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On Lurie's Higher Topos Theory, Prop. 2.1.1.3,

Let $F:C \rightarrow D$ be a functor between categories. Then $C$ is cofibered in groupoids over $D$ if and only if the induced map $N(F):N(C)\rightarrow N(D)$ is a left fibration of simplicial sets.

In line 1 of proof, Lurie states $N(F)$ is an inner fibration - how does this follow from Prop. 1.1.2.2?

We only know that $N(C)$ and $N(D)$ as objects has LLP for inner horns.

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  • $\begingroup$ Proposition 1.1.2.2 says nerves of categories have unique lifts of inner horns. Try using that fact to prove the following claim: Any map from a quasi-category (i.e., what Lurie calls an $\infty$-category: a simplicial set with LLP with respect to inner horns) to the nerve of a category is an inner fibration. $\endgroup$ – Matt Feller Aug 26 '19 at 1:42
  • $\begingroup$ @MattFeller, thanks, I will think about this. $\endgroup$ – Bryan Shih Aug 26 '19 at 23:15
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If you have not already resolve this, I believe this is what Matt means. (Correct me if wrong.)

Consider diagram, $n \ge 2$, $0<i<n$, $S$ a quasicategory, $$ \require{AMScd} \begin{CD} \Lambda^n_i @>h>> S;\\ @VViV @VVfV \\ \Delta^n @>>g> N(C); \end{CD} $$ $g$ is the unique lift of the morphism $i$ with respect to $fh$. (nerves of category have unique lifts)

Given any lift $k:\Delta^n \rightarrow S$ of upper left diagram: $ki=h$ we have $$fki=fh=gi$$ So $fk=g$ from uniqueness. Thus, $f$ is an inner fibration.

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  • $\begingroup$ Yep! That’s what I had in mind. $\endgroup$ – Matt Feller Nov 22 '19 at 13:15

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