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1. • CommentRowNumber1.
   • CommentAuthorsanath
   • CommentTimeFeb 11th 2015
   • (edited Feb 11th 2015)
   • PermaLink
   Author: sanath
   Format: MarkdownItexHow does the definition of a bicategorical equivalence as in Definition 3.5.6 of math.harvard.edu/~lurie/… relate to the definition of a categorical equivalence (as in the Cartesian model structure on $\mathrm{Set}^+_\Delta$)? The reason I'm asking is because I'm trying to understand Theorem 4.2.7 of the Goodwillie paper, by comparing it to Proposition 3.1.3.7 of Higher Topos Theory. I mean, what I can't get to understand is this: how does the left adjoint to the scaled nerve construction relate to the equivalence of $\infty$-categories as in Proposition 3.1.3.3 in Higher Topos Theory? (This might be obvious, maybe I'm just not understanding the left adjoint of $\mathrm{N}^\mathrm{sc}$ properly. Also, this has been asked at the [Homotopy Theory Chat Room](http://chat.stackexchange.com/transcript/message/20006508#20006508) and [MathOverflow](http://mathoverflow.net/questions/196261/cartesian-equivalences-and-bicategorical-equivalences), I was just looking for some more input into this.)

   How does the definition of a bicategorical equivalence as in Definition 3.5.6 of math.harvard.edu/~lurie/… relate to the definition of a categorical equivalence (as in the Cartesian model structure on $\mathrm{Set}^+_\Delta$)? The reason I’m asking is because I’m trying to understand Theorem 4.2.7 of the Goodwillie paper, by comparing it to Proposition 3.1.3.7 of Higher Topos Theory. I mean, what I can’t get to understand is this: how does the left adjoint to the scaled nerve construction relate to the equivalence of $\infty$-categories as in Proposition 3.1.3.3 in Higher Topos Theory?

   (This might be obvious, maybe I’m just not understanding the left adjoint of $\mathrm{N}^\mathrm{sc}$ properly. Also, this has been asked at the Homotopy Theory Chat Room and MathOverflow, I was just looking for some more input into this.)

2. • CommentRowNumber2.
   • CommentAuthorsanath
   • CommentTimeFeb 12th 2015
   • PermaLink
   Author: sanath
   Format: MarkdownItexActually, Adeel helped me figure it out. Basically, the relation is as follows: Suppose $f:X\to Y$ is a bicategorical equivalence. Then $\mathfrak{C}^{\mathrm{sc}}(X)\to\mathfrak{C}^{\mathrm{sc}}(Y)$ is a weak equivalence of enriched $\mathrm{Set}^+_\Delta$-categories. Hence there is a Cartesian equivalence between the mapping spaces of $\mathfrak{C}^{\mathrm{sc}}(X)$ and $\mathfrak{C}^{\mathrm{sc}}(Y)$.

   Actually, Adeel helped me figure it out. Basically, the relation is as follows: Suppose $f:X\to Y$ is a bicategorical equivalence. Then $\mathfrak{C}^{\mathrm{sc}}(X)\to\mathfrak{C}^{\mathrm{sc}}(Y)$ is a weak equivalence of enriched $\mathrm{Set}^+_\Delta$-categories. Hence there is a Cartesian equivalence between the mapping spaces of $\mathfrak{C}^{\mathrm{sc}}(X)$ and $\mathfrak{C}^{\mathrm{sc}}(Y)$.