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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2010
    • (edited Jun 7th 2010)

    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[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…

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2010
    • (edited Jun 8th 2010)

    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 XX the adjunction (X×()[X,])(X \times (-) \dashv [X,-]) is Quillen on the Cech-localization of the projective model structure.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2010
    • (edited Jun 8th 2010)

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