Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 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 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeMar 14th 2012
    • (edited Mar 14th 2012)

    I have ’stumbled on’ something that is quite fun (I doubt it is new). I was working on the higher generation by subgroups stuff, with a plan to go from there to Haefliger’s complexes of groups and then to orbihedra and orbifolds to close up a gap in the nLab linkages and put this also in the Menagerie. I therefore needed to look at group actions on simplicial complexes as that is the starting point of Haefliger’s theory. The notion of regular G-complex is defined in several (probably inequivalent) ways in the sources, so I wanted a categorical one to be clear about things and to try to make sense of the mess. Here is what seems to be the case.

    Simplicial complexes give simplicial sets either by choosing a total order on the vertices or by taking multiple copies of each simplex one for each possible order, and then forming degeneracies. Suppose that GG acts on KK and we denote by K simpK^{simp} the corresponding simplicial set. We can form (K/G) simp(K/G)^{simp} and (K simp)/G(K^{simp})/G and they are not in general isomorphic. There is a simplicial map from (K simp)/G(K^{simp})/G to (K/G) simp(K/G)^{simp} and these would seem to be ’onto’. It it 1-1 (I think) if and only if the action is ’regular’.

    (The action is regular if given elements g 0,,g nGg_0, \ldots, g_n\in G and a simplex, σ={v 0,,v n}\sigma = \{v_0,\ldots, v_n\} of KK such that τ={v 0g 0,,v ng n}\tau = \{v_0 g_0,\ldots, v_n g_n\} is also a simplex of KK, then there is a single element, gGg\in G such that σg=τ\sigma\cdot g = \tau.)

    Any thoughts on the categorical interpretation of ’regular’. Simplicial complexes are strange categorically so it may be not obvious. (The usual treatments of regularity use the geometric realisation not the simplicialisation (or whetever this functor should be called.))