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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 24th 2010
    • (edited Aug 24th 2010)

    Let y:Δ 0Sy:\Delta^0\to S be a vertex of a quasicategory SS. According to the proof of HTT Theorem 2.2.4.1, we can show that St Sy(x)Map [S](x,y)St_S y (x)\cong Map_{\mathfrak{C}[S]}(x,y) for any vertex xx of SS. How can we show this? We can see by definition that St Sy(x):=Map M(x,v)St_S y (x):= Map_M(x,v) where MM is the simplicial category given by the pushout

    [(Δ 0) ] [Δ 0][S],\mathfrak{C}[(\Delta^0)^\triangleright]\coprod_{\mathfrak{C}[\Delta^0]} \mathfrak{C}[S],

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

    (Δ 0) Δ 0SS?(\Delta^0)^\triangleright \coprod_{\Delta^0} S \to S?

    And does this natural map exist whenever, for example in the case X XSX^\triangleright \coprod_X S, XX has a terminal object (or strongly terminal object)?