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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 26th 2010
    • (edited Aug 26th 2010)

    I fixed the definition at over quasi-category so it makes the adjointness relationship clearer between overcategories and joins. In particular, Lurie’s notation and definition makes it very hard to see this. It’s much easier to see what’s going on when we look at things as follows: The join with KK fixed in the first coordinate, Si K KS:KKSS\mapsto i_K^{K\star S}: K\to K\star S, where i K KSi_K^{K\star S} is the canonical inclusion, is a functor SSet(KSSet)SSet\to (K\downarrow SSet). Then the undercategory construction gives the adjoint to this functor sending (KSSet)SSet(K\downarrow SSet) \to SSet. This makes it substantially clearer to understand what’s going on, since Hom SSet(S,X F/):=Hom KSSet(i K KS,F)Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F) is the set of those maps f:KSXf:K\star S\to X such that f|K=fi K KS=Ff|K=f\circ i_K^{K\star S}=F.

    Lurie’s notation Hom SSet(S,X F/):=Hom F(KS,X)Hom_{SSet}(S, X_{F/}):= Hom_F(K\star S, X) is nonstandard and inferior, since it obscures the obvious adjointness property.

    The definition for overcategories is “dual” (by looking at the join of KK on the right).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2010

    Good point.

    I have slightly edited the paragraph further, to make it now read

    a natural bijection of hom-sets

    Hom sSet(S,C F/)(Hom (KSSet))(i K,S,F), Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,

    where i K,S:KKSi_{K,S} : K \to K \star S is the canonical inclusion of KK into its join of simplicial sets with SS.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 26th 2010

    Hey Urs, how do you get that little curved i that stands for “canonical inclusion” in LaTeX?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2010

    Maybe you mean iota ι\iota?

    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 26th 2010

    Yes, that’s the one. Thanks!