A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy counterexamples.


For n=2, let $X=\Delta[2]$ (which is the nerve of a category): there is a map $\Lambda^0[2]\rightarrow\Delta[2]$ that maps vertices $0\mapsto 1$, $1\mapsto 0$, and $2\mapsto 2$, which can't have a filler because there is no arrow from $1$ to $0$ in $\Delta[2]$.


For $n=3$, any small category $X$ which has a morphism $f$ which is not an epimorphism will give us an unfillable $\Lambda^0[3]$. For example, let $X=N(\Delta)$, and consider the maps $f=d^0\colon [0] \rightarrow [1]$ and $g=d^0\circ s^0\colon [1]\rightarrow [1]$. Then we can construct the horn $\Lambda^0[3]$ where the $d\_1$-face witnesses the composition $g\circ f = d^0 \circ s^0 \circ d^0 = d^0 = f$, and the $d\_2$ and $d\_3$ faces witness $\text{id}\_{[1]}\circ f = f$. If there were a filler for this horn, then we'd have $\text{id}\_{[1]}=g\circ\text{id}\_{[1]}=g$, but $g=d^0\circ s^0$ is not the identity.


For $n\geq 4$, a counterexample can't come from the nerve of a category, because nerves of categories are $2$-coskeletal. I've tried to look at other examples of quasicategories—for example, those given in section 8 of [Rezk's notes on quasicategories](https://faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf)—but I believe I can show the existence of outer horns for $n\geq 4$ in every example given there.


So, are there any known examples of quasicategories with an unfillable $\Lambda^0[4]$?



What is an example of a quasicategory with an outer 4-horn which has no filler?
Let $A$ be any $(2,1)$-category containing a composable pair of morphisms $f\colon a\to b$ and $g \colon b \to c$, and a $2$-cell $\alpha \colon g \to g$ such that $\alpha f = \mathrm{id}\_{gf}$. Then the [Roberts--Street--Duskin nerve](http://tac.mta.ca/tac/volumes/9/n10/9-10abs.html) $NA$ of $A$ contains a $(4,0)$-horn, described below, which admits a filler if and only if the $2$-cell $\alpha$ is an identity.


To describe this horn, let's first label a few simplices in $NA$. Let $\beta$ and $\gamma$ denote the $2$-simplices in $NA$ given by the $2$-cells $\alpha \colon g \to 1\_c\circ g$ and $\mathrm{id}\_{gf} \colon gf \to g\circ f$ in $A$. Let $\varepsilon$ denote the $3$-simplex in $NA$ whose boundary faces are
$d\_0(\varepsilon) = \beta$, $d\_1(\varepsilon) = s\_1(gf)$, $d\_2(\varepsilon) = d\_3(\varepsilon) = \gamma$; this $3$-boundary extends to a $3$-simplex in $NA$ thanks to the equation $\alpha f = \mathrm{id}\_{gf}$ in $A$.


The $(4,0)$-horn of interest is as follows:
$$(-,s\_1(s\_1(gf)),\varepsilon,s\_2(\gamma),s\_2(\gamma)): \Lambda^0[4] \to NA.$$ By $3$-coskeletality of $NA$, this horn extends to a $4$-simplex in $NA$ if and only if the $3$-boundary $$(s\_0(s\_0(c)),\beta,s\_1(g),s\_1(g)) : \partial\Delta[3] \to NA$$ extends to a $3$-simplex in $NA$, which is the case if and only if the $2$-cell $\alpha$ in $A$ is an identity.


