∞-categories

Which of these horns are inner? (Choose all that apply.)

How many non-degenerate simplices, of all dimensions, does the horn $\Lambda^3_1 \subset \Delta[3]$ contain?

In the nerve $N(\mathcal{C})$, a horn $\Lambda^2_0$ consists of $f : x_0 \to x_1$ and $h : x_0 \to x_2$. When does it have a filler?

The proof that any two composites $h, h'$ of $g$ and $f$ are homotopic fills a 3-dimensional horn whose faces are $\sigma$ (edges $f, g, h$), $\sigma'$ (edges $f, g, h'$) and the degenerate triangle $s_1 g$. Which horn is it, and why does the argument work in every quasi-category?

Which Lean statement says that the nerve of the poset $\{0 < 1\}$ is not an ∞-groupoid, although it is an ∞-category?

Which of the following simplicial sets are quasi-categories? (Choose all that apply.)

What is the homotopy category $h\,\mathrm{Sing}(T)$ of the singular complex of a space $T$?

Let $X$ be a quasi-category in which every edge $f : x \to y$ has a homotopy inverse (some $g : y \to x$ with $[g][f] = [\mathrm{id}_x]$ and $[f][g] = [\mathrm{id}_y]$ in $hX$). What can you conclude?

A one-object category (a monoid) is given by a list of elements and a function then(f, g) meaning 'first $f$, then $g$'. Write outer_fillers(elements, then, f, h) returning the list of all $g$ (in the order of elements) that fill the outer horn $\Lambda^2_0$ with edges $f$ (from vertex 0 to 1) and $h$ (from vertex 0 to 2), i.e. with then(f, g) == h.