The Lumenary · Research pathways

∞-categories

Name: ______________________

Date: ______________________

You will learn to

Define simplicial sets, horns, Kan complexes and quasi-categories, and recognise nerves of categories and singular complexes of spaces as examples
Prove that nerves of categories have unique inner horn fillers, and that the nerve of a non-groupoid is a quasi-category but not a Kan complex
Explain why composition in a quasi-category is defined only up to homotopy, prove that any two composites are homotopic, and construct the homotopy category
Survey the models of $(\infty, 1)$ -categories (quasi-categories, complete Segal spaces, simplicial categories) and how they are compared
Separate established results (the homotopy hypothesis for Kan complexes, Joyal's theorems) from guiding principles and open problems

1.
Which of these horns are inner? (Choose all that apply.)
- A. $\Lambda^2_1$
- B. $\Lambda^2_0$
- C. $\Lambda^3_2$
- D. $\Lambda^3_3$
- E. $\Lambda^1_1$
2.
How many non-degenerate simplices, of all dimensions, does the horn $\Lambda^3_1 \subset \Delta[3]$ contain?
3.
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?
- A.Exactly when there is $g : x_1 \to x_2$ with $g \circ f = h$
- B.Always: nerves are quasi-categories
- C.Exactly when $f$ is an isomorphism
- D.Exactly when $x_1 = x_2$
4.
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?
- A. $\Lambda^3_1$ — it is inner, so quasi-categories fill it
- B. $\Lambda^3_0$ — it is outer, so we need a Kan complex
- C. $\Lambda^3_3$ — the missing face contains the long edges
- D. $\partial\Delta[3]$ — all four faces are given
5.
Which Lean statement says that the nerve of the poset $\{0 < 1\}$ is not an ∞-groupoid, although it is an ∞-category?
- A.¬ KanComplex (nerve (Fin 2))
- B.¬ (nerve (Fin 2)).Quasicategory
- C.¬ ∃ f : (1 : Fin 2) ⟶ 0, True
- D.KanComplex (nerve (Fin 2)) → False ∨ True
6.
Which of the following simplicial sets are quasi-categories? (Choose all that apply.)
- A.The nerve of any category
- B. $\mathrm{Sing}(T)$ for a topological space $T$
- C.The standard simplex $\Delta[n]$
- D.The horn $\Lambda^2_1$
- E.The boundary $\partial\Delta[2]$
7.
What is the homotopy category $h\,\mathrm{Sing}(T)$ of the singular complex of a space $T$ ?
- A.The fundamental groupoid $\Pi_1(T)$
- B.The set $\pi_0(T)$ of path components, as a discrete category
- C.The homotopy category of spaces
- D.A category whose morphisms remember $\pi_2(T)$
8.
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. $X$ is a Kan complex (Joyal's theorem)
- B. $X$ is the nerve of a groupoid
- C.Only the outer horns of dimension 2 fill; higher outer horns may fail
- D.Nothing: homotopy inverses say nothing about horn filling
9.
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.
Computational — Practice

Answer key — ∞-categories

1.
$\Lambda^2_1$ and $\Lambda^3_2$ are inner; $\Lambda^2_0$ , $\Lambda^3_3$ and $\Lambda^1_1$ are outer. Inner horns exist only in dimensions $n \ge 2$ .
2.
The horn contains all 4 vertices and all 6 edges of $\Delta[3]$ (every edge lies in two faces, at least one of which is not $d_1$ ), and 3 of the 4 triangles. Total $4 + 6 + 3 = 13$ . In general, for $n \ge 3$ a horn $\Lambda^n_k$ contains the whole 1-skeleton, which is why nerves fill horns of dimension $\ge 3$ uniquely when they fill them at all.
3.
A filler is a 2-simplex $(f, g)$ with long edge $g \circ f = h$ , so it exists exactly when $h$ factors through $f$ . Such $g$ always exists, uniquely, when $f$ is invertible; this is why nerves of groupoids satisfy the outer-horn condition in dimension 2 (and in fact are Kan complexes).
4.
The given faces are $d_0, d_2, d_3$ , so the horn is $\Lambda^3_1$ , which is inner. Every quasi-category fills it, and the missing face $d_1$ — on vertices $0, 2, 3$ with edges $h$ , $\mathrm{id}_z$ , $h'$ — is by definition a homotopy $h \simeq h'$ .
5.
¬ KanComplex (nerve (Fin 2)). Its proof glues the edges $0 \to 1$ and $\mathrm{id}_0$ into a horn $\Lambda^2_0$ , fills it using the Kan property, and reads off an arrow $1 \to 0$ , i.e. a proof of $1 \le 0$ , which is absurd.
6.
Nerves, $\Delta[n] = N([n])$ and $\mathrm{Sing}(T)$ are quasi-categories. $\Lambda^2_1$ and $\partial\Delta[2]$ are not: the inner horn formed by their edges $0 \to 1 \to 2$ would need a non-degenerate 2-simplex, which they do not have.
7.
$h\,\mathrm{Sing}(T) \cong \Pi_1(T)$ . Edges are paths, and two paths with the same endpoints are homotopic in the quasi-categorical sense exactly when they are homotopic relative to their endpoints. It is a groupoid, consistent with Joyal's theorem since $\mathrm{Sing}(T)$ is a Kan complex.
8.
By Joyal's theorem (2002), a quasi-category whose homotopy category is a groupoid is a Kan complex: it fills all horns, inner and outer, in every dimension. It need not be a nerve ( $\mathrm{Sing}(S^2)$ is a counterexample), because Kan complexes may have non-unique fillers.
9.
```
def outer_fillers(elements, then, f, h):
    return [g for g in elements if then(f, g) == h]
```
In the group $\Z/3$ every outer horn has exactly one filler ( $g = h - f$ ), as for any groupoid. In the idempotent monoid the horn $(e, 1)$ has none and the horn $(e, e)$ has two, so its nerve is not a Kan complex — while every inner horn still has exactly one filler.