# Does the nerve of a category have the following stronger lifting property?

Definition:

• For integer $$k\geq0$$ we let $$\Delta^k := \mathrm{Hom}_{\text{FinitePosets(Nonstrict)}}(-\;,\;[k]=\{0,1,\ldots,k\})$$ denote the usual simplicial-set model of the $$k$$-simplex.

• For integer $$k>0$$, let $$v_i : \Delta^0 \rightarrow \Delta^1$$ be obtained (via Yoneda) from the maps $$[0] \rightarrow [1]$$ sending $$0$$ to $$i$$ for each $$i\in\{0,1\}$$, and define $$L_k$$ to be the following colimit in simplicial sets: $$L_k := \mathrm{colim} \left( \Delta^1 \xleftarrow{v_1} \Delta^0 \xrightarrow{v_0} \Delta^1 \xleftarrow{v_1} \Delta^0 \xrightarrow{v_0} \Delta^1 \cdots \Delta^1 \xleftarrow{v_1} \Delta^0 \xrightarrow{v_0} \Delta^1 \right)$$ where the number of $$\Delta^0$$ factors is $$k-1$$ (so the number of $$\Delta^1$$'s is $$k$$), and the colimit is taken in simplicial sets. (Rmk: we can also define $$L_0 := \Delta^0$$ if we wish.)

So, $$L_k$$ is the simplicial-set model for the directed graph $$\bullet \to \bullet \to \cdots \to \bullet$$ where there are $$k$$ edges (so $$k+1$$ vertices).

• For each integer $$k$$, for integer $$j$$ with $$0\leq j\leq k-1$$, let $$e_j : \Delta^1 \rightarrow \Delta^n$$ be obtained (via Yoneda) from the map $$[1] \rightarrow [n]$$ which sends $$0,1$$ to $$j,j+1$$. These data combine to give a simplicial-set morphism $$L_k \rightarrow \Delta^k$$. This should be a simplicial-set mononomorphism.

We can view $$L_k$$ as a "maximal directed 1-chain (with no vertex repetition)", in $$\Delta^k$$.

• Let us say a given simplicial set $$X$$ has property $$\mathscr{P}$$ if for every integer $$k\geq0$$, every simplicial-set morphism $$L_k \rightarrow X$$ can lift along $$L_k \rightarrow \Delta^k$$ to give some simplicial-set morphism $$\Delta^k \rightarrow X$$.

My question is:

If $$X = N(C)$$ is the standard nerve of a 1-category $$C$$, then does $$X$$ have property $$\mathscr{P}$$?

Notes:

• We already know that if $$X = N(C)$$ then $$X$$ has the lifting property for the inner-horn inclusions $$\Lambda_i^k \rightarrow \Delta^k$$, for $$0. I.e., $$N(C)$$ is a quasicategory.
• We note also that if $$k\geq2$$ and $$0, or $$k\geq3$$ and $$0\leq i \leq k$$, then $$L_k$$ "sits inside" $$\Lambda_i^k$$.
• If $$X = N(C)$$ does have property $$\mathscr{P}$$, then in fact the lifting of a morphism $$L_k \rightarrow N(C)$$ to a morphism $$\Delta^k \rightarrow N(C)$$ should be unique. This is because every $$k$$-cell in $$N(C)$$ is a composable chain of $$k$$ many arrows of $$C$$, the cell determined by the list of arrows.

Sounds like you're describing the spine of a simplex. The inclusion of the spine into the simplex is inner anodyne, so there is a bijection $$\operatorname{Hom}(\Delta^k, X) \to \operatorname{Hom}(L_k, X)$$ whenever $$X$$ is the nerve of an ordinary category; see here. In other words, given a map $$L_k \to X$$, an extension to $$\Delta^k \to X$$ exists and is unique. Briefly, the proof is just to repeatedly use the horn-filling property you mentioned to fill in the rest of the simplex from the spine.