# HTT, 2.1.1.3, Lurie

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.

• 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. – Matt Feller Aug 26 '19 at 1:42
• @MattFeller, thanks, I will think about this. – Bryan Shih Aug 26 '19 at 23:15

## 1 Answer

If you have not already resolve this, I believe this is what Matt means. (Correct me if wrong.)

Consider diagram, $$n \ge 2$$, $$0, $$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.

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