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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexUrs created [[submersion]] and I added a little more. Still a bit stubby, though.

   Urs created submersion and I added a little more. Still a bit stubby, though.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2010
   • (edited Jul 20th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded links (we have [[transversal maps]], for instance). We should say something about regular epimorphisms.

   added links (we have transversal maps, for instance).

   We should say something about regular epimorphisms.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexEdited [[transversal maps]] to change the direct sum sign to just a sum in the definition - as it should be.

   Edited transversal maps to change the direct sum sign to just a sum in the definition - as it should be.

4. • CommentRowNumber4.
   • CommentAuthorKevin Lin
   • CommentTimeJul 20th 2010
   • (edited Jul 20th 2010)
   • PermaLink
   Author: Kevin Lin
   Format: HtmlI added a few words to [[submersion]], which lead to adding a few words to [[proper morphism]], which lead to the creation of [[separated morphism]] and [[Ehresmann's theorem]], which lead to adding a few words to [[Riemann-Hilbert correspondence]] and the creation of [[higher direct image]] and [[hypercohomology]].

   I added a few words to submersion, which lead to adding a few words to proper morphism, which lead to the creation of separated morphism and Ehresmann's theorem, which lead to adding a few words to Riemann-Hilbert correspondence and the creation of higher direct image and hypercohomology.
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> which lead to [...] which lead to [...] which lead to [...] Thanks! I added the $n$POV to [[hypercohomology]], The very existence of the notion "hypercohomology" is a strong point in favor of the nPOV on cohomology.

   which lead to […] which lead to […] which lead to […]

   Thanks!

   I added the $n$POV to hypercohomology,

   The very existence of the notion “hypercohomology” is a strong point in favor of the nPOV on cohomology.

6. • CommentRowNumber6.
   • CommentAuthorKevin Lin
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Kevin Lin
   Format: HtmlReally cool stuff!

   Really cool stuff!
7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Really cool stuff! Yes. And in principle known since before 1973. A mystery why this isn't taught in high school in the 21 century...

   Really cool stuff!

   Yes. And in principle known since before 1973. A mystery why this isn’t taught in high school in the 21 century…

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI added the example of [[Deligne cohomology]] to [[hypercohomology]].

   I added the example of Deligne cohomology to hypercohomology.

9. • CommentRowNumber9.
   • CommentAuthorKevin Lin
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Kevin Lin
   Format: HtmlDo you know any good references for basic stuff about Eilenberg-Mac Lane objects?

   Do you know any good references for basic stuff about Eilenberg-Mac Lane objects?
10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2010
    • (edited Jul 20th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI like the discussion in _Higher Topos Theory_ . But, as we have seen elsewhere, I am strange and different... ;-) More seriously, I am not aware of many references that would describe Eilenberg-macLane objects in a higher sheaf topos really conceptually. Often it is just by decree that one takes these to be things represented by chain complexes concentrated in one degree. That's fine of course, because that's what it boils down to anyway. But a good conceptual description is in HTT.

    I like the discussion in Higher Topos Theory . But, as we have seen elsewhere, I am strange and different… ;-)

    More seriously, I am not aware of many references that would describe Eilenberg-macLane objects in a higher sheaf topos really conceptually. Often it is just by decree that one takes these to be things represented by chain complexes concentrated in one degree. That’s fine of course, because that’s what it boils down to anyway. But a good conceptual description is in HTT.

11. • CommentRowNumber11.
    • CommentAuthorKevin Lin
    • CommentTimeJul 20th 2010
    • PermaLink
    Author: Kevin Lin
    Format: HtmlOkay, I'll try to look at HTT.

    Okay, I'll try to look at HTT.
12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeJul 21st 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI added an arrow-theoretic characterisation of [[transversal maps]].

    I added an arrow-theoretic characterisation of transversal maps.

13. • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJul 21st 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI put in a coordinate-dependent definition at [[submersion]] on the grounds that it applies in a more general situation.

    I put in a coordinate-dependent definition at submersion on the grounds that it applies in a more general situation.

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 4th 2013
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexRe #12: I'm a little baffled by this "slick category-theoretic" way to express transversality. As pointed out in a comment by Tom Goodwillie at MO ([link](http://mathoverflow.net/questions/71902/is-every-representable-map-a-submersion/71942#71942) visible for those with 10k of rep), if we have $f = g: 1 \to M$, then the pullback of $f$ and $g$ surely exists (the pullback of a mono against itself trivially exists in any category, taking the pullback projections to be identity maps). Moreover, the tangent bundle functor $T$ preserves the terminal object $1$, so $T$ takes $f$ and $g$ to a map (necessarily monic, being a map out of the terminal) $T f = T g: 1 \to T M$, and therefore the pullback diagram, pulling back $f$ along $g$, is preserved by the tangent bundle functor.

    Re #12: I’m a little baffled by this “slick category-theoretic” way to express transversality. As pointed out in a comment by Tom Goodwillie at MO (link visible for those with 10k of rep), if we have $f = g: 1 \to M$, then the pullback of $f$ and $g$ surely exists (the pullback of a mono against itself trivially exists in any category, taking the pullback projections to be identity maps). Moreover, the tangent bundle functor $T$ preserves the terminal object $1$, so $T$ takes $f$ and $g$ to a map (necessarily monic, being a map out of the terminal) $T f = T g: 1 \to T M$, and therefore the pullback diagram, pulling back $f$ along $g$, is preserved by the tangent bundle functor.

15. • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeJun 5th 2013
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexWhy does one need rep to see this comment? I can read the question, 6 comments on it, and an answer, nothing by Goodwillie. Are there hidden comments???

    Why does one need rep to see this comment? I can read the question, 6 comments on it, and an answer, nothing by Goodwillie. Are there hidden comments???

16. • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeJun 5th 2013
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI\'m trying to remember how that condition came about; I think that it\'s missing a clause.

    I'm trying to remember how that condition came about; I think that it's missing a clause.

17. • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2013
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex@Toby: Sorry -- the explanation is that the answer that I linked to, and which has Goodwillie's comment, had been deleted; such answers on MO cannot be seen except by those with 10k of rep. (To such users the deleted answer appears in a pink box.) I just tried the link and was sent directly to the deleted answer; I guess other users are sent to the page anyway, but just can't see the deleted answer I really meant to link to.

    @Toby: Sorry – the explanation is that the answer that I linked to, and which has Goodwillie’s comment, had been deleted; such answers on MO cannot be seen except by those with 10k of rep. (To such users the deleted answer appears in a pink box.) I just tried the link and was sent directly to the deleted answer; I guess other users are sent to the page anyway, but just can’t see the deleted answer I really meant to link to.

18. • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeJun 5th 2013
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexYeah, that\'s what I see.

    Yeah, that's what I see.

19. • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 5th 2013
    • (edited Jun 5th 2013)
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    Author: DavidRoberts
    Format: MarkdownItexThe answer was by me, and Goodwillie pointed out the obvious counterexample to the claim in my answer (which I think I got from the nLab page). My answer is: *** Edit: This is wrong as it stands, I'm leaving it here in case someone (perhaps me) can fill in the missing detail. --- Here's a sketch. Fix the map $f:C\to D$. The pullback of $f$ along $g:E \to D$ exists if and only if $f$ is [transversal](http://en.wikipedia.org/wiki/Transversality_%28mathematics%29) to $g$, namely given $c\in C$ and $e\in E$ such that $f(c) = g(e) = d$, we have $$ T_{f(c)} D = f_\ast T_c C + g_\ast T_e E $$ (not direct sum, notice). (see, e.g. the [nLab article](http://ncatlab.org/nlab/show/transversal+maps) ) (EDIT: $f$ is transversal to $g$ iff the pullback exists and is preserved by the tangent bundle functor!) Thus for all pullbacks of $f$ to exist along arbitrary maps, including from $E = *$, we must have $T_{f(c)} D = f_* T_c C$, for all $c\in C$, i.e. $f$ is a submersion.

    The answer was by me, and Goodwillie pointed out the obvious counterexample to the claim in my answer (which I think I got from the nLab page). My answer is:

    ---

    Edit: This is wrong as it stands, I’m leaving it here in case someone (perhaps me) can fill in the missing detail.

    ---

    Here’s a sketch.

    Fix the map $f:C\to D$. The pullback of $f$ along $g:E \to D$ exists if and only if $f$ is transversal to $g$, namely given $c\in C$ and $e\in E$ such that $f(c) = g(e) = d$, we have

    $$T_{f(c)} D = f_\ast T_c C + g_\ast T_e E$$

    (not direct sum, notice). (see, e.g. the nLab article ) (EDIT: $f$ is transversal to $g$ iff the pullback exists and is preserved by the tangent bundle functor!)

    Thus for all pullbacks of $f$ to exist along arbitrary maps, including from $E = *$, we must have $T_{f(c)} D = f_* T_c C$, for all $c\in C$, i.e. $f$ is a submersion.

20. • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJun 5th 2013
    • (edited Jun 5th 2013)
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexOK, so David thought that $f \pitchfork g$ iff the pullback exists, I added the condition that the tangent bundle preserves it, and now we see that there is something more! I\'m pretty sure that I teased this result out of Lang, so I should go look there again. In the meantime, there is a warning that the result is incomplete.

    OK, so David thought that $f \pitchfork g$ iff the pullback exists, I added the condition that the tangent bundle preserves it, and now we see that there is something more! I'm pretty sure that I teased this result out of Lang, so I should go look there again. In the meantime, there is a warning that the result is incomplete.