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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2009
   • (edited Dec 15th 2009)
   • PermaLink
   Author: Urs
   Format: Markdowncreated a section <a href="http://ncatlab.org/nlab/show/lined+topos#ContractibleObjects">Contractible objects</a> at [[lined topos]]. This introduces and discusses a bit a notion of objects being contractible with respect to a specified line object (maybe the section deserves to be at [[interval object]] instead, not sure). This notion is something I made up, so review critically. I am open for suggestions of different terminology. The concept itself, simple as it is (though not entirely trivial), I need for the discussion of path oo-groupoids of oo-stacks on my personal web: if a lined Grothendieck topos <latex> (\mathcal{T} = Sh(C),R) </latex> is such that all representable objects are contractible with respect to the line object <latex> R</latex>, then the path oo-groupoid functor <latex> \Pi : SSh(C) \to SSh(C) </latex> on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks (i.e. respects the local weak equivalences).

   created a section Contractible objects at lined topos.

   This introduces and discusses a bit a notion of objects being contractible with respect to a specified line object (maybe the section deserves to be at interval object instead, not sure).

   This notion is something I made up, so review critically. I am open for suggestions of different terminology. The concept itself, simple as it is (though not entirely trivial), I need for the discussion of path oo-groupoids of oo-stacks on my personal web:

   if a lined Grothendieck topos is such that all representable objects are contractible with respect to the line object , then the path oo-groupoid functor

   

   on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks (i.e. respects the local weak equivalences).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2009
   • PermaLink
   Author: Urs
   Format: Markdown<blockquote> if a lined Grothendieck topos is such that all representable objects are contractible with respect to the line object R, then the path oo-groupoid functor on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks </blockquote> The proof for that, by the way, is now typed up in the section <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#the_local_quillen_adjunction_13">Local Quillen adjunction</a> at <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid">path oo-groupoid</a> on my personal web.

   This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> if a lined Grothendieck topos is such that all representable objects are contractible with respect to the line object R, then the path oo-groupoid functor on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks </blockquote> <p>The proof for that, by the way, is now typed up in the section <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#the_local_quillen_adjunction_13">Local Quillen adjunction</a> at <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid">path oo-groupoid</a> on my personal web.</p> </div>
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 22nd 2009
   • PermaLink
   Author: Urs
   Format: Markdownadded one more example at [[lined topos]] in the section <a href="http://ncatlab.org/nlab/show/lined+topos#ContractibleObjects">Contractible objects</a>: on contractible objects in [[smooth topos]]es

   added one more example at lined topos in the section Contractible objects: on contractible objects in smooth toposes