Definition:
- By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The category of simplicial sets is denoted $\mathrm{sSet}$. For integer $k\geq0$ let $\Delta^k$ denote the simplicial set represented as a presheaf by $[k] = (\{0,1,2,\ldots,k\},\leq)$; we call this the (simplicial set model of the) $k$-simplex.
- For integer $k\geq0$ let $L_k \subset \Delta^k$ denote the spine of the $k$-simplex.
- Let us say a given simplicial set $X$ has property $\mathscr{P}$ (resp. $\mathscr{Q}$) if the restriction function $\mathrm{Hom}_{\text{sSet}}(\Delta^k , X) \rightarrow \mathrm{Hom}_{\text{sSet}}(L_k , X)$ is surjective (resp. injective).
My question is, could anyone provide an example of a quasicategory $X$ which fails to have property $\mathscr{P}$? What about property $\mathscr{Q}$?
Note: according to this answer, if $X = N(C)$ is the nerve of an ordinary 1-category $C$, then $N(C)$ has both properties $\mathscr{P}$ and $\mathscr{Q}$.