∞-categories · The Lumenary

MotivationH

Why the homotopy category is not enough

Homotopy theory studies spaces up to homotopy equivalence, chain complexes up to quasi-isomorphism, and many similar situations where "sameness" is weaker than isomorphism. The classical move is to pass to a homotopy category: keep the objects and replace morphisms by homotopy classes. This is useful, but it throws away essential information.

Homotopy colimits are not colimits in the homotopy category. Gluing two discs along their boundary circle gives $S^2$ , the homotopy pushout of $* \leftarrow S^1 \rightarrow *$ . Every map $S^2 \to Y$ restricts to null-homotopic maps on the two discs, so in the homotopy category all maps $S^2 \to Y$ look the same on the pieces — yet for simply connected $Y$ their homotopy classes correspond to the elements of $\pi_2(Y)$ . The missing information is which homotopy glues the pieces along the circle. In fact the homotopy category of spaces does not have all pushouts.
Constructions are not functorial. In a triangulated category the cone of a morphism is unique only up to non-unique isomorphism (see Stable ∞-categories).
Gluing fails. Derived categories of sheaves on the pieces of an open cover cannot, in general, be glued to the derived category of the whole space at the level of triangulated categories.

The cure is to remember the homotopies, the homotopies between homotopies, and so on: an $(\infty, 1)$ -category has objects, morphisms, 2-morphisms between morphisms, 3-morphisms, …, with all $k$ -morphisms for $k \ge 2$ invertible. Concretely, between any two objects there is a space of morphisms, and composition is defined up to a contractible space of choices. This lesson builds the most widely used model, quasi-categories (Boardman and Vogt's weak Kan complexes, developed by Joyal and Lurie), which are special simplicial sets. It assumes Higher categories and Algebraic topology.

IntuitionH

Triangles that witness composition

In a category, $h = g \circ f$ is a property of three morphisms. In an ∞-category it is data: a 2-dimensional cell $\sigma$ with boundary $f$ , $g$ , $h$ , a "witness" (or homotopy) that $h$ is a composite of $f$ and $g$ . Witnesses for associativity are 3-dimensional cells, and so on. Simplices are the natural shapes for this bookkeeping: a 2-simplex is a triangle with edges $f : x_0 \to x_1$ , $g : x_1 \to x_2$ , $h : x_0 \to x_2$ .

Left: an inner horn Λ²₁ — two composable edges f and g. Filling it means choosing a composite h together with a triangle σ witnessing it. Right: an outer horn Λ²₀ — edges f and h out of the same vertex. Filling it means finding g with g∘f = h, i.e. dividing by f, which is possible for all f only when f can be inverted.

This picture explains the whole definition. A Kan complex is a simplicial set in which every horn can be filled: a model for spaces, or ∞-groupoids, where every morphism is invertible up to homotopy. A quasi-category asks only for inner horns to be filled: composites always exist, but morphisms need not be invertible. Fillers are not required to be unique — only, as we shall see, unique up to a contractible space of choices.

DefinitionsH

Simplicial sets, horns and quasi-categories

Definition— simplex category and simplicial set

The simplex category $\Delta$ has objects the finite ordinals $[n] = \{0 < 1 < \dots < n\}$ for $n \ge 0$ and morphisms the order-preserving maps. A simplicial set is a functor $X : \Delta^{\mathrm{op}} \to \mathbf{Set}$ . We write $X_n = X([n])$ for the set of $n$ -simplices, $d_i : X_n \to X_{n-1}$ for the face maps (induced by the injection $[n-1] \to [n]$ that skips $i$ ) and $s_i : X_n \to X_{n+1}$ for the degeneracy maps (induced by the surjection $[n+1] \to [n]$ that repeats $i$ ). A simplex is degenerate if it is in the image of some $s_i$ .

Definition— standard simplex, boundary and horn

The standard $n$ -simplex is the representable simplicial set $\Delta[n] = \mathrm{Hom}_\Delta(-, [n])$ . By the Yoneda lemma, maps $\Delta[n] \to X$ correspond to $n$ -simplices of $X$ . The boundary $\partial\Delta[n] \subset \Delta[n]$ is the union of the faces $d_i\Delta[n]$ , and for $0 \le k \le n$ the horn $\Lambda^n_k \subset \Delta[n]$ is the union of all faces except the $k$ -th. A horn is inner if $0 < k < n$ and outer if $k = 0$ or $k = n$ . A map $\Lambda^n_k \to X$ is a family of $(n-1)$ -simplices $(x_i)_{i \ne k}$ with $d_i x_j = d_{j-1} x_i$ for all $i < j$ with $i, j \ne k$ . A filler is an $n$ -simplex $x$ with $d_i x = x_i$ for $i \ne k$ .

Definition— Kan complex and quasi-category

A simplicial set $X$ is a Kan complex if every horn $\Lambda^n_k \to X$ ( $n \ge 1$ , $0 \le k \le n$ ) has a filler, and a quasi-category (or ∞-category, in Joyal and Lurie's terminology) if every inner horn $\Lambda^n_k \to X$ ( $0 < k < n$ ) has a filler.

Definition— nerve and singular complex

The nerve $N(\mathcal{C})$ of a category has $N(\mathcal{C})_n =$ the set of functors $[n] \to \mathcal{C}$ , that is, strings $c_0 \to c_1 \to \dots \to c_n$ of $n$ composable morphisms; $d_i$ composes at $c_i$ (or drops $c_0$ or $c_n$ ), and $s_i$ inserts an identity. The singular complex $\mathrm{Sing}(T)$ of a topological space has $\mathrm{Sing}(T)_n =$ the set of continuous maps $|\Delta^n| \to T$ from the topological $n$ -simplex.

Definition— objects, morphisms and homotopy in a quasi-category

Let $X$ be a quasi-category. Its objects are its vertices; a morphism $f : x \to y$ is an edge with $d_1 f = x$ and $d_0 f = y$ ; the identity of $x$ is $s_0 x$ . Two morphisms $f, f' : x \to y$ are homotopic, $f \simeq f'$ , if there is a 2-simplex $\sigma$ with $d_2 \sigma = f$ , $d_1 \sigma = f'$ and $d_0 \sigma = \mathrm{id}_y$ . Given $f : x \to y$ and $g : y \to z$ , a composite of $g$ and $f$ is any edge $h = d_1\sigma$ where $\sigma$ fills the inner horn $(d_0, d_2) = (g, f)$ .

Definition— homotopy category

The homotopy category $hX$ of a quasi-category $X$ has the vertices of $X$ as objects and homotopy classes of morphisms as morphisms, with $[g] \circ [f] = [h]$ for any composite $h$ of $g$ and $f$ . A morphism of $X$ is an equivalence if its class is an isomorphism in $hX$ .

Worked examplesH

Examples and non-examples

Example— Nerves: quasi-categories with unique fillers

In $N(\mathcal{C})$ a horn $\Lambda^2_1$ is a composable pair $(f, g)$ , and its only filler is the string $(f, g)$ itself, whose long edge is $g \circ f$ . Theorem 1 below shows that every inner horn in a nerve has exactly one filler. Outer horns are another matter: a horn $\Lambda^2_0$ given by $f : x_0 \to x_1$ and $h : x_0 \to x_2$ fills exactly when there is $g$ with $g \circ f = h$ . In a groupoid take $g = h \circ f^{-1}$ ; in the poset $[1] = \{0 < 1\}$ the horn with $f : 0 \to 1$ and $h = \mathrm{id}_0$ has no filler, since there is no morphism $1 \to 0$ .

Example— Singular complexes: Kan complexes

For any space $T$ , $\mathrm{Sing}(T)$ is a Kan complex: a horn is a continuous map from a union of faces of $|\Delta^n|$ , and there is a retraction $|\Delta^n| \to |\Lambda^n_k|$ along which it extends. Its homotopy category is the fundamental groupoid $\Pi_1(T)$ : objects are points, and morphisms are homotopy classes of paths relative to their endpoints. $\mathrm{Sing}(T)$ is not a nerve once $T$ has a non-constant path: "concatenate at the midpoint" and "concatenate at time one third" give different edges, and both are long edges of fillers of the same inner horn.

Example— Simplicial groups

Every simplicial group (a functor $\Delta^{\mathrm{op}} \to \mathbf{Grp}$ ) is a Kan complex (Moore). Through the Dold–Kan correspondence, simplicial abelian groups are equivalent to chain complexes concentrated in non-negative degrees, so chain complexes give a large supply of Kan complexes whose homotopy groups are homology groups.

Example— Non-examples

The horn $\Lambda^2_1$ itself, viewed as a simplicial set, is not a quasi-category: the identity map $\Lambda^2_1 \to \Lambda^2_1$ is an inner horn with no filler (there is no non-degenerate 2-simplex, and the long edge is missing).
The boundary $\partial \Delta[2]$ is not a quasi-category: the inner horn $\Lambda^2_1 \hookrightarrow \partial\Delta[2]$ would need a 2-simplex of $\partial\Delta[2]$ with three distinct vertices, and every 2-simplex of $\partial\Delta[2]$ is degenerate. (The long edge exists but nothing witnesses that it is a composite.)
$N([1])$ is a quasi-category (a nerve) but not a Kan complex (an outer horn fails, as above). This is proved in Lean below.

ProofH

Central theorems

Theorem— Theorem 1 (nerves and unique fillers)

A simplicial set is isomorphic to the nerve of a category if and only if every inner horn $\Lambda^n_k \to X$ ( $0 < k < n$ ) has a unique filler. In particular, nerves are quasi-categories.

Proof

We prove that nerves have unique inner fillers, and sketch the converse.

Spines. An $n$ -simplex of $N(\mathcal{C})$ is a functor $[n] \to \mathcal{C}$ , which is determined by its spine: the objects $c_0, \dots, c_n$ and the $n$ arrows $c_{i-1} \to c_i$ (all other arrows $c_i \to c_j$ are composites). So two simplices of the nerve with the same spine are equal.

Uniqueness. Let $0 < k < n$ and let $h : \Lambda^n_k \to N(\mathcal{C})$ be a horn. Each spine edge $\{i-1, i\}$ of $\Delta[n]$ lies in a face $d_j\Delta[n]$ with $j \notin \{i-1, i\}$ and $j \ne k$ : for $n = 2$ , $k = 1$ the edges $\{0,1\}$ and $\{1,2\}$ lie in $d_2$ and $d_0$ ; for $n \ge 3$ there are at least two vertices outside $\{i-1, i\}$ , and one of them differs from $k$ . So $h$ determines the whole spine, and any two fillers have the same spine; hence there is at most one filler.

Existence. Let $x$ be the $n$ -simplex whose spine is the one determined by $h$ . We must show $d_j x = h_j$ for every $j \ne k$ ; since both are simplices of the nerve, it suffices to compare spines. If $j = 0$ or $j = n$ , the spine of $d_j x$ consists of spine edges of $x$ , which agree with $h$ . If $0 < j < n$ , the spine of $d_j x$ also contains the edge $\{j-1, j+1\}$ , which in $x$ is the composite of the edges $\{j-1, j\}$ and $\{j, j+1\}$ . In $h$ this edge is also that composite, because the triangle $\{j-1, j, j+1\}$ lies in the horn: it is contained in a face $d_m\Delta[n]$ with $m \notin \{j-1, j, j+1\}$ and $m \ne k$ (for $n \ge 4$ there are at least two choices of $m$ ; for $n = 3$ the only choice is $m \in \{0, 3\}$ , which is not $k$ because $k$ is inner; for $n = 2$ the case $0 < j < n$ , $j \neq k$ does not occur), and $h$ restricted to it is a 2-simplex of the nerve, that is, a commuting triangle. So $d_j x$ and $h_j$ have the same spine, and are equal.

Converse (sketch). If every inner horn of $X$ has a unique filler, define a category with objects $X_0$ , morphisms $X_1$ , identities $s_0 x$ , and $g \circ f :=$ the long edge of the unique filler of the horn $(f, g)$ . The unit laws come from degenerate 2-simplices, and associativity from unique fillers of $\Lambda^3_1$ (or $\Lambda^3_2$ ): the four faces of the filling 3-simplex show that the two bracketings agree. Unique filling in higher dimensions then shows by induction that every simplex of $X$ is determined by its spine, which identifies $X$ with the nerve. (Details: Lurie, Higher Topos Theory, §1.1.2, or Joyal's notes.)

∎

Theorem— Theorem 2 (composites are unique up to homotopy)

Let $X$ be a quasi-category, $f : x \to y$ and $g : y \to z$ . If $h$ and $h'$ are both composites of $g$ and $f$ , then $h \simeq h'$ .

Proof

Let $\sigma$ and $\sigma'$ be 2-simplices with $(d_2, d_0, d_1)$ equal to $(f, g, h)$ and $(f, g, h')$ . Build a horn $\Lambda^3_1 \to X$ on vertices $x, y, z, z$ by prescribing three faces:

$d_3 = \sigma$ on vertices $\{0, 1, 2\}$ , with edges $f$ , $g$ , $h$ ;
$d_2 = \sigma'$ on vertices $\{0, 1, 3\}$ , with edges $f$ , $g$ , $h'$ ;
$d_0 = s_1 g$ on vertices $\{1, 2, 3\}$ , the degenerate 2-simplex with edges $g$ , $\mathrm{id}_z$ , $g$ (by the simplicial identities $d_2 s_1 g = d_1 s_1 g = g$ and $d_0 s_1 g = s_0 d_0 g = \mathrm{id}_z$ ).

They agree on shared edges: $\{0, 1\}$ is $f$ in $\sigma$ and $\sigma'$ ; $\{1, 2\}$ is $g$ in $\sigma$ and $s_1 g$ ; $\{1, 3\}$ is $g$ in $\sigma'$ and $s_1 g$ . The horn $\Lambda^3_1$ is inner, so it has a filler $\tau$ . Its missing face $d_1 \tau$ , on vertices $\{0, 2, 3\}$ , has edges $d_2 = h$ (edge $\{0,2\}$ ), $d_0 = \mathrm{id}_z$ (edge $\{2, 3\}$ ) and $d_1 = h'$ (edge $\{0, 3\}$ ): by definition it is a homotopy $h \simeq h'$ .

∎

Arguments of the same kind, with other horns $\Lambda^3_1$ and $\Lambda^3_2$ , show that $\simeq$ is an equivalence relation and that composition of homotopy classes is well defined, associative and unital (Boardman and Vogt, 1973; Joyal). So $hX$ is a category. For a nerve, $h N(\mathcal{C}) \cong \mathcal{C}$ ; for $\mathrm{Sing}(T)$ , $h\,\mathrm{Sing}(T) \cong \Pi_1(T)$ . The functor $X \mapsto hX$ is left adjoint to the nerve.

Theorem— Theorem 3 (Joyal; stated)

A simplicial set $X$ is a quasi-category if and only if the restriction map $\mathrm{Fun}(\Delta[2], X) \to \mathrm{Fun}(\Lambda^2_1, X)$ is a trivial Kan fibration.

Here $\mathrm{Fun}(A, X)$ is the simplicial set of maps $A \times \Delta[n] \to X$ . The fibre over a composable pair $(f, g)$ is the space of all pairs (composite, witnessing 2-simplex), and the theorem says it is contractible (non-empty, connected, with trivial higher homotopy). This is the precise meaning of "composition is defined up to a contractible space of choices". See Lurie, Higher Topos Theory, §2.3, where it is attributed to Joyal.

Theorem— Theorem 4 (Joyal, 2002; stated)

A quasi-category $X$ is a Kan complex if and only if its homotopy category $hX$ is a groupoid.

So Kan complexes are exactly the ∞-categories in which every morphism is an equivalence: ∞-groupoids. One direction is easy to believe from the picture of outer horns: filling $\Lambda^2_0$ produces left inverses up to homotopy. The converse — that homotopy invertibility of edges forces all outer horns in all dimensions to fill — is Joyal's theorem ("Quasi-categories and Kan complexes", J. Pure Appl. Algebra 175, 2002).

Theorem— Theorem 5 (the homotopy hypothesis for Kan complexes; stated)

Geometric realisation and the singular complex form a Quillen equivalence between simplicial sets with the Kan–Quillen model structure (fibrant objects: Kan complexes) and topological spaces (Quillen, 1967). Consequently the homotopy theory of Kan complexes is equivalent to the homotopy theory of CW complexes.

This is the established, precise form of Grothendieck's homotopy hypothesis, "∞-groupoids are spaces", for one particular definition of ∞-groupoid. For other definitions (algebraic weak ∞-groupoids in the style of Batanin or Grothendieck) the corresponding statement is a guiding principle, proved in some cases and open in others.

Theorem— Theorem 6 (models of (∞,1)-categories; stated)

The following model categories are connected by chains of Quillen equivalences: quasi-categories (Joyal's model structure on simplicial sets), complete Segal spaces (Rezk, 2001), Segal categories, and simplicial categories (Bergner's model structure). The comparisons are due to Joyal and Tierney (2007), Bergner (2007) and Lurie (Higher Topos Theory, 2009), the last via the homotopy coherent nerve of Cordier.

Toën (2005) characterised the homotopy theory of $(\infty, 1)$ -categories axiomatically, so that any model satisfying his axioms is equivalent to these. This is why results proved in one model can be used in another, and why Riehl and Verity can develop ∞-category theory "model-independently" in an ∞-cosmos.

ExplanationH

The landscape: what is known and what is open

Established. Inside quasi-categories, Joyal and Lurie developed the main tools of category theory: joins and slices, limits and colimits, adjunctions, Kan extensions, the Yoneda lemma, presentable ∞-categories and ∞-topoi, and the straightening–unstraightening equivalence (an ∞-categorical Grothendieck construction) (Lurie, Higher Topos Theory, 2009; Cisinski, 2019; Riehl–Verity, 2022; Lurie's online Kerodon). This machinery underlies Lurie's Higher Algebra (stable ∞-categories, $E_n$ -algebras), derived and spectral algebraic geometry (Toën–Vezzosi; Lurie), modern algebraic K-theory, factorization homology and much of contemporary homotopy theory.

Established but subtle. The ∞-categorical Yoneda lemma and the existence of (co)limits are theorems, but their proofs in quasi-categories are long; the same statements are short in an ∞-cosmos (Riehl–Verity), at the price of building the framework first.

Open or in progress.

$(\infty, n)$ -categories. There are many models (Rezk's $\Theta_n$ -spaces, $n$ -fold complete Segal spaces, complicial sets, …), and Barwick and Schommer-Pries (2021) gave axioms that characterise the homotopy theory of $(\infty, n)$ -categories up to equivalence; comparison of all proposed models is ongoing.
Synthetic and formal foundations. Riehl and Shulman (2017) proposed a simplicial homotopy type theory in which ∞-categories are types with a "Segal" property; developing it fully, and formalising ∞-category theory in proof assistants (Mathlib has quasi-categories, nerves and the homotopy category; a community project is formalising Riehl–Verity's ∞-cosmoi in Lean), are active, open-ended projects.
Weak ∞-groupoids and the homotopy hypothesis for algebraic definitions of ∞-groupoid (Theorem 5 and the remark after it).

Formal mathematicsF

Quasi-categories in Lean and Mathlib

Mathlib defines simplicial sets (SSet), standard simplices Δ[n], horns Λ[n, i], Kan complexes (as fibrant objects of the Kan–Quillen model structure) and quasi-categories (SSet.Quasicategory, inner horn filling). It proves that Kan complexes are quasi-categories and that nerves are quasi-categories (via the strict Segal condition: simplices are determined by spines, as in Theorem 1), and it constructs the homotopy category functor, left adjoint to the nerve. In Mathlib the nerve composes diagrammatically: the 2-simplex with edges $f : x_0 \to x_1$ , $g : x_1 \to x_2$ , $h : x_0 \to x_2$ (an Edge.CompStruct) witnesses f ≫ g = h, which is $g \circ f = h$ .

The definition, unfolded, and the two basic library facts:

quasicategory_iffLean 4Proved here from basic factsFormally verified

/-- **What a quasi-category is.** A simplicial set `S` is a quasi-category exactly when every
*inner* horn `Λ[n, i] ⟶ S` (with `0 < i < n`) extends along the inclusion `Λ[n, i] ⟶ Δ[n]`.
Mathlib's class `SSet.Quasicategory` states this for `n + 2` (inner horns only exist from
dimension 2); here it is unfolded into the form used in the lesson. -/
theorem quasicategory_iff (S : SSet) :
    S.Quasicategory ↔
      ∀ (n : ℕ) (i : Fin (n + 1)) (σ₀ : (Λ[n, i] : SSet) ⟶ S),
        0 < i → i < Fin.last n → ∃ σ : Δ[n] ⟶ S, σ₀ = Λ[n, i].ι ≫ σ := by
  constructor
  · intro hS n i σ₀ h0 hn
    exact Quasicategory.hornFilling h0 hn σ₀
  · intro h
    exact ⟨fun n i σ₀ h0 hn => h (n + 2) i σ₀ h0 hn⟩