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.
   • CommentAuthorUrs
   • CommentTimeDec 28th 2014
   • (edited Dec 28th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added to _[[regular epimorphism]]_ the statement ([here](http://ncatlab.org/nlab/show/regular+epimorphism#PullbackSquareOfRegularEpisIsPushout)) that a pullback square of regular epis is also a pushout. This must be true for effective epimorphisms in an $\infty$-topos, too. What's the proof?

   have added to regular epimorphism the statement (here) that a pullback square of regular epis is also a pushout.

   This must be true for effective epimorphisms in an $\infty$-topos, too. What’s the proof?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeDec 29th 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIf I'm not mistaken, the $\infty$-analogue is false. For instance, $$\begin{array}{ccc} 4 & \rightarrow & 2 \\ \downarrow & & \downarrow \\ 2 & \rightarrow & 1 \end{array}$$ is a (homotopy) pullback square of effective epimorphisms (of sets) in $\infty Grpd$ but not a homotopy pushout square. (The homotopy pushout is a circle.)

   If I’m not mistaken, the $\infty$-analogue is false. For instance,

   $$\begin{array}{ccc} 4 & \rightarrow & 2 \\ \downarrow & & \downarrow \\ 2 & \rightarrow & 1 \end{array}$$

   is a (homotopy) pullback square of effective epimorphisms (of sets) in $\infty Grpd$ but not a homotopy pushout square. (The homotopy pushout is a circle.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2014
   • (edited Dec 29th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's true of course, stupid me. Thanks. I'll need to consider some further conditions.

   That’s true of course, stupid me. Thanks.

   I’ll need to consider some further conditions.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2015
   • (edited Jan 7th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo concretely, I am looking at the following situation: $X$ a $V$-manifold (as [here](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)), $T_{inf}X$ its infinitesimal disk bundle (as [here](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles)), $U \to X$ a formally étale 1-epi, and the pullback in question being the one that exhibits a local trivialization of the formal disk bundle $$ \array{ && U \times \mathbb{D} \\ & \swarrow && \searrow \\ T_{inf}X && && U \\ & \searrow && \swarrow \\ && X } $$ So $T_{inf}X \to X$ as well as $U \times \mathbb{D} \to U$ are $\Re$-equivalences ($\Re$ the [[reduction modality]]). This means that applying $\Re$ to this diagram it does become a homotopy pushout. Since $\Re$ preserves homotopy pushouts, this means that the homotopy pushout $T_{inf} X \underset{U \times \mathbb{D}}{\coprod} U $ differs from $X$ at most in some infinitesimal extension. So for achieving the desired $X \simeq T_{inf} X \underset{U\times \mathbb{D}}{\coprod} U $ I am reduced to arguing (or else to try to arrange by further assumptions) that also this infinitesimal difference vanishes. This should follow since $U \times \mathbb{D}\to T_{inf}X $ is something like a fiberwise equivalence. Hm...

   So concretely, I am looking at the following situation: $X$ a $V$-manifold (as here), $T_{inf}X$ its infinitesimal disk bundle (as here), $U \to X$ a formally étale 1-epi, and the pullback in question being the one that exhibits a local trivialization of the formal disk bundle

   $$\array{ && U \times \mathbb{D} \\ & \swarrow && \searrow \\ T_{inf}X && && U \\ & \searrow && \swarrow \\ && X }$$

   So $T_{inf}X \to X$ as well as $U \times \mathbb{D} \to U$ are $\Re$-equivalences ($\Re$ the reduction modality). This means that applying $\Re$ to this diagram it does become a homotopy pushout. Since $\Re$ preserves homotopy pushouts, this means that the homotopy pushout $T_{inf} X \underset{U \times \mathbb{D}}{\coprod} U$ differs from $X$ at most in some infinitesimal extension.

   So for achieving the desired $X \simeq T_{inf} X \underset{U\times \mathbb{D}}{\coprod} U$ I am reduced to arguing (or else to try to arrange by further assumptions) that also this infinitesimal difference vanishes. This should follow since $U \times \mathbb{D}\to T_{inf}X$ is something like a fiberwise equivalence. Hm…

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2015
   • (edited Jan 28th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItex...

   …

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2015
   • (edited Jan 28th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItex...

   …

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 8th 2015
   • (edited Jan 8th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn more detail, what I am after is described around def. 4.7 of this note on parameterized WZW terms ([pdf](https://dl.dropboxusercontent.com/u/12630719/cwzw.pdf)). My real goal is to get a useful condition that makes the necessary obstruction of theorem 4.5 also a sufficient obstruction, as in corollary 4.11. Possibly there is an altogether better way to do this than I have there right now.

   In more detail, what I am after is described around def. 4.7 of this note on parameterized WZW terms (pdf). My real goal is to get a useful condition that makes the necessary obstruction of theorem 4.5 also a sufficient obstruction, as in corollary 4.11. Possibly there is an altogether better way to do this than I have there right now.