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Sep 3, 2017 at 20:02 comment added Daniel Gerigk Perhaps it's worth mentioning that the functor in $(\operatorname{Set}_{\Delta})_{/S} \to (\operatorname{Set}_{\Delta})_{/S^{\triangleright}}$ that sends an object $(X,p : X \to S)$ to the object $(S \sqcup_X X^{\triangleright},\dots)$ is fully faithful, and its essential image contains precisely those objects $(Y, q : Y \to S^{\triangleright})$ that have the property that $q$ is an isomorphism over $S \sqcup \{*\}$.
Aug 28, 2011 at 13:14 comment added Akhil Mathew This is what I was looking for. Thanks.
Aug 28, 2011 at 13:14 vote accept Akhil Mathew
Aug 28, 2011 at 12:28 history edited Jacob Lurie CC BY-SA 3.0
added 5 characters in body
Aug 28, 2011 at 12:20 history answered Jacob Lurie CC BY-SA 3.0