Infinity-Categories and Quasi-Categories

⚡ Key Takeaways

Infinity-categories ( $\infty$ -categories, more precisely $(\infty,1)$ -categories) are categorical structures with objects, morphisms, and an infinite hierarchy of higher morphisms between morphisms, where all $k$ -morphisms for $k \geq 2$ are weakly invertible. Quasi-categories are a concrete, combinatorially tractable model for $(\infty,1)$ -categories given by simplicial sets satisfying the weak Kan (inner horn filling) condition. Key result: the homotopy theory of quasi-categories is a model for the homotopy theory of homotopy theories, equivalent to simplicial categories, complete Segal spaces, and relative categories. Used in derived algebraic geometry, stable homotopy theory, topological field theory, and the theory of $\infty$ -topoi. Introduced by Boardman–Vogt (1973) as "weak Kan complexes," developed by Joyal, and made into a comprehensive theory by Lurie (2009).

💡 The Core Insight

Replace the set of morphisms $\operatorname{Hom}(x,y)$ by a space (an $\infty$ -groupoid), and accept that composition is a contractible choice rather than a function. A quasi-category packages this with a single elegant lifting condition: composability data and witnesses to composition live in the same simplicial set, and the inner-horn-filling axiom simultaneously (a) guarantees composites exist, (b) makes them unique up to a contractible space of choices, and (c) supplies all higher associativity coherences for free. The distinction between " $0$ " and " $1$ " in $(\infty,1)$ is exactly that $1$ -morphisms need not be invertible while everything above is.

DefinitionSimplicial set

A simplicial set is a functor $X \colon \Delta^{\mathrm{op}} \to \mathbf{Set}$ , where $\Delta$ is the simplex category with objects $[n] = \{0 < 1 < \cdots < n\}$ and morphisms the order-preserving maps. We write $X_n := X([n])$ for the set of $n$ -simplices, with face maps $d_i \colon X_n \to X_{n-1}$ and degeneracy maps $s_i \colon X_n \to X_{n+1}$ satisfying the simplicial identities:

1

\quad d_i d_j = d_{j-1} d_i \ (i<j), \qquad s_i s_j = s_{j+1} s_i \ (i \le j), \qquad d_i s_j = \begin{cases} s_{j-1} d_i & i<j \\ \mathrm{id} & i \in \{j, j+1\} \\ s_j d_{i-1} & i > j+1. \end{cases}

Derivation hint: each identity is the image under $X$ of the corresponding relation among coface $\delta^i$ and codegeneracy $\sigma^j$ maps in $\Delta$ , obtained by composing order-preserving maps and reading off which factorization applies.

DefinitionHorns and the inner horn condition

The standard $n$ -simplex is the representable functor $\Delta^n := \operatorname{Hom}_\Delta(-,[n])$ . Its boundary $\partial\Delta^n$ is the union of all faces, and the $k$ -th horn $\Lambda^n_k \subset \Delta^n$ is the union of all faces except the $k$ -th:

2

\quad \Lambda^n_k \;=\; \bigcup_{\,i \ne k\,} d_i(\Delta^{n-1}) \;\subset\; \partial\Delta^n \;\subset\; \Delta^n.

Derivation hint: $\partial\Delta^n$ is the colimit over proper monomorphisms $[m] \hookrightarrow [n]$ ; remove the single face $d_k$ (image of $\delta^k\colon [n-1]\to[n]$ ) from this colimit to obtain the horn. A horn is inner if $0 < k < n$ , and outer if $k \in \{0, n\}$ . A simplicial set $X$ is a quasi-category (or weak Kan complex, or $\infty$ -category) if every inner horn admits a filler: for all $0 < k < n$ and every map $\sigma_0 \colon \Lambda^n_k \to X$ there exists $\sigma \colon \Delta^n \to X$ with $\sigma|_{\Lambda^n_k} = \sigma_0$ :

3

\quad \begin{array}{ccc} \Lambda^n_k & \xrightarrow{\;\sigma_0\;} & X \\ \big\downarrow & \nearrow_{\exists\,\sigma} & \\ \Delta^n & & \end{array} \qquad (0 < k < n).

Derivation hint: a map out of $\Lambda^n_k$ is, by the Yoneda lemma and the horn's colimit description, exactly a compatible family of $(n-1)$ -simplices on the included faces; the filler $\sigma$ supplies the missing face $d_k\sigma$ together with the interior $n$ -simplex.

DefinitionKan complex vs. quasi-category vs. nerve

If $X$ satisfies the filling condition for all horns ( $0 \le k \le n$ ), it is a Kan complex — a model for an $\infty$ -groupoid (a space).
The nerve $N(\mathcal{C})$ of an ordinary category $\mathcal{C}$ has $N(\mathcal{C})_n = \operatorname{Fun}([n],\mathcal{C})$ , the set of length- $n$ composable chains. Then $N(\mathcal{C})$ is a quasi-category, and inner horns have unique fillers.

4

\quad X \text{ is the nerve of a category} \iff \text{every inner horn has a } \textit{unique} \text{ filler.}

Derivation hint: the unique filler condition is the Segal condition with uniqueness, forcing $X_n \xrightarrow{\sim} X_1 \times_{X_0} \cdots \times_{X_0} X_1$ to be a bijection; this exhibits $X$ as determined by objects $X_0$ and a strictly associative composition on $X_1$ .

TheoremJoyal: existence of the Joyal model structure

There is a model structure on the category $\mathbf{sSet}$ of simplicial sets, the Joyal model structure, whose:

cofibrations are the monomorphisms,
fibrant objects are exactly the quasi-categories,
weak equivalences are the categorical equivalences (maps inducing equivalences of homotopy categories after enrichment),
fibrations between fibrant objects are the isofibrations.

The fibrant objects model $(\infty,1)$ -categories, and this model structure is Quillen-equivalent to the model structures on simplicial categories, complete Segal spaces, and Segal categories.

TheoremThe homotopy category

Every quasi-category $X$ has a homotopy category $\mathrm{h}X$ , an ordinary category with $\operatorname{Ob}(\mathrm{h}X) = X_0$ and morphisms the homotopy classes of edges. Two edges $f, g \colon x \to y$ are homotopic if there is a $2$ -simplex exhibiting $f \simeq g$ via a degenerate edge. Composition in $\mathrm{h}X$ is defined by choosing fillers of $\Lambda^2_1$ :

5

\quad g \circ f := d_1(\sigma), \qquad \text{where } \sigma \colon \Delta^2 \to X,\ d_2\sigma = f,\ d_0\sigma = g.

Derivation hint: a $2$ -simplex $\sigma$ has faces $d_2\sigma = f$ ( $0\to1$ ), $d_0\sigma = g$ ( $1\to2$ ), $d_1\sigma$ ( $0\to2$ ); the inner horn $\Lambda^2_1$ is exactly $(f,g)$ , and any filler's long edge $d_1\sigma$ is a composite, well-defined up to homotopy by uniqueness-up-to-contractibility of fillers.

⚠️ Common mistake

The composite $g \circ f$ in [5] is not a function on the nose: it depends on a choice of filler $\sigma$ . What the inner-horn axiom guarantees is that the space of fillers (and more generally the space of all higher coherence data) is contractible, so $g \circ f$ is well-defined in the homotopy category $\mathrm{h}X$ and coherent up to all higher homotopies in $X$ itself. Treating $g\circ f$ as a strict operation is the single most common error.

ProofComposition is associative up to coherent homotopy

We show that the inner-horn condition forces associativity to hold up to homotopy, with the homotopy itself canonical.

Step 1 (Composites exist). Given $f\colon x\to y$ and $g\colon y\to z$ , the pair $(f,g)$ defines a map $\Lambda^2_1 \to X$ . Inner-horn filling yields $\sigma$ with $d_1\sigma =: g\circ f$ . ∎(existence)

Step 2 (Three composables give a $\Lambda^3_1$ or $\Lambda^3_2$ ). Given $f\colon x\to y$ , $g\colon y\to z$ , $h\colon z\to w$ , form the $2$ -simplices $\sigma_{012}$ (witnessing $g\circ f$ ), $\sigma_{123}$ (witnessing $h\circ g$ ), and choose composites to fill in the remaining faces of a $3$ -simplex. Two of the four faces of $\Delta^3$ together with the chosen composites assemble into an inner horn $\Lambda^3_1$ .

Step 3 (Fill the $3$ -horn). Inner-horn filling gives $\tau\colon\Delta^3 \to X$ . Its faces are $2$ -simplices; reading off $d_0\tau$ and $d_3\tau$ identifies the two bracketings:

6

\quad d_2(d_1 \tau) = h\circ(g\circ f), \qquad d_0(d_3\tau)\ \text{relates}\ (h\circ g)\circ f,

and the interior of $\tau$ is precisely a homotopy $h\circ(g\circ f) \simeq (h\circ g)\circ f$ . Derivation hint: enumerate the four faces $d_i\tau$ of the filled $3$ -simplex; each is a $2$ -simplex asserting one composition relation. The Pachner-style "matching" of the four faces around the long edge $0\to3$ is the associator.

Step 4 (Higher coherence). Pentagon-type coherences arise identically from inner $4$ -horns $\Lambda^4_k$ , $0<k<4$ , and inductively all coherences from $\Lambda^n_k$ . The contractibility of the space of fillers (Joyal's theorem) ensures these homotopies are themselves coherent, not merely existent. $\blacksquare$

∎

TheoremMapping spaces are Kan complexes

For a quasi-category $X$ and objects $x,y \in X_0$ , the mapping space $\operatorname{Map}_X(x,y)$ — defined e.g. as the (left) hom-space $\operatorname{Hom}^L_X(x,y)$ or via the simplicial enrichment after fibrant replacement — is a Kan complex. Equivalently, $X$ is a category enriched (up to coherence) in $\infty$ -groupoids.

7

\quad \pi_0 \operatorname{Map}_X(x,y) \;\cong\; \operatorname{Hom}_{\mathrm{h}X}(x,y).

Derivation hint: model $\operatorname{Map}_X(x,y)$ as the fiber of $X^{\Delta^1} \to X^{\partial\Delta^1} = X\times X$ over $(x,y)$ ; the inner-horn condition on $X$ upgrades the inner lifting to all horn lifting on this fiber because both endpoints are pinned, making it a Kan complex.

mathematician · Jacob Lurie | b. 1977

Lurie's two foundational monographs reshaped the subject. Higher Topos Theory (2009) took Joyal's quasi-categories and built the entire apparatus of category theory — (co)limits, adjoint functors, presentable and accessible $\infty$ -categories, $\infty$ -topoi, the $\infty$ -categorical Yoneda lemma — at full generality and rigor. Higher Algebra developed symmetric monoidal $\infty$ -categories, $\mathbb{E}_n$ -algebras, stable $\infty$ -categories, and the foundations of derived/spectral algebraic geometry. Earlier, André Joyal (b. 1943) is the architect of the quasi-categorical model itself: he established the Joyal model structure, the theory of the homotopy coherent nerve, and most of the basic combinatorics, building on the 1973 "weak Kan complexes" of J. Michael Boardman and Rainer Vogt.

🧠 Intuition

Picture an ordinary category as a graph with composition law: dots (objects) and arrows (morphisms), where any two head-to-tail arrows have a unique composite. Now imagine you are no longer allowed to name the composite — instead, whenever you have $f$ then $g$ , you must draw a triangle whose third side is "a composite." Different triangles give different but homotopic composites. An $n$ -simplex in a quasi-category is a coherent diagram of all composites of $n$ arrows, recording not just the answers but the witnesses, the witnesses-of-witnesses, and so on forever. The inner/outer distinction is illuminating: filling an inner horn $\Lambda^2_1 = (f, g)$ means "compose two given composable arrows" — always possible. Filling an outer horn $\Lambda^2_0 = (f, h)$ with $f\colon x\to y$ , $h\colon x\to z$ would mean "find $g\colon y\to z$ with $g\circ f = h$ " — i.e., invert/divide, which a general category cannot do. That outer horns may fail is exactly what makes $1$ -morphisms non-invertible; demanding all horns fill collapses you to a Kan complex (an $\infty$ -groupoid), where every morphism is an equivalence.

Figure

An inner

2

-horn and its filler: composition as triangle-filling

ExampleCounting horn-filling data in the nerve of a poset

Given. Let $P = \{0 < 1 < 2\}$ as a poset (a category). Compute its nerve $N(P)$ low-dimensionally and verify the unique inner-horn condition.

Steps. 1. $N(P)_0 = \operatorname{Fun}([0],P) = \{0,1,2\}$ — three objects. 2. $N(P)_1 = \operatorname{Fun}([1],P) = \{$ order-preserving $[1]\to P\}$ . These are pairs $(a\le b)$ : there are $\binom{3}{2}+3 = 3 + 3 = 6$ (the $3$ nondegenerate edges $0\!\to\!1,\,1\!\to\!2,\,0\!\to\!2$ plus $3$ degenerate identities). 3. $N(P)_2 = \operatorname{Fun}([2],P) = \{(a\le b\le c)\}$ . The unique nondegenerate $2$ -simplex is $(0\le1\le2)$ . 4. Inner horn $\Lambda^2_1$ with $d_2 = (0\to1)$ , $d_0 = (1\to2)$ : the only filler is $(0\le1\le2)$ , and its third face $d_1 = (0\to2)$ .

Boxed answer.

\boxed{\;N(P)_0 = 3,\quad N(P)_1 = 6,\quad \#\{\text{nondeg. } 2\text{-simplices}\} = 1,\quad \text{the } \Lambda^2_1\text{-filler is unique.}\;}

The uniqueness confirms [4]: $N(P)$ is the nerve of a strict category. (Units: pure counts, dimensionless.)

ExampleThe mapping space in a Kan complex:

\pi_1

recovered

Given. Let $X = N(BG)$ where $BG$ is the one-object groupoid of a discrete group $G$ — equivalently view the Kan complex $\overline{W}G$ modeling the classifying space $BG$ . Take the unique object $\ast$ . Compute $\operatorname{Map}_X(\ast,\ast)$ and its homotopy.

Steps. 1. $BG$ has one object and morphisms $G$ , all invertible, so $N(BG)$ is a Kan complex (every morphism is an equivalence ⟹ outer horns fill). 2. By [7], $\pi_0\operatorname{Map}_X(\ast,\ast) = \operatorname{Hom}_{\mathrm{h}X}(\ast,\ast) = G$ as a set. 3. Composition in $\mathrm{h}X$ is the group multiplication; the mapping space is the discrete simplicial set on $G$ , so $\pi_n\operatorname{Map} = 0$ for $n\ge1$ . 4. The space $X$ itself satisfies $\pi_1(X,\ast) = G$ and $\pi_n(X) = 0$ for $n\neq1$ — it is a $K(G,1)$ .

Boxed answer.

\boxed{\;\operatorname{Map}_{N(BG)}(\ast,\ast) \simeq G \ (\text{discrete}),\qquad \pi_1(N(BG)) \cong G,\quad \pi_{n}=0\ (n\ge2).\;}

For $G = \mathbb{Z}/2$ : $|\operatorname{Map}| = 2$ points, $\pi_1 = \mathbb{Z}/2$ . (Units: group elements / homotopy groups, dimensionless.)

⚠️ Common mistake

"A quasi-category is just a simplicial set with extra structure." No — it is a simplicial set satisfying a property (the inner-horn lifting condition). Unlike a topological category or a strictly associative enriched category, no composition function is part of the data. The composition is recovered from the property, and only up to contractible choice. This is why quasi-categories are so flexible: limits, functor categories ( $Y^X = \operatorname{Fun}(X,Y)$ is again a quasi-category when $Y$ is), and slices are constructed by purely simplicial operations.

⚠️ Common mistake

Confusing equivalences in $X$ with isomorphisms in $\mathrm{h}X$ . An edge $f$ in a quasi-category is an equivalence iff it becomes an isomorphism in $\mathrm{h}X$ — but you must check invertibility up to homotopy with witnesses, not on the nose. Joyal's theorem: $X$ is a Kan complex iff $\mathrm{h}X$ is a groupoid iff every edge of $X$ is an equivalence. Do not conflate " $f$ has a one-sided homotopy inverse" with " $f$ is an equivalence" until you have checked the coherence.

CounterexampleOuter horns genuinely fail

Take the nerve $N(\mathcal{C})$ of the category $\mathcal{C} = (x \xrightarrow{f} y)$ with two objects and one nonidentity arrow. Consider the outer horn $\Lambda^2_0 \to N(\mathcal{C})$ given by $d_2 = f\colon x\to y$ and $d_1 = \mathrm{id}_x\colon x\to x$ . A filler would require an edge $d_0\colon y \to x$ with appropriate $2$ -simplex, i.e. a morphism $y\to x$ in $\mathcal{C}$ — but none exists. Thus $N(\mathcal{C})$ fills all inner horns yet fails this outer one, confirming it is a quasi-category but not a Kan complex. The failure encodes exactly that $f$ is not invertible.

CounterexampleA simplicial set that is NOT a quasi-category

Let $X = \Delta^1 \sqcup_{\Delta^0} \Delta^1$ — two edges $f\colon x\to y$ and $g\colon y\to z$ glued at $y$ , with no $2$ -simplices added (the literal $1$ -skeleton union). The map $\Lambda^2_1 \to X$ sending the horn to $(f,g)$ has no filler, since $X$ has no $2$ -simplices besides degeneracies and no edge $x\to z$ . Hence $X$ is not a quasi-category. Its fibrant replacement in the Joyal model structure freely adds the composite and all higher coherences, yielding (up to equivalence) $\Delta^2$ , i.e. the nerve of $(x\to y\to z)$ .

connection

Quasi-categories sit at the crossroads of several frameworks. Via the homotopy coherent nerve $\mathfrak{N}$ , they are equivalent to Simplicial Categories (categories enriched in simplicial sets); $\mathfrak{N}$ is a right Quillen equivalence from the Bergner model structure to the Joyal model structure. They are equivalent to Complete Segal Spaces (Rezk's bisimplicial model) and to Segal Categories, all four being models of The Homotopy Theory of Homotopy Theories. Restricting to Kan complexes recovers Infinity-Groupoids, and by the homotopy hypothesis these are spaces — so quasi-categories generalize both ordinary Category Theory (via the nerve) and Homotopy Theory (via Kan complexes). Stable examples connect to Stable Infinity-Categories and Spectra; presentable examples to Infinity-Topoi and Derived Algebraic Geometry. The straightening/unstraightening equivalence relates them to Cartesian Fibrations and the $\infty$ -Grothendieck construction.

🔬 Where this appears

⏳ History

- 1957 — Kan. Daniel Kan introduces the extension condition (horn filling for all horns); Kan complexes become the combinatorial model for spaces.

- 1967 — Quillen. Homotopical Algebra introduces model categories, providing the abstract homotopy-theoretic foundation later used to compare

\infty

-categorical models.

- 1973 — Boardman & Vogt. In Homotopy Invariant Algebraic Structures on Topological Spaces, they define weak Kan complexes (inner-horn fillers) and the homotopy coherent nerve — the first appearance of quasi-categories.

- 1980s — Dwyer–Kan. Simplicial localization and the theory of simplicial categories provide an early "homotopy theory of homotopy theories."

- 1991 / 2001 — Joyal. André Joyal develops the theory systematically, constructs the Joyal model structure, proves quasi-categories support full category theory; circulates the influential Theory of Quasi-Categories notes.

- 2001 — Rezk. Charles Rezk introduces complete Segal spaces, a Segal-condition model later shown equivalent to quasi-categories.

- 2007 — Bergner. Julia Bergner establishes the model structure on simplicial categories and proves equivalences among the four main models (with Joyal–Tierney, Lurie).

- 2009 — Lurie. Higher Topos Theory gives a comprehensive, rigorous development: presentable

\infty

-categories,

\infty

-topoi, the

\infty

-Yoneda lemma, straightening/unstraightening.

- 2012–2017 — Lurie; Cisinski; Riehl–Verity. Higher Algebra; Cisinski's model-categorical treatment; Riehl–Verity's $\infty$ -cosmoi giving a model-independent "synthetic" approach.

open-problem

Model-independence and synthetic foundations. Although all major models are Quillen-equivalent, transporting a theorem proved in one model to another can be delicate; the Riehl–Verity $\infty$ -cosmos program and Homotopy Type Theory-flavored "directed type theory" aim for genuinely model-independent or formal foundations. Open and active questions include:

A fully satisfactory directed/simplicial type theory that internally axiomatizes $\infty$ -categories (progress: Riehl–Shulman's simplicial HoTT, but a complete computer-checkable foundation is incomplete).
Efficient, computable models for $(\infty,n)$ -categories for general $n$ — complicial sets, $\Theta_n$ -spaces, and $n$ -fold Segal spaces all exist, but practical manipulation remains hard.
The precise scope of strictification: when can coherence data be replaced by strict structure? (Power's and Lack's strictification theorems give partial answers; the $\infty$ case is subtle.)
Foundations and size issues: large quasi-categories, universes, and the interaction of $\infty$ -categorical logic with set-theoretic foundations are still being clarified.

Infinity-Categories and Quasi-Categories

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also