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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 $0<i<k$. I.e., $N(C)$ is a quasicategory.
  • We note also that if $k\geq2$ and $0<i <k$, 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.
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1 Answer 1

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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.

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  • $\begingroup$ Thank you, this is very helpful! $\endgroup$ Aug 16, 2022 at 17:20

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