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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 24th 2010
   • (edited Aug 24th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexLet $y:\Delta^0\to S$ be a vertex of a quasicategory $S$. According to the proof of HTT Theorem 2.2.4.1, we can show that $St_S y (x)\cong Map_{\mathfrak{C}[S]}(x,y)$ for any vertex $x$ of $S$. How can we show this? We can see by definition that $St_S y (x):= Map_M(x,v)$ where $M$ is the simplicial category given by the pushout $$\mathfrak{C}[(\Delta^0)^\triangleright]\coprod_{\mathfrak{C}[\Delta^0]} \mathfrak{C}[S],$$ and $v$ is the image of the cone point. Also, what is the natural map that we have $$(\Delta^0)^\triangleright \coprod_{\Delta^0} S \to S?$$ And does this natural map exist whenever, for example in the case $X^\triangleright \coprod_X S$, $X$ has a terminal object (or strongly terminal object)?

   Let be a vertex of a quasicategory . According to the proof of HTT Theorem 2.2.4.1, we can show that for any vertex of . How can we show this? We can see by definition that where is the simplicial category given by the pushout

   and is the image of the cone point. Also, what is the natural map that we have

   And does this natural map exist whenever, for example in the case , has a terminal object (or strongly terminal object)?