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 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 nforum 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 sheaves 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.
   • CommentAuthorUrs
   • CommentTimeOct 15th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownI continued working my way through the lower realms of the [[Whitehead tower]] of the [[orthogonal group]] by creating [[special orthogonal group]] and, yes, [[orthogonal group]]. For the time being the material present there just keeps repeating the Whitehead-tower story. But I want more there, eventually: I have a query box at [[orthogonal group]]. The most general sensible-nonsense context to talk about the orthogonal group should be any [[lined topos]]. I am wondering if there is anything interesting to be said, from that perspective. Incidentally, I was prepared in this context to also have to create [[general linear group]], only to find to my pleasant surprise that Zoran had already created that some time back. And in fact, Zoran discusses there an algebro-geometric perspective on GL(n) which, I think, is actually usefully thought of as the perspective of GL(n) in the lined topos of, at least, presheaves on <latex> CRing^op </latex>. Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as <latex> \mathbb{Z} </latex> and <latex> \mathbb{R} / \mathbb{Z} </latex> and <latex> GL(n) </latex> etc. in [[lined topos]]es and [[smooth topos]]es. Inspired not the least by the wealth of cool structure that even just <latex> \mathbb{Z} </latex> carries in cases such as the <latex> \mathbb{B} </latex>-topos in [[Models for Smooth Infinitesimal Analysis]].

   I continued working my way through the lower realms of the Whitehead tower of the orthogonal group by creating special orthogonal group and, yes, orthogonal group.

   For the time being the material present there just keeps repeating the Whitehead-tower story.

   But I want more there, eventually: I have a query box at orthogonal group. The most general sensible-nonsense context to talk about the orthogonal group should be any lined topos.

   I am wondering if there is anything interesting to be said, from that perspective. Incidentally, I was prepared in this context to also have to create general linear group, only to find to my pleasant surprise that Zoran had already created that some time back. And in fact, Zoran discusses there an algebro-geometric perspective on GL(n) which, I think, is actually usefully thought of as the perspective of GL(n) in the lined topos of, at least, presheaves on .

   Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as and and etc. in lined toposes and smooth toposes. Inspired not the least by the wealth of cool structure that even just carries in cases such as the -topos in Models for Smooth Infinitesimal Analysis.