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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 26th 2010
   • (edited Aug 26th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI 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 $K$ fixed in the first coordinate, $S\mapsto i_K^{K\star S}: K\to K\star S$, where $i_K^{K\star S}$ is the canonical inclusion, is a functor $SSet\to (K\downarrow SSet)$. Then the undercategory construction gives the adjoint to this functor sending $(K\downarrow SSet) \to SSet$. This makes it substantially clearer to understand what's going on, since $Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F)$ is the set of those maps $f:K\star S\to X$ such that $f|K=f\circ i_K^{K\star S}=F$. Lurie's notation $Hom_{SSet}(S, X_{F/}):= Hom_F(K\star S, X)$ is nonstandard and inferior, since it obscures the obvious adjointness property. The definition for overcategories is "dual" (by looking at the join of $K$ on the right).

   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 $K$ fixed in the first coordinate, $S\mapsto i_K^{K\star S}: K\to K\star S$, where $i_K^{K\star S}$ is the canonical inclusion, is a functor $SSet\to (K\downarrow SSet)$. Then the undercategory construction gives the adjoint to this functor sending $(K\downarrow SSet) \to SSet$. This makes it substantially clearer to understand what’s going on, since $Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F)$ is the set of those maps $f:K\star S\to X$ such that $f|K=f\circ i_K^{K\star S}=F$.

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

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2010
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   Author: Urs
   Format: MarkdownItexGood point. I have slightly edited the paragraph further, to make it now read > a natural bijection of hom-sets > $$ Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,$$ > where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its [[join of simplicial sets]] with $S$.

   Good point.

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

   a natural bijection of hom-sets

   $$Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,$$

   where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its join of simplicial sets with $S$.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 26th 2010
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   Author: Harry Gindi
   Format: MarkdownItexHey Urs, how do you get that little curved i that stands for "canonical inclusion" in LaTeX?

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe you mean iota $\iota$?

   Maybe you mean iota $\iota$?

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 26th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexYes, that's the one. Thanks!

   Yes, that’s the one. Thanks!