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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 )? 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 -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 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 is a bicategorical equivalence. Then is a weak equivalence of enriched -categories. Hence there is a Cartesian equivalence between the mapping spaces of and .