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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI created a stub for [[stabilizer subgroup]] as it was needed by [[orbit]]. This could do with a nPOV section, but I am not sure what to say for that. (I am still not sure how to `do' an nPOV version of all the stuff around group actions. I would love to have it clear in my head as then a straightforward nPOV adaption of Grothendieck in SGA1 would be feasible. The treatment of SGA1 in the nLab still has some holes in it... e.g. the transition from the prorepresenting $P$ to the profinite group $Aut(P)$. I hope to sketch a bit more detail there soon. In fact a lot of the detail is skated over in the original SGA1. Any thoughts anyone?)

   I created a stub for stabilizer subgroup as it was needed by orbit.

   This could do with a nPOV section, but I am not sure what to say for that. (I am still not sure how to ‘do’ an nPOV version of all the stuff around group actions. I would love to have it clear in my head as then a straightforward nPOV adaption of Grothendieck in SGA1 would be feasible. The treatment of SGA1 in the nLab still has some holes in it… e.g. the transition from the prorepresenting $P$ to the profinite group $Aut(P)$. I hope to sketch a bit more detail there soon. In fact a lot of the detail is skated over in the original SGA1. Any thoughts anyone?)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> This could do with a nPOV section I have added it [here](http://ncatlab.org/nlab/show/stabilizer+subgroup#GeneralAbstract).

   This could do with a nPOV section

   I have added it here.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, the server went down briefly after my edit. So here is roughly what I typed: a group action $\rho : G \times X \to X$ is equivalently encoded in the corresponding [[action groupoid]], which is the $\rho$-associated bundle to the the universal $G$-bundle $X \to X//G \to \mathbf{B}G$. The stabilizer group of an element $x : * \to X$ is the first homotopy group of $X//G$ at this point. Or, what is equivalent for 1-groups, it is the [[loop space object]] $\Omega_{x} X//G $. (Hm, of course also the formulas don't all display as long as the nLab server is down...) In this last formulation the notion directly generalizes to stabilizer $\infty$-groups of $\infty$-group actions. And all of this goes through for cohesive $\infty$-groups, too. This is also in section 2.3.7 [here](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos). (Which of course is, however, also not available as long as the server is down.)

   Hm, the server went down briefly after my edit.

   So here is roughly what I typed: a group action $\rho : G \times X \to X$ is equivalently encoded in the corresponding action groupoid, which is the $\rho$-associated bundle to the the universal $G$-bundle $X \to X//G \to \mathbf{B}G$. The stabilizer group of an element $x : * \to X$ is the first homotopy group of $X//G$ at this point. Or, what is equivalent for 1-groups, it is the loop space object $\Omega_{x} X//G$.

   (Hm, of course also the formulas don’t all display as long as the nLab server is down…)

   In this last formulation the notion directly generalizes to stabilizer $\infty$-groups of $\infty$-group actions. And all of this goes through for cohesive $\infty$-groups, too.

   This is also in section 2.3.7 here. (Which of course is, however, also not available as long as the server is down.)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 10th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs there any need to revisit this [thread](http://golem.ph.utexas.edu/category/2009/03/the_stabilizer_of_a_subcategor.html), e.g., Mike's [suggestion](http://golem.ph.utexas.edu/category/2009/03/the_stabilizer_of_a_subcategor.html#c022247)?

   Is there any need to revisit this thread, e.g., Mike’s suggestion?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWhile stabilizer group, isotropy group, isotropy subgroup, stabilizer subgroup and small group are also used, I think that the modern default is the shortest: just [[stabilizer]]. The latter term subsumes also the categorified versions.

   While stabilizer group, isotropy group, isotropy subgroup, stabilizer subgroup and small group are also used, I think that the modern default is the shortest: just stabilizer. The latter term subsumes also the categorified versions.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Is there any need to revisit this [thread](http://golem.ph.utexas.edu/category/2009/03/the_stabilizer_of_a_subcategor.html), e.g., Mike's [suggestion](http://golem.ph.utexas.edu/category/2009/03/the_stabilizer_of_a_subcategor.html#c022247)? Okay, sure. I had forgotten about this. First, over there is discussed the stabilizer of a general morphism $Y \to X$, whereas I was here talking about stabilizers of global elements $* \to X$ only. Second, there the action in $n$-categories is at least alluded to, whereas what I wrote here is for $(n,1)$-categories. So there is room to generalize. What I don't see right now is how Mike's suggestion gives a group object, in general. What Mike defines one can think of as producing the homotopy fixed point data fibered over the group (over each element $g$ of the group the data that makes $Y \to X$ a homotopy fixed point under $g$.) How is that a group object? (I am genuinely asking, I don't see it right now.)

   Is there any need to revisit this thread, e.g., Mike’s suggestion?

   Okay, sure. I had forgotten about this.

   First, over there is discussed the stabilizer of a general morphism $Y \to X$, whereas I was here talking about stabilizers of global elements $* \to X$ only.

   Second, there the action in $n$-categories is at least alluded to, whereas what I wrote here is for $(n,1)$-categories.

   So there is room to generalize.

   What I don’t see right now is how Mike’s suggestion gives a group object, in general. What Mike defines one can think of as producing the homotopy fixed point data fibered over the group (over each element $g$ of the group the data that makes $Y \to X$ a homotopy fixed point under $g$.) How is that a group object?

   (I am genuinely asking, I don’t see it right now.)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> First, over there is discussed the stabilizer of a general morphism $Y \to X$, whereas I was here talking about stabilizers of global elements $* \to X$ only. Of course the definition of stabilizer $\infty$-groups as [here](http://ncatlab.org/nlab/show/stabilizer+subgroup#GeneralAbstract) immediately generalizes to this case: for $G$ an action on $X$, and $Y$ any other object, there is an induced action on the internal hom $[Y, X]$ leading to an action $\infty$-groupoid $[Y,X]//G$. So now for given $f : Y \to X$ we can form the stabilizer $\infty$-group $\Omega_f ( [Y,X]//G )$. edit: as action $\infty$-groupoids we obtain $[Y,X]//G \to \mathbf{B}G$ from $X//G \to \mathbf{B}G$ as the $\infty$-pullback $$ \array{ [Y,X]//G &\to& [Y, X//G] \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& [Y,\mathbf{B}G] } \,, $$ where the bottom morphism picks the trivial cocycle and its global automorphisms (meaning that $G$ acts by the same element on all points of $Y$ at once, as it should). So the bottom morphism is the hom-adjunct to the projection $Y \times \mathbf{B}G \to \mathbf{B}G$.

   First, over there is discussed the stabilizer of a general morphism $Y \to X$, whereas I was here talking about stabilizers of global elements $* \to X$ only.

   Of course the definition of stabilizer $\infty$-groups as here immediately generalizes to this case:

   for $G$ an action on $X$, and $Y$ any other object, there is an induced action on the internal hom $[Y, X]$ leading to an action $\infty$-groupoid $[Y,X]//G$. So now for given $f : Y \to X$ we can form the stabilizer $\infty$-group $\Omega_f ( [Y,X]//G )$.

   edit:

   as action $\infty$-groupoids we obtain $[Y,X]//G \to \mathbf{B}G$ from $X//G \to \mathbf{B}G$ as the $\infty$-pullback

   $$\array{ [Y,X]//G &\to& [Y, X//G] \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& [Y,\mathbf{B}G] } \,,$$

   where the bottom morphism picks the trivial cocycle and its global automorphisms (meaning that $G$ acts by the same element on all points of $Y$ at once, as it should). So the bottom morphism is the hom-adjunct to the projection $Y \times \mathbf{B}G \to \mathbf{B}G$.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2012
   • (edited Jan 10th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFinally, I should say, for the definition that I was looking at, that and how there is a canonical morphism from the stabilizer $\infty$-group to the $\infty$-group $G$. It's given by the top left morphism $i_x$ in the pasting diagram of homotopy pullbacks $$ \array{ Stab_G(x) &\stackrel{i_x}{\to}& G &\to& * \\ \downarrow && \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X &\to& X//G \\ && \downarrow && \downarrow^{\rho} \\ && * &\to& \mathbf{B}G } \,. $$ edit: what is maybe noteworthy here is that this is just producing the long fiber sequence for the case where we regard $X$ and $X//G$ as objects pointed by $x$. So $i_x$ here is simply the looping of the representation itself: $$ i_x \simeq \Omega_x \rho \,. $$

   Finally, I should say, for the definition that I was looking at, that and how there is a canonical morphism from the stabilizer $\infty$-group to the $\infty$-group $G$. It’s given by the top left morphism $i_x$ in the pasting diagram of homotopy pullbacks

   $$\array{ Stab_G(x) &\stackrel{i_x}{\to}& G &\to& * \\ \downarrow && \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X &\to& X//G \\ && \downarrow && \downarrow^{\rho} \\ && * &\to& \mathbf{B}G } \,.$$

   edit:

   what is maybe noteworthy here is that this is just producing the long fiber sequence for the case where we regard $X$ and $X//G$ as objects pointed by $x$. So $i_x$ here is simply the looping of the representation itself:

   $$i_x \simeq \Omega_x \rho \,.$$
9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJan 10th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI believe the two definitions are actually the same, both in the case of global points and of arbitrary morphisms $Y\to X$. I actually found it easiest to prove this using HoTT, although quite possibly there is an easy arrow-theoretic proof too. Note that the following code uses some lemmas and tactics from my fork of the HoTT repository that I haven't gotten around to merging back into the main trunk yet. Require Import Homotopy. Let's assume a pointed type $B G$, with the group object $G$ defined as its loop space. Hypothesis BG : Type. Hypothesis pt : BG. Definition G := pt == pt. Section OfAPoint. Similarly, let's assume the action groupoid as a fibration $X\sslash G \to B G$, with the type $X$ defined as its fiber over the basepoint. Note that the action of $G$ on $X$ is then via `transport`. Hypothesis XG : BG -> Type. Let X := XG pt. Let's consider first the stabilizer of a point of $X$. Hypothesis x : X. Urs's definition of the stabilizer of a point $x\colon X$ is as the loop space of that point, regarded as living in the total space $X\sslash G$ rather than in $X$. Definition Urs_stab_pt : Type := (existT XG pt x) == (existT XG pt x). On the other hand, my definition specializes to the following. Definition Mike_stab_pt : Type := { g : G & transport g x == x }. And their equivalence is just a special case of the characterization of path-spaces of total spaces. Definition stab_pt_equiv : Urs_stab_pt <~> Mike_stab_pt := total_paths_equiv BG XG (existT XG pt x) (existT XG pt x). End OfAPoint. Section OfAMap. Now let's consider the stabilizer of a map $f\colon Y\to X$. Again we assume $X$ via its action groupoid. Hypothesis XG : BG -> Type. Let X := XG pt. Hypothesis Y : Type. Hypothesis f : Y -> X. The following definition is the fibration $[Y, X\sslash G] \to [Y, BG]$. Let Y_XG : (Y -> BG) -> Type := fun h => (forall y:Y, XG (h y)). As a sanity check, we verify that its total space is, in fact, the function space $[Y, X\sslash G]$. Lemma Y_XG_total : (Y -> sigT XG) <~> sigT Y_XG. Proof. apply section_total_equiv. Defined. Now this is the action groupoid $[Y,X]\sslash G$, defined with a pullback as in comment #7 above. Let YX_G : BG -> Type := fun bg => Y_XG (fun _ => bg). Urs's definition of the stabilizer of $f\colon Y\to X$ is its stabilizer as a point of this action of $G$ on $[Y,X]$. Definition Urs_stab_map : Type := Urs_stab_pt YX_G f. Whereas here's what my definition would give. Definition Mike_stab_map : Type := { g : G & (transport g) o f == f }. In this case, the proof of equivalence is a little longer, but it really only has two steps. First, we apply the characterization of path-spaces of total spaces to identify Urs's definition with the total space of some fibration over $G$. Definition stab_map_equiv : Urs_stab_map <~> Mike_stab_map. Proof. apply @equiv_compose with (B := { g : G & transport (P := fun bg => Y -> XG bg) g f == f }). refine (total_paths_equiv BG YX_G _ _). Now we just need to show that this fibration (`g` $\mapsto$ `transport g f == f`) is fiberwise equivalent to `g` $\mapsto$ `transport g` $\circ$ `f == f`, and for this it suffices to show that `transport g f` is equivalent to `(transport g)` $\circ$ `f`. assert (H : forall g:G, transport (P := fun bg => Y -> XG bg) g f == (transport g) o f). This should probably be a lemma in the standard library, but I don't think it's there yet, so we can just prove it here. intros g; apply funext; intros y. path_change ((fun h => h y) (transport (P := fun bg => Y -> XG bg) g f)). path_change (transport g ((fun h => h y) f)). refine (trans_map _ (fun bg (h: Y -> XG bg) => h y) _). Now we can complete the proof. apply total_equiv with (g := fun g => concat (!H g)). intros g; apply concat_is_equiv_left. Defined. End OfAMap.

   I believe the two definitions are actually the same, both in the case of global points and of arbitrary morphisms $Y\to X$. I actually found it easiest to prove this using HoTT, although quite possibly there is an easy arrow-theoretic proof too.

   Note that the following code uses some lemmas and tactics from my fork of the HoTT repository that I haven’t gotten around to merging back into the main trunk yet.

   Require Import Homotopy.
   

   Let’s assume a pointed type $B G$, with the group object $G$ defined as its loop space.

   Hypothesis BG : Type.
   Hypothesis pt : BG.
   Definition G := pt == pt.
   
   Section OfAPoint.
   

   Similarly, let’s assume the action groupoid as a fibration $X\sslash G \to B G$, with the type $X$ defined as its fiber over the basepoint. Note that the action of $G$ on $X$ is then via transport.

     Hypothesis XG : BG -> Type.
     Let X := XG pt.
   

   Let’s consider first the stabilizer of a point of $X$.

     Hypothesis x : X.
   

   Urs’s definition of the stabilizer of a point $x\colon X$ is as the loop space of that point, regarded as living in the total space $X\sslash G$ rather than in $X$.

     Definition Urs_stab_pt : Type
       := (existT XG pt x) == (existT XG pt x).
   

   On the other hand, my definition specializes to the following.

     Definition Mike_stab_pt : Type
       := { g : G & transport g x == x }.
   

   And their equivalence is just a special case of the characterization of path-spaces of total spaces.

     Definition stab_pt_equiv : Urs_stab_pt <~> Mike_stab_pt
       := total_paths_equiv BG XG (existT XG pt x) (existT XG pt x).
   
   End OfAPoint.
   
   Section OfAMap.
   

   Now let’s consider the stabilizer of a map $f\colon Y\to X$. Again we assume $X$ via its action groupoid.

     Hypothesis XG : BG -> Type.
     Let X := XG pt.
   
     Hypothesis Y : Type.
     Hypothesis f : Y -> X.
   

   The following definition is the fibration $[Y, X\sslash G] \to [Y, BG]$.

     Let Y_XG : (Y -> BG) -> Type
       := fun h => (forall y:Y, XG (h y)).
   

   As a sanity check, we verify that its total space is, in fact, the function space $[Y, X\sslash G]$.

     Lemma Y_XG_total : (Y -> sigT XG) <~> sigT Y_XG.
     Proof.
       apply section_total_equiv.
     Defined.
   

   Now this is the action groupoid $[Y,X]\sslash G$, defined with a pullback as in comment #7 above.

     Let YX_G : BG -> Type
       := fun bg => Y_XG (fun _ => bg).
   

   Urs’s definition of the stabilizer of $f\colon Y\to X$ is its stabilizer as a point of this action of $G$ on $[Y,X]$.

     Definition Urs_stab_map : Type
       := Urs_stab_pt YX_G f.
   

   Whereas here’s what my definition would give.

     Definition Mike_stab_map : Type
       := { g : G & (transport g) o f == f }.
   

   In this case, the proof of equivalence is a little longer, but it really only has two steps. First, we apply the characterization of path-spaces of total spaces to identify Urs’s definition with the total space of some fibration over $G$.

     Definition stab_map_equiv : Urs_stab_map <~> Mike_stab_map.
     Proof.
       apply @equiv_compose with
         (B := { g : G & transport (P := fun bg => Y -> XG bg) g f == f }).
       refine (total_paths_equiv BG YX_G _ _).
   

   Now we just need to show that this fibration (g $\mapsto$ transport g f == f) is fiberwise equivalent to g $\mapsto$ transport g $\circ$ f == f, and for this it suffices to show that transport g f is equivalent to (transport g) $\circ$ f.

       assert (H : forall g:G,
         transport (P := fun bg => Y -> XG bg) g f == (transport g) o f).
   

   This should probably be a lemma in the standard library, but I don’t think it’s there yet, so we can just prove it here.

       intros g; apply funext; intros y.
       path_change ((fun h => h y) 
         (transport (P := fun bg => Y -> XG bg) g f)).
       path_change (transport g ((fun h => h y) f)).
       refine (trans_map _ (fun bg (h: Y -> XG bg) => h y) _).
   

   Now we can complete the proof.

       apply total_equiv with
         (g := fun g => concat (!H g)).
       intros g; apply concat_is_equiv_left.
     Defined.
   
   End OfAMap.
   

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 10th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexBy the way, I think that $X\sslash G$ (iTex `\sslash`) looks much nicer than $X//G$.

    By the way, I think that $X\sslash G$ (iTex \sslash) looks much nicer than $X//G$.

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 10th 2012
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexIt saddens me that I find #9 almost completely unreadable.

    It saddens me that I find #9 almost completely unreadable.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I believe the two definitions are actually the same, [...] I actually found it easiest to prove this using HoTT, Mike, that is quite impressive. I literally have to run now, but thanks for this. And thanks for \sslash I didn't know that code and didn't bother to search for it.

    I believe the two definitions are actually the same, […] I actually found it easiest to prove this using HoTT,

    Mike, that is quite impressive. I literally have to run now, but thanks for this.

    And thanks for

    \sslash
    

    I didn’t know that code and didn’t bother to search for it.

13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJan 10th 2012
    • PermaLink
    Author: zskoda
    Format: MarkdownItexI can not read sslash in firefox on XP.

    I can not read sslash in firefox on XP.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 10th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexZoran: That's weird; I presume you have all the usual fonts installed. Can someone who knows more than I do suggest any reason why Zoran's fonts might be missing that character?

    Zoran: That’s weird; I presume you have all the usual fonts installed. Can someone who knows more than I do suggest any reason why Zoran’s fonts might be missing that character?

15. • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 11th 2012
    • (edited Jan 11th 2012)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexI have the same problem as Zoran. And, while I'm thinking of it, $\preceq$ doesn't render for me either. (That's \preceq, in case it doesn't render for you either.)

    I have the same problem as Zoran. And, while I’m thinking of it, $\preceq$ doesn’t render for me either. (That’s \preceq, in case it doesn’t render for you either.)

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexTodd, are you also in Firefox on Windows? Does it look okay in other browsers, or if you change the itex mode on the forum?

    Todd, are you also in Firefox on Windows? Does it look okay in other browsers, or if you change the itex mode on the forum?

17. • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 11th 2012
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThanks for asking, Mike. Yes, I nomally use Firefox on Windows XP. The \preceq, but not the \sslash, shows up in IE 8.0.6001.18702. (There might be something I need to enable which would allow me to see \sslash?) Sorry, I don't know what it means to change itex mode, unless you mean to change the filter for formatting comments (?).

    Thanks for asking, Mike. Yes, I nomally use Firefox on Windows XP. The \preceq, but not the \sslash, shows up in IE 8.0.6001.18702. (There might be something I need to enable which would allow me to see \sslash?) Sorry, I don’t know what it means to change itex mode, unless you mean to change the filter for formatting comments (?).

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have briefly added some of the above stuff to [[stabilizer group]] (not yet Mike's stuff, though). Maybe I'll add more later. But now I need to be taking care of something else.

    I have briefly added some of the above stuff to stabilizer group (not yet Mike’s stuff, though). Maybe I’ll add more later. But now I need to be taking care of something else.

19. • CommentRowNumber19.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2012
    • PermaLink
    Author: zskoda
    Format: MarkdownItexI usually see all the fonts in $n$Lab and $n$Forum, sslash is one of rare exception, and example preceq from 15 as well.

    I usually see all the fonts in $n$Lab and $n$Forum, sslash is one of rare exception, and example preceq from 15 as well.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex@Todd: Go to "Account" at the top of the nForum page. On the left, click "change iTex mode", select different things and see how it looks. This changes whether math is displayed as images or as MathML (which the browser has to understand — currently Firefox is the only major browser that supports MathML, so the "auto" setting selects MathML if you're using Firefox and images otherwise. I think). You can also test whether math is currently displaying as MathML or as images by trying to select part, but not all, of a mathematical formula with the cursor. This is possible with MathML, but not with an image.

    @Todd: Go to “Account” at the top of the nForum page. On the left, click “change iTex mode”, select different things and see how it looks. This changes whether math is displayed as images or as MathML (which the browser has to understand — currently Firefox is the only major browser that supports MathML, so the “auto” setting selects MathML if you’re using Firefox and images otherwise. I think).

    You can also test whether math is currently displaying as MathML or as images by trying to select part, but not all, of a mathematical formula with the cursor. This is possible with MathML, but not with an image.

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexAt _[[stabilizer group]]_ I have expanded the $\infty$-topos-theoretic discussion [here](http://ncatlab.org/nlab/show/stabilizer+group#ForInfinityGroupActions) such as to highlight that the map from the stabilizer comes from a 1-image factorization. Then I noticed that in the example _[Stabilizers of Shapes -- Klein geometry](http://ncatlab.org/nlab/show/stabilizer+group#KleinGeometry)_ we used to have an ad-hoc definition of the right conjugation action on a hom space (from a trivial action into a nontrivial action). This was before we had the systematic discussion of conjugation actions simply via the internal homs in the slice $\mathbf{H}_{/\mathbf{B}G}$. I have now added [a proposition](http://ncatlab.org/nlab/show/stabilizer+group#ReformulationOfRightSidedConjugationAction), and its proof, that this ad-hoc definition is indeed equivalent to the systematic one. (This should eventually go to a general discussion at _[[infinity-action]]_).

    At stabilizer group I have expanded the $\infty$-topos-theoretic discussion here such as to highlight that the map from the stabilizer comes from a 1-image factorization.

    Then I noticed that in the example Stabilizers of Shapes – Klein geometry we used to have an ad-hoc definition of the right conjugation action on a hom space (from a trivial action into a nontrivial action). This was before we had the systematic discussion of conjugation actions simply via the internal homs in the slice $\mathbf{H}_{/\mathbf{B}G}$.

    I have now added a proposition, and its proof, that this ad-hoc definition is indeed equivalent to the systematic one. (This should eventually go to a general discussion at infinity-action).

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2014
    • (edited Dec 18th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHave added to the list of examples the [remark](http://ncatlab.org/nlab/show/stabilizer+group#ForCanonicalActionOnCosetSpace) that for any homomorphism of $\infty$-groups $H \to G$, the homotopy quotient $G/\!/H$ is the canonical space acted on by $G$ for which $H$ is the stabilizer of points. Also added a brief [remark](http://ncatlab.org/nlab/show/stabilizer+group#StabilizersOfCoShapes) about an amusing dual version of Klein geometry: where Klein considered for a $G$-space $X$ stabilizers of maps (figures) $A \to X$ in the $G$-space $[A,X]$, we may also consider, dually, stabilizers of maps ("co-figures") $X \to A$ in the $G$-space $[X,A]$. This is the situation considered at [[general covariance]]. If here $A= \mathbf{B}^n U(1)_{conn}$ (and if we replace $[X,A]$ by its differential concretification) then stabilizers of co-shapes are [[quantomorphism n-groups]].

    Have added to the list of examples the remark that for any homomorphism of $\infty$-groups $H \to G$, the homotopy quotient $G/\!/H$ is the canonical space acted on by $G$ for which $H$ is the stabilizer of points.

    Also added a brief remark about an amusing dual version of Klein geometry:

    where Klein considered for a $G$-space $X$ stabilizers of maps (figures) $A \to X$ in the $G$-space $[A,X]$, we may also consider, dually, stabilizers of maps (“co-figures”) $X \to A$ in the $G$-space $[X,A]$. This is the situation considered at general covariance.

    If here $A= \mathbf{B}^n U(1)_{conn}$ (and if we replace $[X,A]$ by its differential concretification) then stabilizers of co-shapes are quantomorphism n-groups.

23. • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 18th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexDoes the stabilizer of shape/coshape story meet up with the invariants/coinvariants of a G-action story? Though I had something about (co)invariants of the induced action on (co)shapes, but it's gone. And now off for 'celebration' of our REF results, remember g+ discussion. (We didn't do very well.)

    Does the stabilizer of shape/coshape story meet up with the invariants/coinvariants of a G-action story? Though I had something about (co)invariants of the induced action on (co)shapes, but it’s gone.

    And now off for ’celebration’ of our REF results, remember g+ discussion. (We didn’t do very well.)

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItex> Does the stabilizer of shape/coshape story meet up with the invariants/coinvariants of a G-action story? It's a "co" going in a different direction. A stabilizer group is always a thing of co-invariants, namely it is the loop space of the space of coinvariants on the given element being stabilized. But as we consider actions on mapping spaces between spaces with $G$-action, there are two opposite extreme cases: that where the source or that where there domain carries the trivial $G$-action. Coinvariants in these two cases give stabilizers "of shapes" and "of co-shapes", respectively.

    Does the stabilizer of shape/coshape story meet up with the invariants/coinvariants of a G-action story?

    It’s a “co” going in a different direction.

    A stabilizer group is always a thing of co-invariants, namely it is the loop space of the space of coinvariants on the given element being stabilized.

    But as we consider actions on mapping spaces between spaces with $G$-action, there are two opposite extreme cases: that where the source or that where there domain carries the trivial $G$-action. Coinvariants in these two cases give stabilizers “of shapes” and “of co-shapes”, respectively.