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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeFeb 14th 2012
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexI created [[Pi-closed morphism]]. This material is in [differential cohomology in a cohesive (∞,1)-topos](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos), too.

   I created Pi-closed morphism. This material is in differential cohomology in a cohesive (∞,1)-topos, too.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2012
   • (edited Feb 14th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. Good idea to make your observation about admissibility structures explicit here. I have briefly added some further hyperlinks. Where you point to the catlab-discussion, one should maybe add a remark that this is about 1-categorical factorization, while here its $\infty$-categorical. Of course the central statements carry over. Maybe, if you have the time, add a pointer _to_ this entry from some relevant general entry on cohesive $(\infty,1)$-toposes.

   Thanks. Good idea to make your observation about admissibility structures explicit here.

   I have briefly added some further hyperlinks.

   Where you point to the catlab-discussion, one should maybe add a remark that this is about 1-categorical factorization, while here its $\infty$-categorical. Of course the central statements carry over.

   Maybe, if you have the time, add a pointer to this entry from some relevant general entry on cohesive $(\infty,1)$-toposes.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexYou know, you have a very good point here. This admissibility structure should be regarded in conjunction with the statements in [section 2.3.11 Universal coverings and geometric Whitehead towers](https://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=165). In particular prop. 2.3.118, which asserts that every object in a cohesive oo-topos has a cover (now in your sense!) by null-homotopic patches. There is a good story to be told here. Let's talk about it on Thursday!

   You know, you have a very good point here. This admissibility structure should be regarded in conjunction with the statements in section 2.3.11 Universal coverings and geometric Whitehead towers. In particular prop. 2.3.118, which asserts that every object in a cohesive oo-topos has a cover (now in your sense!) by null-homotopic patches.

   There is a good story to be told here. Let’s talk about it on Thursday!

4. • CommentRowNumber4.
   • CommentAuthorStephan A Spahn
   • CommentTimeFeb 14th 2012
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex>Where you point to the catlab-discussion, one should maybe add a remark that this is about 1-categorical factorization, while here its -categorical. Of course the central statements carry over. > Maybe, if you have the time, add a pointer to this entry from some relevant general entry on cohesive -toposes. I corrected this and linked the article from [cohesive (∞,1)-topos - factorization systems for Pi](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos#FactorizationSystemsForPi). > This admissibility structure should be regarded in conjunction with the statements in [section 2.3.11 Universal coverings and geometric Whitehead towers](https://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=165). Thanks for the hint. I will look at this.

   Where you point to the catlab-discussion, one should maybe add a remark that this is about 1-categorical factorization, while here its -categorical. Of course the central statements carry over.

   Maybe, if you have the time, add a pointer to this entry from some relevant general entry on cohesive -toposes.

   I corrected this and linked the article from cohesive (∞,1)-topos - factorization systems for Pi.

   This admissibility structure should be regarded in conjunction with the statements in section 2.3.11 Universal coverings and geometric Whitehead towers.

   Thanks for the hint. I will look at this.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexStephan, I have started adding some of the necessary stuff to _[[orthogonal factorization system]]_.

   Stephan,

   I have started adding some of the necessary stuff to orthogonal factorization system.

6. • CommentRowNumber6.
   • CommentAuthorStephan A Spahn
   • CommentTimeFeb 14th 2012
   • (edited Feb 14th 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexHi Urs, thanks. I didn´t do it myself since in the 1-categorical case the catlab is very detailed and I didn't mean to reproduce that. May be I look in Higher Topos Theory if there is more material concerning the (∞,1)-case. > This admissibility structure should be regarded in conjunction with the statements in [section 2.3.11 Universal coverings and geometric Whitehead towers](https://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=165). Yes! It seems this cover morphisms coming from the $X^{(\infty)}$ are Pi-closed...

   Hi Urs, thanks. I didn´t do it myself since in the 1-categorical case the catlab is very detailed and I didn’t mean to reproduce that. May be I look in Higher Topos Theory if there is more material concerning the (∞,1)-case.

   This admissibility structure should be regarded in conjunction with the statements in section 2.3.11 Universal coverings and geometric Whitehead towers.

   Yes! It seems this cover morphisms coming from the $X^{(\infty)}$ are Pi-closed…

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I didn´t do it myself since in the 1-categorical case the catlab is very detailed and I didn't mean to reproduce that. Sure, but it will be good to have some of the central statements collected on the $n$Lab. Makes referencing and future expansion easier. > Yes! It seems this cover morphisms coming from the $X^{(\infty)}$ are $\Pi$-closed... Yes, they are $\Pi$-closed, because they are by definition pullbacks of $Disc(-)$ of a morphism. What is good about it is that this "universal $\infty$-connected cover" $X^{(\infty)}$ of $X$ does indeed behave like a _[[good cover]]_ (in that all its components are null-homotopic), whereas a general covering space will be far from from being "good" in this sense. This makes me think that also the admissibility structure that you point out will be "good". I need to think more about it, though.

   I didn´t do it myself since in the 1-categorical case the catlab is very detailed and I didn’t mean to reproduce that.

   Sure, but it will be good to have some of the central statements collected on the $n$Lab. Makes referencing and future expansion easier.

   Yes! It seems this cover morphisms coming from the $X^{(\infty)}$ are $\Pi$-closed…

   Yes, they are $\Pi$-closed, because they are by definition pullbacks of $Disc(-)$ of a morphism.

   What is good about it is that this “universal $\infty$-connected cover” $X^{(\infty)}$ of $X$ does indeed behave like a good cover (in that all its components are null-homotopic), whereas a general covering space will be far from from being “good” in this sense.

   This makes me think that also the admissibility structure that you point out will be “good”. I need to think more about it, though.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexWell, there is some mild size-issue. For a given $X \in \mathbf{H}$, which is geometrically connected, we are looking at the morphism of discrete $\infty$-groupoids $$ * \to \Pi(X) \,. $$ Its homotopy fibers are the loop space $\Omega \Pi(X)$. So $* \to \Pi(X)$ is classified by a map into the $\kappa$-compact [[object classifier]] of [[∞Grpd]] if $\Omega \Pi X$ is $\kappa$-compact. So if we want to be precise we need to be introducing concepts of the following kind: 1. say an object $X$ is _geometrically $\kappa$-compact_ if $\Omega \Pi X$ is $\kappa$-compact; 1. consider your admissibility structures over geometrically $\kappa$-compact objects for given regular cardinal $\kappa$. Not a big deal, but this will be necessary for being precise.

   Well, there is some mild size-issue.

   For a given $X \in \mathbf{H}$, which is geometrically connected, we are looking at the morphism of discrete $\infty$-groupoids

   $$* \to \Pi(X) \,.$$

   Its homotopy fibers are the loop space $\Omega \Pi(X)$. So $* \to \Pi(X)$ is classified by a map into the $\kappa$-compact object classifier of ∞Grpd if $\Omega \Pi X$ is $\kappa$-compact.

   So if we want to be precise we need to be introducing concepts of the following kind:

   1. say an object $X$ is geometrically $\kappa$-compact if $\Omega \Pi X$ is $\kappa$-compact;

   2. consider your admissibility structures over geometrically $\kappa$-compact objects

   for given regular cardinal $\kappa$. Not a big deal, but this will be necessary for being precise.