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