with “connected” and”simply-connected” will always suffer from a counting ambiguity.

   Not sure what we should do.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJan 18th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>If I understand correctly, you are suggesting to say "$n$-simply connected" instead of "$n$-connected" to make the counting match the default for $n=1$ instead of $n=0$. Right? Right. But then one can decide to abbreviate "$n$-simply connected" as "$n$-connected" afterwards. I would have found things less confusing if I saw the longer term first and was taught the latter as an abbreviation. On the other hand, the numbering of "$n$-connective" is more sensible *anyway*. Only the actual word is funny.

   If I understand correctly, you are suggesting to say “$n$-simply connected” instead of “$n$-connected” to make the counting match the default for $n=1$ instead of $n=0$. Right?

   Right. But then one can decide to abbreviate “$n$-simply connected” as “$n$-connected” afterwards. I would have found things less confusing if I saw the longer term first and was taught the latter as an abbreviation.

   On the other hand, the numbering of “$n$-connective” is more sensible anyway. Only the actual word is funny.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 18th 2011
   • (edited Jan 18th 2011)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexUnfortunately, "$n$-connective" also violates the rule of "a foo is a 1-foo": classically a "connective spectrum" is one with vanishing *negative* homotopy groups, i.e. what (it sounds like) Lurie would call "0-connective". But it does seem less likely to cause confusion. > Do the numbers have to match at all? While you're right that being locally k-connective as an (n,1)-topos has nothing to do with being k-whatever-connected as an (n,1)-category, there should certainly be a way to define "locally k-connective (n,1)-topos", or even an (n,1)-geometric morphism, which makes sense for arbitrary k and n. The definition is just simplest when k = n. If k > n, we can define an (n,1)-topos to be k-connective if the (k,1)-topos of k-sheaves on it is so. And if k < n, we can define an (n,1)-topos to be k-connective if its k-localic reflection, regarded as a (k,1)-topos, is k-connective. The latter condition ought to be equivalent to various other characterizations, such as are known in the classical case k=0, n=1 of "open 1-topos." For instance, a 1-geometric morphism is open (= locally 0-connective) if its inverse image part is a Heyting functor, i.e. it preserves [[dependent products]] of 0-truncated objects. So one might expect an n-geometric morphism to be locally k-connected, for k<n, if its inverse image part preserves dependent products of k-truncated objects.

   Unfortunately, “$n$-connective” also violates the rule of “a foo is a 1-foo”: classically a “connective spectrum” is one with vanishing negative homotopy groups, i.e. what (it sounds like) Lurie would call “0-connective”. But it does seem less likely to cause confusion.

   Do the numbers have to match at all?

   While you’re right that being locally k-connective as an (n,1)-topos has nothing to do with being k-whatever-connected as an (n,1)-category, there should certainly be a way to define “locally k-connective (n,1)-topos”, or even an (n,1)-geometric morphism, which makes sense for arbitrary k and n. The definition is just simplest when k = n. If k > n, we can define an (n,1)-topos to be k-connective if the (k,1)-topos of k-sheaves on it is so. And if k < n, we can define an (n,1)-topos to be k-connective if its k-localic reflection, regarded as a (k,1)-topos, is k-connective.

   The latter condition ought to be equivalent to various other characterizations, such as are known in the classical case k=0, n=1 of “open 1-topos.” For instance, a 1-geometric morphism is open (= locally 0-connective) if its inverse image part is a Heyting functor, i.e. it preserves dependent products of 0-truncated objects. So one might expect an n-geometric morphism to be locally k-connected, for k<n, if its inverse image part preserves dependent products of k-truncated objects.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeJan 18th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>Unfortunately, "$n$-connective" also violates the rule of "a foo is a 1-foo" Drats, I hoped that Lurie had just invented that as a synonym for "connected"; I didn't think of connective spectra. >So one might expect an n-geometric morphism to be locally k-truncated, for k<n, if its inverse image part preserves dependent products of k-truncated objects. You mean that the $n$-geometric morphism is expected to be locally $k$-*connected* here. (Also, this and the analogous "if" are "iff", right?)

   Unfortunately, “$n$-connective” also violates the rule of “a foo is a 1-foo”

   Drats, I hoped that Lurie had just invented that as a synonym for “connected”; I didn’t think of connective spectra.

   So one might expect an n-geometric morphism to be locally k-truncated, for k<n, if its inverse image part preserves dependent products of k-truncated objects.

   You mean that the $n$-geometric morphism is expected to be locally $k$-connected here. (Also, this and the analogous “if” are “iff”, right?)

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 18th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes, "truncated" is fixed now, thanks. And yes, I was using "if" in the definitional (i.e. "iff") sense.

   Yes, “truncated” is fixed now, thanks. And yes, I was using “if” in the definitional (i.e. “iff”) sense.