Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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 object 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.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 22nd 2010
   • (edited Oct 22nd 2010)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexhere is something I've been fighting with lately. it should be completely n-classical in terms of 2-groups and crossed modules, but since I am sistematically lost in the 2-groups diagrammatics I'll write here what I have in mind, in the hope someone will tell me how this construction is n-classically named :) given an abelian group extension $1\to A\to G\to H\to 1$, we can consider the following 3-coskeletal simplicial set: there is exactly one vertex $\bullet$; there is an edge $\bullet\stackrel{g}{\to}\bullet$ for each element in $G$; there is a 2-simplex with edges $g_1,g_2,g_3$ and face $a\in A$ for any such elements such that $g_1g_2g_3=a$; there is a 3-simplex with faces $a_1,a_2,a_3,a_4$ (and elements from $G$ on the edges) whenever $\sum_i(-1)^i a_i=0$. let me call $\mathbf{B}(G//A)$ this simplicial set. it should be a Kan complex (which is immediate, if I did not miss some point here) so it is the nerve of (the deloping of) a 2-group $G//A$. moreover there is a natural morphism $\mathbf{B}(G//A)\to \mathbf{B}H$ (a weak equivalence, I guess) which fits into a commutative diagram $ \begin{matrix} \mathbf{B}(G//A)&\hookrightarrow& \mathbf{cosk}_1\mathbf{B}G\\ \downarrow &&\downarrow\\ \mathbf{B}H &\to &\mathbf{cosk}_1\mathbf{B}H \end{matrix} $

   here is something I’ve been fighting with lately. it should be completely n-classical in terms of 2-groups and crossed modules, but since I am sistematically lost in the 2-groups diagrammatics I’ll write here what I have in mind, in the hope someone will tell me how this construction is n-classically named :)

   given an abelian group extension $1\to A\to G\to H\to 1$, we can consider the following 3-coskeletal simplicial set: there is exactly one vertex $\bullet$; there is an edge $\bullet\stackrel{g}{\to}\bullet$ for each element in $G$; there is a 2-simplex with edges $g_1,g_2,g_3$ and face $a\in A$ for any such elements such that $g_1g_2g_3=a$; there is a 3-simplex with faces $a_1,a_2,a_3,a_4$ (and elements from $G$ on the edges) whenever $\sum_i(-1)^i a_i=0$.

   let me call $\mathbf{B}(G//A)$ this simplicial set. it should be a Kan complex (which is immediate, if I did not miss some point here) so it is the nerve of (the deloping of) a 2-group $G//A$. moreover there is a natural morphism $\mathbf{B}(G//A)\to \mathbf{B}H$ (a weak equivalence, I guess) which fits into a commutative diagram $\begin{matrix} \mathbf{B}(G//A)&\hookrightarrow& \mathbf{cosk}_1\mathbf{B}G\\ \downarrow &&\downarrow\\ \mathbf{B}H &\to &\mathbf{cosk}_1\mathbf{B}H \end{matrix}$

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeOct 22nd 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexMy thought is the following: The inclusion of $A$ into $G$ is a crossed module (therefore gives a 2-group). The simplicial set you give is the clasifying space of that crossed module (bar one or two details). The other facts follow in general as this 2-group has trivial second homotopy group (= kernel). There are numerous ways of looking at these things, e.g. classifying spaces of general crossed modules are discussed in the [[Menagerie]], but that is not the only terminology one can use for these things.

   My thought is the following:

   The inclusion of $A$ into $G$ is a crossed module (therefore gives a 2-group). The simplicial set you give is the clasifying space of that crossed module (bar one or two details). The other facts follow in general as this 2-group has trivial second homotopy group (= kernel).

   There are numerous ways of looking at these things, e.g. classifying spaces of general crossed modules are discussed in the Menagerie, but that is not the only terminology one can use for these things.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 22nd 2010
   • (edited Oct 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, so that 2-group I'd notationally identify with the [[crossed module]] $(A \to G)$ as Tim says. This is a groupoid with $G$ as its objects and morphisms labeled by $A$. It gives a one-object 2-groupoid $\mathbf{B}(A \to G)$, yes. The simplicial set you indicate is indeed the [[Duskin nerve]] $N \mathbf{B}(A \to G)$. And, yes, the obvious 2-functor $$ \mathbf{B}(A \to G) \to \mathbf{B}H $$ is an equivalence of 2-groupoids. You can check this for instance noticing that this is a [[k-surjective functor]] for all $k$, or equivalently of course by noticing that $$ N\mathbf{B}(A \to G) \to N \mathbf{B}H $$ is an acyclic Kan fibration. This weak equivalence, by the way, is the tool to extend the short exact sequence to the corresponding fiber sequence. In the $\infty$-category of $\infty$-groupoids this goes $$ A \to G \to H \to \mathbf{B}A \to \mathbf{B}G \to \mathbf{B}H \to \mathbf{B}^2 A \,. $$ The last step is _modeled_ in terms of strict functors by the 2-anafunctor $$ \array{ \mathbf{B}(A \to G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ {}^{\mathrlap{\simeq}}\downarrow \\ \mathbf{B}H } $$

   Right, so that 2-group I’d notationally identify with the crossed module $(A \to G)$ as Tim says. This is a groupoid with $G$ as its objects and morphisms labeled by $A$. It gives a one-object 2-groupoid $\mathbf{B}(A \to G)$, yes.

   The simplicial set you indicate is indeed the Duskin nerve $N \mathbf{B}(A \to G)$.

   And, yes, the obvious 2-functor

   $$\mathbf{B}(A \to G) \to \mathbf{B}H$$

   is an equivalence of 2-groupoids. You can check this for instance noticing that this is a k-surjective functor for all $k$, or equivalently of course by noticing that

   $$N\mathbf{B}(A \to G) \to N \mathbf{B}H$$

   is an acyclic Kan fibration.

   This weak equivalence, by the way, is the tool to extend the short exact sequence to the corresponding fiber sequence. In the $\infty$-category of $\infty$-groupoids this goes

   $$A \to G \to H \to \mathbf{B}A \to \mathbf{B}G \to \mathbf{B}H \to \mathbf{B}^2 A \,.$$

   The last step is modeled in terms of strict functors by the 2-anafunctor

   $$\array{ \mathbf{B}(A \to G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ {}^{\mathrlap{\simeq}}\downarrow \\ \mathbf{B}H }$$
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 22nd 2010
   • (edited Oct 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added this discussion to <a href="http://ncatlab.org/nlab/show/group+extension#CentralExtensions">group extension -- Central extensions</a>

   I have added this discussion to group extension – Central extensions

5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 22nd 2010
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItex> I have added this discussion to group extension -- Central extensions just looked at it: isn't $\mathbf{B}(A\to G)$ 3-coskeletal? (and not 2-coskeletal, I mean)

   I have added this discussion to group extension – Central extensions

   just looked at it: isn’t $\mathbf{B}(A\to G)$ 3-coskeletal? (and not 2-coskeletal, I mean)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 22nd 2010
   • (edited Oct 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, thanks. I have fixed it now.

   Yes, thanks. I have fixed it now.