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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 7th 2010
   • (edited Jun 7th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexA discussion of the cartesian closed monoidal structure on an (oo,1)-topos is currently missing on the nLab. I started making a first step in the direction of including it: * at [[model structure on simplicial presheaves]] I added a section <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#MonoidalStructure">Closed monoidal structure</a> with a pointer to Toen's lectures (where the following is an exercise) and a statement and proof of how $[C^{op},sSet]_{proj}$ is a monoidal model category by the Cartesian product. * as a lemma for that I added to [[Quillen bifunctor]] the statement that on cofib generated model cats a Quillen bifunctor property is checked already on generating cofibrations (<a href="http://ncatlab.org/nlab/show/Quillen+bifunctor#Properties">here</a>). More later...

   A discussion of the cartesian closed monoidal structure on an (oo,1)-topos is currently missing on the nLab.

   I started making a first step in the direction of including it:

   • at model structure on simplicial presheaves I added a section Closed monoidal structure with a pointer to Toen’s lectures (where the following is an exercise) and a statement and proof of how $[C^{op},sSet]_{proj}$ is a monoidal model category by the Cartesian product.

   • as a lemma for that I added to Quillen bifunctor the statement that on cofib generated model cats a Quillen bifunctor property is checked already on generating cofibrations (here).

   More later…

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 8th 2010
   • (edited Jun 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexa little more: * added to [[Quillen adjunction]] in the section <a href="http://ncatlab.org/nlab/show/Quillen+adjunction#sSet">For sSet-adjunctions</a> the powerful lemma that for an sSet-adjunction between simplicial model categories into a left proper one to be Quillen it is sufficient that the left adjoint preserves cofibrations and the right adjoint just fibrant objects; * used this to add at [[model structure on simplicial presheaves]] in the section <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#MonoidalStructure">Closed monoidal structure</a> the proof of the statement that for cofibrant $X$ the adjunction $(X \times (-) \dashv [X,-])$ is Quillen on the Cech-localization of the projective model structure.

   a little more:

   • added to Quillen adjunction in the section For sSet-adjunctions the powerful lemma that for an sSet-adjunction between simplicial model categories into a left proper one to be Quillen it is sufficient that the left adjoint preserves cofibrations and the right adjoint just fibrant objects;

   • used this to add at model structure on simplicial presheaves in the section Closed monoidal structure the proof of the statement that for cofibrant $X$ the adjunction $(X \times (-) \dashv [X,-])$ is Quillen on the Cech-localization of the projective model structure.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 8th 2010
   • (edited Jun 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex... and finally used this to prove that on a site with products and geometrically contractible objects <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#UnstructuredProperties">the path oo-groupoid functor preserves products</a>.

   … and finally used this to prove that on a site with products and geometrically contractible objects the path oo-groupoid functor preserves products.