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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.