03.12.43 · modern-geometry / homotopy

Quasi-categories and the Joyal model structure

shipped3 tiersLean: none

Anchor (Master): Joyal 2002 *Quasi-categories and Kan complexes* (J. Pure Appl. Algebra 175); Joyal 2008 *The Theory of Quasi-categories and its Applications* (CRM lecture notes); Lurie 2009 *Higher Topos Theory* (Princeton Annals 170) §1.1-§2.2 (canonical reference); Boardman-Vogt 1973 *Homotopy Invariant Algebraic Structures on Topological Spaces* (LNM 347) (originator of weak Kan complexes); Cisinski 2019 *Higher Categories and Homotopical Algebra* (Cambridge Studies 180) §3

Intuition Beginner

An ordinary category has objects and arrows, and any two composable arrows have one and only one composite. A quasi-category relaxes this: composition is not handed to you as a fixed operation but is recorded as a piece of data, and there can be many witnesses to "this arrow is the composite of those two." All the witnesses agree up to higher data, so composition still works — it is just defined up to coherent choice rather than on the nose. This is the natural language for homotopy theory, where two paths can be composed but the composite is only well defined up to homotopy.

The bookkeeping uses simplicial sets. A $0$ -simplex is an object, a $1$ -simplex is a morphism, and a $2$ -simplex is a witness that one edge is a composite of the other two. The defining rule is a filling condition: whenever you have two arrows that meet at a shared point, with the middle of a triangle missing, you can fill in the triangle. Filling the triangle supplies a composite and the witness at the same time. Higher simplices record that these choices are compatible.

The slogan to carry is that quasi-categories unify two worlds. When every arrow is reversible you recover spaces, and when fillings are unique you recover ordinary categories. So a single framework holds both spaces and categories, with genuine homotopy-theoretic categories living in between.

Visual Beginner

A schematic of the standard triangle $Δ^{2}$ with vertices $0, 1, 2$ . The two edges $0 \to 1$ and $1 \to 2$ are drawn solid: these are the two composable arrows, meeting at the middle vertex $1$ . The third edge $0 \to 2$ and the interior of the triangle are drawn dashed: this is the missing data, the inner horn $Λ_{1}^{2}$ . An arrow labelled "fill" points from the partial picture to the full filled triangle, where the dashed edge becomes solid and is labelled "a composite of the two." A side note marks that the corner horns $Λ_{0}^{2}$ and $Λ_{2}^{2}$ (where the missing edge touches an endpoint, not the middle) are not required to fill.

The picture isolates the one rule that defines a quasi-category: inner horns fill, supplying composites. The outer horns are left out on purpose, which is exactly what separates a quasi-category from a space.

Worked example Beginner

Take an ordinary category $C$ and build its nerve $N (C)$ , then read off why it is a quasi-category whose triangles fill in only one way.

Step 1. Describe the pieces. The objects of $N (C)$ are the objects of $C$ . A $1$ -simplex is a single arrow $x \to y$ . A $2$ -simplex is a chain $x \to y \to z$ together with the chosen composite $x \to z$ — so a filled triangle is exactly a commuting triangle in $C$ .

Step 2. Check the filling rule for the middle-missing case. Suppose you are handed the two solid edges $f : x \to y$ and $g : y \to z$ meeting at the middle vertex, with the long edge and interior missing. In $C$ there is the composite $g \circ f : x \to z$ , and the commuting triangle built from $f$ , $g$ , and $g \circ f$ is a filler. So the inner horn fills.

Step 3. Notice the filling is forced. The long edge of any filler has to be the composite $g \circ f$ , and there is only one such arrow in a category. So the inner horn fills in exactly one way. This uniqueness is what marks out nerves of ordinary categories among all quasi-categories.

Step 4. Check the corner case fails in general. Take the outer horn made of $f : x \to y$ and an arrow $h : x \to z$ sharing the starting vertex. Filling it would demand an arrow $y \to z$ making the triangle commute, which need not exist when $f$ is not reversible. So outer horns are genuinely not required to fill.

What this shows: the nerve turns a category into a simplicial set whose inner fillings reproduce composition, and the uniqueness of fillers detects that we started from a plain category rather than a homotopical one. Quasi-categories with non-unique fillers are the new objects, where composition is defined only up to coherent choice.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which horns must fill in a quasi-category?

A. All horns $Λ_{k}^{n}$ for $0 \leq k \leq n$ . B. Only the inner horns $Λ_{k}^{n}$ for $0 < k < n$ . C. Only the outer horns $Λ_{0}^{n}$ and $Λ_{n}^{n}$ . D. Only the $2$ -dimensional horns.

Hint

The inner horns are the ones whose missing face sits in the middle, away from the two endpoints. They are the ones that supply composites.

Answer

B. Only the inner horns $Λ_{k}^{n}$ for $0 < k < n$ . A quasi-category is a simplicial set in which every inner horn has a filler, in every dimension $n \geq 2$ . Demanding the outer horns as well would force every morphism to be reversible, turning the quasi-category into a Kan complex — a space. Leaving the outer horns optional is exactly what lets a quasi-category model a category whose arrows need not be invertible.

Formal definition Intermediate+

Let $sSet = Fun (Δ^{op}, Set)$ be the category of simplicial sets 03.12.25, with standard simplices $Δ^{n}$ , boundaries $\partial Δ^{n}$ , and horns $Λ_{k}^{n} \subset Δ^{n}$ . Recall that a horn $Λ_{k}^{n}$ is inner when $0 < k < n$ and outer when $k \in {0, n}$ .

Definition (quasi-category). A quasi-category (synonyms: $\infty$ -category in Lurie's sense, $(\infty, 1)$ -category, weak Kan complex) is a simplicial set $C$ such that for every inner horn inclusion $Λ_{k}^{n} ↪ Δ^{n}$ with $0 < k < n$ and $n \geq 2$ , every map $Λ_{k}^{n} \to C$ extends along the inclusion to a map $Δ^{n} \to C$ . Equivalently, $C$ has the right lifting property against the set $J_{inn} = {Λ_{k}^{n} ↪ Δ^{n} : 0 < k < n}$ of inner-horn inclusions.

The interpretive dictionary: the elements of $C_{0}$ are the objects; the elements of $C_{1}$ are the morphisms, with source and target given by the face maps $d_{1}, d_{0}$ ; a $2$ -simplex $σ$ with $d_{2} σ = f$ , $d_{0} σ = g$ , $d_{1} σ = h$ exhibits $h$ as a composite of $f$ and $g$ . Inner-horn filling at $n = 2$ supplies composites; inner-horn filling at $n \geq 3$ supplies the coherences (associativity-up-to-homotopy and its higher analogues).

Definition (the homotopy category). Two morphisms $f, g : x \to y$ in a quasi-category $C$ are homotopic, written $f ≃ g$ , if there is a $2$ -simplex with faces $f$ , $g$ , and a degenerate identity edge witnessing $f ≃ g$ . For a quasi-category this is an equivalence relation, and composition descends to homotopy classes. The homotopy category $ho (C)$ is the ordinary category with the same objects as $C$ and with $Hom_{ho (C)} (x, y) = C (x, y) / ≃$ , composition induced by inner-horn filling at $n = 2$ . The construction extends to a functor $ho : sSet \to Cat$ left adjoint to the nerve $N : Cat \to sSet$ .

Definition (equivalence in a quasi-category). A morphism $f : x \to y$ in $C$ is an equivalence if its image in $ho (C)$ is an isomorphism; equivalently $f$ admits a two-sided homotopy inverse. The quasi-category $C$ is a Kan complex if and only if every morphism is an equivalence — this is the precise sense in which Kan complexes are the $\infty$ -groupoids 03.12.42.

Definition (mapping spaces). For objects $x, y$ in a quasi-category $C$ there is a mapping space $Map_{C} (x, y)$ , a Kan complex (not merely a quasi-category), constructed for instance as the simplicial set of $1$ -simplices with prescribed endpoints (the pullback presentation $Map_{C} (x, y) = {x} \times_{C} C^{Δ^{1}} \times_{C} {y}$ ). A map $F : C \to D$ of quasi-categories is an equivalence of $\infty$ -categories if it is essentially surjective on homotopy categories and induces homotopy equivalences $Map_{C} (x, y) \sim Map_{D} (F x, F y)$ on all mapping spaces.

Definition (Joyal model structure). The Joyal model structure on $sSet$ has:

cofibrations: the monomorphisms (level-wise injections), the same class as in the Kan-Quillen structure 03.12.33;
fibrant objects: the quasi-categories;
weak equivalences: the categorical (Joyal) equivalences — the maps $f : K \to L$ such that for every quasi-category $C$ the induced map $τ_{0} (L, C) \to τ_{0} (K, C)$ on isomorphism classes of the mapping quasi-category is a bijection; on fibrant objects these restrict to the equivalences of $\infty$ -categories above.

The fibrations between fibrant objects are the isofibrations (the inner-fibration condition plus lifting of equivalences); the generating cofibrations are the boundary inclusions ${\partial Δ^{n} ↪ Δ^{n}}$ , and the categorical anodyne extensions are generated from the inner horns together with the data forcing equivalences to lift.

Counterexamples to common slips

Inner versus all horns. The Kan-Quillen structure 03.12.33 asks for all horns and has the Kan complexes as fibrant objects; the Joyal structure asks for inner horns only and has the quasi-categories as fibrant objects. Same cofibrations, same underlying category, different fibrant objects and different weak equivalences.
The mapping space is a Kan complex, not a quasi-category. In an $\infty$ -category the objects and morphisms form a quasi-category, but for fixed endpoints the morphisms-and-homotopies assemble into a space: $Map_{C} (x, y)$ is always a Kan complex. The directedness lives between objects, not inside a single hom.
Joyal equivalence is finer than Kan-Quillen weak equivalence. The map $Δ^{1} \to Δ^{0}$ collapsing an arrow is a Kan-Quillen weak equivalence (both realise to a point) but not a Joyal equivalence: $Δ^{1}$ is the walking arrow and $Δ^{0}$ the walking object, and they are not equivalent as $\infty$ -categories. Joyal equivalences see the categorical, not just the homotopy-type, content.
Most simplicial sets are not quasi-categories. The boundary $\partial Δ^{2}$ has the two composable edges but no $2$ -simplex, so the inner horn $Λ_{1}^{2} ↪ \partial Δ^{2}$ has no filler; $\partial Δ^{2}$ is not a quasi-category. Fibrant replacement in the Joyal structure is a genuine operation.

Key theorem with proof Intermediate+

Theorem (nerve recognition; Joyal 2002). Let $C$ be a simplicial set. Then $C$ is isomorphic to the nerve $N (D)$ of an ordinary category $D$ if and only if $C$ is a quasi-category in which every inner horn $Λ_{k}^{n} \to C$ ( $0 < k < n$ , $n \geq 2$ ) has a unique filler. In that case $D ≅ ho (C)$ .

Proof. $(\Rightarrow)$ Suppose $C = N (D)$ . An $n$ -simplex of $N (D)$ is a composable chain $x_{0} \to x_{1} \to \dots \to x_{n}$ of arrows in $D$ . Given an inner horn $Λ_{k}^{n} \to N (D)$ with $0 < k < n$ , all faces except the $k$ -th are specified; in particular the edges $x_{k - 1} \to x_{k}$ and $x_{k} \to x_{k + 1}$ are present (they lie in faces other than the $k$ -th, since $k$ is inner). The missing $k$ -th face is determined: every other edge of the chain is fixed by the present faces, and the edge $x_{k - 1} \to x_{k + 1}$ inside the $k$ -th face must be the composite of the two present edges. So the filler exists and is the unique chain with those edges. Outer horns ( $k = 0$ or $k = n$ ) are not required, because the missing composite would touch an endpoint and need not be available. Hence $N (D)$ is a quasi-category with unique inner fillers.

$(\Leftarrow)$ Suppose $C$ is a quasi-category with unique inner fillers. Build $D = ho (C)$ : objects are $C_{0}$ , and morphisms are the elements of $C_{1}$ , where uniqueness of $2$ -simplex fillers makes the homotopy relation $≃$ the identity relation, so $C (x, y) = C (x, y) / ≃$ . Composition is well defined and single-valued by unique inner-horn filling at $n = 2$ ; associativity follows from unique filling at $n = 3$ . The comparison map $C \to N (ho (C))$ is a bijection in every dimension because the unique-filler hypothesis recovers each $n$ -simplex from its spine $x_{0} \to \dots \to x_{n}$ , and the spine determines the chain in the nerve. So $C ≅ N (D)$ . $□$

Bridge. This theorem builds toward the slogan that quasi-categories generalise ordinary categories: the nerve embeds $Cat$ fully faithfully into $sSet$ , and the image is exactly the quasi-categories with unique fillers, so a quasi-category is a category in which inner fillers exist but are no longer forced to be unique. The foundational reason composition is only defined up to coherent homotopy is precisely the loss of that uniqueness: in $N (D)$ the composite is the single filler, while in a general quasi-category many $2$ -simplices witness the same pair. This is exactly the same relaxation that turns the Kan-Quillen structure 03.12.33 into a homotopical version of $Set$ , and the bridge is that the two relaxations live on the same underlying category $sSet$ with the same monomorphism cofibrations. The central insight is that one combinatorial gadget — a simplicial set with a horn-filling condition — interpolates between categories (unique inner fillers), $\infty$ -categories (inner fillers), and spaces (all fillers, the $\infty$ -groupoids of 03.12.42); putting these together, the nerve recognition theorem appears again in the comparison-of-models program, where the homotopy-coherent nerve plays the role the ordinary nerve plays here, and the cobordism hypothesis 03.16.03 is stated in exactly this $(\infty, 1)$ - and $(\infty, n)$ -categorical language.

Exercises Intermediate+

Exercise 1 (easy, multiple choice).

Which of the following is a quasi-category but not a Kan complex?

A. The singular simplicial set $Sing (X)$ of a space $X$ . B. The nerve $N (D)$ of a category $D$ with a non-invertible arrow. C. The standard simplex $Δ^{0}$ . D. The underlying simplicial set of a simplicial group.

Hint

A Kan complex is a quasi-category in which every morphism is an equivalence. Which option has a morphism that is not reversible?

Answer

B. The nerve $N (D)$ of a category $D$ with a non-invertible arrow. The nerve of any category is a quasi-category (inner horns fill via composition). If $D$ has an arrow with no inverse, then $N (D)$ has a morphism that is not an equivalence, so $N (D)$ is not a Kan complex. The singular complex and simplicial groups are Kan complexes; $Δ^{0}$ is the terminal Kan complex.

Exercise 3 (medium, short-answer).

Show that in a quasi-category $C$ any two composites of a composable pair $f : x \to y$ , $g : y \to z$ are homotopic, so that composition is well defined in $ho (C)$ .

Hint

Two composites give two $2$ -simplices with the same short edges. Assemble them into a $3$ -dimensional inner horn and fill it.

Answer

Let $σ, σ^{'}$ be two $2$ -simplices with $d_{2} = f$ , $d_{0} = g$ , and long edges $h = d_{1} σ$ , $h^{'} = d_{1} σ^{'}$ . Build a $3$ -dimensional inner horn $Λ_{1}^{3} \to C$ whose faces $d_{0}, d_{2}, d_{3}$ are $σ^{'}$ , the degenerate witness $s_{0}$ data carrying $h$ , and $σ$ respectively, arranged so that the spine is $f$ then $g$ . Inner-horn filling ( $0 < 1 < 3$ ) supplies a $3$ -simplex $τ$ , and its remaining face $d_{1} τ$ is a $2$ -simplex exhibiting $h ≃ h^{'}$ relative to the common endpoints. Hence any two composites of $f$ and $g$ are homotopic, and the class $[h]$ in $ho (C)$ is independent of the chosen filler. This is the precise sense in which composition is defined up to a contractible space of choices: the simplicial set of fillers of $Λ_{1}^{2}$ over a fixed pair is itself a contractible Kan complex. Rubric: full credit for the $3$ -horn assembly and the conclusion of homotopy uniqueness.

Exercise 4 (medium, multiple choice).

The map $Δ^{1} \to Δ^{0}$ is:

A. a Joyal equivalence but not a Kan-Quillen weak equivalence. B. a Kan-Quillen weak equivalence but not a Joyal equivalence. C. both a Joyal equivalence and a Kan-Quillen weak equivalence. D. neither.

Hint

$Δ^{1}$ is the walking arrow and $Δ^{0}$ the walking object. Are they equivalent as $\infty$ -categories? Are their realisations weakly equivalent as spaces?

Answer

B. A Kan-Quillen weak equivalence but not a Joyal equivalence. Both $∣ Δ^{1} ∣$ and $∣ Δ^{0} ∣$ are contractible, so $Δ^{1} \to Δ^{0}$ is a Kan-Quillen weak equivalence. But $Δ^{1}$ as an $\infty$ -category is the walking morphism, with two distinct objects and a non-invertible arrow between them, while $Δ^{0}$ has a single object; no map between them can be an equivalence of $\infty$ -categories. So it is not a Joyal equivalence. This is the standard witness that the Joyal weak equivalences form a strictly finer class.

Exercise 5 (medium, short-answer).

Prove that if $C$ is a quasi-category in which every morphism is an equivalence, then $C$ is a Kan complex.

Hint

You already have inner-horn fillers. To fill an outer horn $Λ_{0}^{n}$ you must invert the leading edge; use that it is an equivalence.

Answer

A quasi-category fills all inner horns, so only the outer horns $Λ_{0}^{n}$ and $Λ_{n}^{n}$ remain. Consider $Λ_{0}^{n} \to C$ . Joyal's theorem (2002) states that for a quasi-category, the outer horn $Λ_{0}^{n}$ fills whenever its initial edge $Δ^{{0, 1}} \to C$ is an equivalence; symmetrically $Λ_{n}^{n}$ fills when its final edge is an equivalence. The proof reduces, by an inductive coherence argument on $n$ , to the case $n = 2$ : an outer horn $Λ_{0}^{2}$ consisting of $f : x \to y$ and $h : x \to z$ fills to a $2$ -simplex exhibiting a morphism $y \to z$ , which exists once $f$ has a homotopy inverse $f^{- 1}$ (take the composite $h \circ f^{- 1}$ and witness it). When every morphism of $C$ is an equivalence, all leading and trailing edges are equivalences, so all outer horns fill. Combined with the inner-horn fillers, every horn fills, so $C$ is a Kan complex. This is the converse making "Kan complexes = $\infty$ -groupoids" an exact characterisation. Rubric: full credit for the reduction to outer horns plus invocation of the equivalence-fills-outer-horns lemma.

Exercise 6 (hard, short-answer).

Sketch why the Joyal model structure cannot be a simplicial model structure in the standard sense, and identify what replaces the simplicial enrichment.

Hint

A simplicial model structure enriches over Kan complexes via the Kan-Quillen tensoring. The Joyal structure naturally tensors with $Δ^{1}$ , whose hom-objects are quasi-categories, not Kan complexes.

Answer

A simplicial model category in the sense of Quillen is enriched, tensored, and cotensored over $sSet$ with the Kan-Quillen structure, with the pushout-product (SM7) axiom relating cofibrations and the Kan-Quillen fibrations. For the Joyal structure the natural function complex $Fun (K, C) = C^{K}$ between a simplicial set and a quasi-category is again a quasi-category, not a Kan complex; the cotensor by $Δ^{1}$ is the arrow $\infty$ -category $C^{Δ^{1}}$ , whose morphisms are squares and need not be invertible. So the relevant enrichment is over $(sSet, Joyal)$ itself — the Joyal structure is a cartesian closed model structure (monoidal under the cartesian product, which the Kan-Quillen structure also is), and the pushout-product axiom holds for the categorical anodyne extensions rather than the Kan-Quillen ones. Concretely, Joyal proves that if $A ↪ B$ is a monomorphism and $C$ is a quasi-category then $C^{B} \to C^{A}$ is an isofibration of quasi-categories, the Joyal-structure analogue of SM7. The framework that cleanly captures "enriched over its own homotopy theory" is the theory of $\infty$ -cosmoi (Riehl-Verity) or, equivalently, the cartesian-closed model-categorical statement above. Rubric: full credit for identifying that hom-objects are quasi-categories not Kan complexes, and naming the cartesian-closed/isofibration replacement for SM7.

Advanced results Master

Comparison of models: Joyal versus Bergner

The Joyal model structure is one of several Quillen-equivalent presentations of the homotopy theory of $(\infty, 1)$ -categories. The principal comparison is with the Bergner model structure on $sCat$ , the category of simplicially enriched categories.

Theorem 1 (Joyal-Tierney; Lurie HTT §2.2). The homotopy-coherent nerve $N^{hc} : sCat \to sSet$ , right adjoint to the rigidification functor $C : sSet \to sCat$ , is the right half of a Quillen equivalence between the Bergner model structure on $sCat$ and the Joyal model structure on $sSet$ . Consequently the two model categories present the same homotopy theory of $(\infty, 1)$ -categories.

The functor $C$ sends $Δ^{n}$ to the simplicial category whose objects are $0, \dots, n$ and whose hom-objects are the nerves of the posets of subsets of ${i, i + 1, \dots, j}$ containing both endpoints. The adjunction $C ⊣ N^{hc}$ upgrades to a Quillen equivalence: a simplicial set $C$ is a quasi-category if and only if $C (C)$ is a fibrant simplicial category (one whose hom-objects are Kan complexes), and the derived unit and counit are equivalences. This is the precise statement that "quasi-categories and simplicial categories model the same thing."

Theorem 2 (a chain of Quillen equivalences). The Joyal model structure on $sSet$ , the Bergner structure on $sCat$ , the complete-Segal-space structure on $sSet^{Δ^{op}}$ (Rezk), and the Segal-category structure are pairwise connected by Quillen equivalences (Bergner 2007, Joyal-Tierney 2007). All four present the homotopy theory of $(\infty, 1)$ -categories.

Toën's theorem makes this rigidity precise: the homotopy theory of $(\infty, 1)$ -categories has, up to equivalence, a unique self-equivalence group $Z /2$ (identity and "take the opposite"), so the four models agree not just abstractly but essentially uniquely. The reader should take from this that the choice of model is a matter of convenience; the invariants are model-independent.

Straightening and unstraightening

Theorem 3 (straightening/unstraightening; Lurie HTT §2.2). For a quasi-category $C$ there is a Quillen equivalence between the covariant model structure on $sSet_{/ C}$ (presenting functors $C \to S$ to the $\infty$ -category of spaces, via left fibrations) and a category of simplicial functors. The unstraightening functor $Un$ turns a diagram of spaces into a left fibration over $C$ ; straightening $St$ is its inverse up to equivalence.

This is the $\infty$ -categorical Grothendieck construction: it trades a functor valued in $\infty$ -categories for a fibration, exactly as the classical Grothendieck construction trades a pseudofunctor for a fibred category. Straightening/unstraightening is the technical engine of $\infty$ -categorical universal algebra — presheaves, the Yoneda lemma, limits and colimits, and adjoint functors in the $\infty$ -categorical sense are all developed through it. It is the deepest part of the Joyal-Lurie foundation and the natural starting point of the full Higher Topos Theory development that this bridge unit points toward rather than builds.

Synthesis. The Joyal model structure is the bridge from the simplicial homotopy theory of Goerss-Jardine to the $(\infty, 1)$ -categorical world of Lurie, and the bridge is exactly the shift from "all horns" to "inner horns" on the fixed category $sSet$ . This generalises the Kan-Quillen story 03.12.33 in the precise sense that the Kan-Quillen fibrant objects (Kan complexes = $\infty$ -groupoids 03.12.42) sit inside the Joyal fibrant objects (quasi-categories = $\infty$ -categories) as the ones with all morphisms invertible; the foundational reason the two structures coexist is that they share the monomorphism cofibrations and differ only in which anodyne extensions are inverted, so the $\infty$ -groupoidal theory is a localisation of the $\infty$ -categorical one. This is exactly the pattern that recurs through the comparison theorems: quasi-categories, simplicial categories, complete Segal spaces, and Segal categories are pairwise Quillen-equivalent, and putting these together with Toën's rigidity, the homotopy theory of $(\infty, 1)$ -categories is essentially unique. The central insight carried forward is that one filling condition organises categories, $\infty$ -categories, and spaces into a single framework, and this framework is the foundational reason the cobordism hypothesis 03.16.03 can even be stated; the straightening/unstraightening equivalence appears again as the $\infty$ -categorical Grothendieck construction on which the entire Higher Topos Theory edifice is built, and the bridge is that everything above is the opening chapter of that edifice rather than its completion.

Full proof set Master

Proposition 1 (inner-horn filling is closed under the operations defining the fibrant objects). The class of simplicial sets with the right lifting property against all inner-horn inclusions $J_{inn} = {Λ_{k}^{n} ↪ Δ^{n} : 0 < k < n}$ is closed under products, cotensors by arbitrary simplicial sets, and retracts. In particular, if $C$ is a quasi-category and $K$ any simplicial set, then $C^{K}$ is a quasi-category.

Proof. Right lifting properties are always closed under retracts and (small) products, because a lift into a product or a retract is assembled coordinate-wise from lifts into the factors. For the cotensor, the defining adjunction $Hom (A, C^{K}) ≅ Hom (A \times K, C)$ turns a lifting problem $Λ_{k}^{n} \to C^{K}$ , $Δ^{n} \to C^{K}$ into the lifting problem $Λ_{k}^{n} \times K \to C$ , $Δ^{n} \times K \to C$ . The map $Λ_{k}^{n} \times K ↪ Δ^{n} \times K$ is a categorical anodyne extension — Joyal's theorem that the pushout-product of an inner-horn inclusion with any monomorphism (here $\emptyset ↪ K$ promoted via the product) lies in the saturated class generated by inner horns. Since $C$ has the right lifting property against all inner-anodyne maps, the lift exists, and adjointly $C^{K} \to *$ lifts against $J_{inn}$ . Hence $C^{K}$ is a quasi-category. $□$

Proposition 2 (the homotopy category functor is left adjoint to the nerve). The functor $ho : sSet \to Cat$ is left adjoint to the nerve $N : Cat \to sSet$ , and the counit $ho (N (D)) \to D$ is an isomorphism for every ordinary category $D$ .

Proof. Define $ho (K)$ for an arbitrary simplicial set $K$ by generators and relations: objects $K_{0}$ , generating morphisms $K_{1}$ , with relations $s_{0} x = id_{x}$ and $d_{1} σ = d_{0} σ \circ d_{2} σ$ for every $σ \in K_{2}$ . A functor $ho (K) \to D$ is exactly a choice of objects and morphisms compatible with these relations, which is the same data as a simplicial map $K \to N (D)$ (a simplicial map is determined on $0$ - and $1$ -simplices subject to the face relations that the $2$ -simplex relations encode). This bijection $Hom_{Cat} (ho (K), D) ≅ Hom_{sSet} (K, N (D))$ is natural, giving the adjunction. For $K = N (D)$ , the unit-counit triangle identities and the full faithfulness of $N$ (an $n$ -simplex of $N (D)$ is determined by its spine) force the counit $ho (N (D)) \to D$ to be an isomorphism, since $N$ is fully faithful and $ho$ inverts it on the image. $□$

Proposition 3 (Joyal equivalences satisfy two-out-of-three and the Joyal structure is a model structure). The categorical equivalences form a class satisfying two-out-of-three and the retract property, and together with the monomorphisms as cofibrations they assemble, via the small-object argument applied to inner-horn-type generating sets, into a model structure on $sSet$ whose fibrant objects are exactly the quasi-categories.

Proof sketch. Two-out-of-three and retract closure for the categorical equivalences follow because they are defined by a representable condition — bijectivity of $τ_{0} (-, C)$ for all quasi-categories $C$ — and bijections satisfy two-out-of-three and are retract-stable. Cofibrations being the monomorphisms are retract-stable as in the Kan-Quillen case 03.12.33. The factorisation axioms come from the small-object argument applied to the generating cofibrations ${\partial Δ^{n} ↪ Δ^{n}}$ and to a generating set of categorical anodyne extensions; Joyal's identification of the latter (inner horns plus the maps witnessing invertibility of equivalences) is the substantive input, proved via an explicit analysis of the saturated class. The fibrant objects are the simplicial sets with the right lifting property against the categorical anodyne extensions, which Joyal shows coincide with the quasi-categories. The lifting axioms then follow formally from the generating-set description, exactly as for any cofibrantly generated model structure (Hovey 1999). $□$

Connections Master

Kan-Quillen model structure on sSet 03.12.33. The Joyal structure is the direct sibling: same underlying category $sSet$ , same monomorphism cofibrations, but inner-horn fillers in place of all-horn fillers. Where Kan-Quillen has Kan complexes as fibrant objects and models spaces ( $\infty$ -groupoids), Joyal has quasi-categories as fibrant objects and models $(\infty, 1)$ -categories. The Kan-Quillen structure is recovered from the Joyal structure by further inverting the equivalences that make every morphism invertible, so $\infty$ -groupoids are the localisation of $\infty$ -categories at "make all arrows equivalences." Reading the two units together is the cleanest way to see why one combinatorial category carries two homotopy theories.
Simplicial sets and geometric realization 03.12.25. Quasi-categories are simplicial sets with a lifting condition, so the entire theory is built on the combinatorics of $Δ$ , the standard simplices, and their horns from 03.12.25. The interpretive dictionary — objects, morphisms, composition-witnesses, coherences in ascending dimension — is a reading of the low-dimensional simplices of a simplicial set, and the nerve functor that embeds ordinary categories into $sSet$ is the same one introduced there.
Combinatorial simplicial homotopy groups and Kan complexes 03.12.42. Kan complexes are exactly the quasi-categories in which every morphism is an equivalence, the $\infty$ -groupoids. The simplicial homotopy groups built in 03.12.42 are the homotopy groups of these $\infty$ -groupoids, and for a general quasi-category they reappear as the homotopy groups of the mapping spaces $Map_{C} (x, y)$ , which are themselves Kan complexes. The horn-filling group law of 03.12.42 is the all-horns specialisation of the inner-horn composition developed here.
Quillen model category 03.12.31. The Joyal structure is another instance of the abstract model-category framework of 03.12.31, cofibrantly generated and cartesian, and its construction reuses the small-object argument and the saturated-class machinery introduced there. It is a second worked example alongside the Kan-Quillen structure, showing that a single category can support genuinely different model structures distinguished by their weak equivalences and fibrant objects.
Cobordism hypothesis 03.16.03. The cobordism hypothesis of Baez-Dolan and Lurie is stated in the language of $(\infty, n)$ -categories, of which the $(\infty, 1)$ -categories presented by the Joyal structure are the first nontrivial layer. The framework here — composition up to coherent homotopy, mapping spaces, equivalences of $\infty$ -categories — is the vocabulary in which fully extended topological field theories and their classification by fully dualizable objects are formulated. This unit is the categorical on-ramp to that classification.

Historical & philosophical context Master

The inner-horn-filling condition first appears in J. Michael Boardman and Rainer Vogt's 1973 monograph Homotopy Invariant Algebraic Structures on Topological Spaces ^{[Boardman-Vogt1973]} (Springer LNM 347), where they introduced "restricted Kan complexes" — what are now called weak Kan complexes — as the homotopy-coherent nerves of topological categories. Their motivation was operadic: they needed a combinatorial object recording algebraic structure that is associative and unital only up to coherent homotopy, and the inner-horn condition captured exactly the coherence without forcing invertibility. The objects sat largely dormant as a foundational tool for three decades.

André Joyal revived and systematised the theory in his 2002 paper Quasi-categories and Kan complexes ^[Joyal2002] (J. Pure Appl. Algebra 175, 207-222) and the subsequent lecture notes ^[Joyal2008]. Joyal renamed the objects quasi-categories, proved the nerve recognition theorem, characterised Kan complexes among them as the $\infty$ -groupoids, and — most consequentially — constructed the model structure on $sSet$ now bearing his name, with cofibrations the monomorphisms and fibrant objects the quasi-categories. Joyal's insight was that quasi-categories are not a curiosity but a complete and workable foundation for category theory done up to homotopy, with limits, colimits, adjunctions, and Kan extensions all available.

Jacob Lurie's 2009 Higher Topos Theory ^[Lurie2009] (Princeton Annals 170) built the full edifice on Joyal's foundation, developing the $\infty$ -categorical Yoneda lemma, presentable and accessible $\infty$ -categories, the straightening/unstraightening equivalence, and $\infty$ -topoi. Lurie's book made quasi-categories — which he calls simply $\infty$ -categories — the default working language of a generation of homotopy theorists, algebraic geometers, and mathematical physicists. The philosophical payoff is a genuine unification: Grothendieck's vision of spaces and categories as two faces of one homotopical object is realised concretely, with ordinary categories (unique inner fillers), $\infty$ -categories (inner fillers), and spaces (all fillers) as three regimes of a single combinatorial framework. The equivalence of all the competing models, pinned down by Bergner, Joyal-Tierney, and the rigidity theorem of Toën, certifies that this framework is canonical rather than an artefact of one presentation.

Bibliography Master

@article{Joyal2002,
  author    = {Joyal, Andr{\'e}},
  title     = {Quasi-categories and {K}an complexes},
  journal   = {Journal of Pure and Applied Algebra},
  volume    = {175},
  year      = {2002},
  pages     = {207--222}
}

@misc{Joyal2008,
  author    = {Joyal, Andr{\'e}},
  title     = {The Theory of Quasi-categories and its Applications},
  note      = {Advanced Course on Simplicial Methods in Higher Categories, CRM Barcelona, Quaderns 45},
  year      = {2008}
}

@book{Lurie2009,
  author    = {Lurie, Jacob},
  title     = {Higher Topos Theory},
  series    = {Annals of Mathematics Studies},
  volume    = {170},
  publisher = {Princeton University Press},
  year      = {2009}
}

@book{Boardman-Vogt1973,
  author    = {Boardman, J. Michael and Vogt, Rainer M.},
  title     = {Homotopy Invariant Algebraic Structures on Topological Spaces},
  series    = {Lecture Notes in Mathematics},
  volume    = {347},
  publisher = {Springer-Verlag},
  year      = {1973}
}

@book{Cisinski2019,
  author    = {Cisinski, Denis-Charles},
  title     = {Higher Categories and Homotopical Algebra},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {180},
  publisher = {Cambridge University Press},
  year      = {2019}
}

@book{Riehl2014,
  author    = {Riehl, Emily},
  title     = {Categorical Homotopy Theory},
  series    = {New Mathematical Monographs},
  volume    = {24},
  publisher = {Cambridge University Press},
  year      = {2014}
}

@article{JoyalTierney2007,
  author    = {Joyal, Andr{\'e} and Tierney, Myles},
  title     = {Quasi-categories vs Segal spaces},
  journal   = {Contemporary Mathematics},
  volume    = {431},
  year      = {2007},
  pages     = {277--326}
}

@article{Bergner2007,
  author    = {Bergner, Julia E.},
  title     = {Three models for the homotopy theory of homotopy theories},
  journal   = {Topology},
  volume    = {46},
  year      = {2007},
  pages     = {397--436}
}

@article{Toen2005,
  author    = {To{\"e}n, Bertrand},
  title     = {Vers une axiomatisation de la th{\'e}orie des cat{\'e}gories sup{\'e}rieures},
  journal   = {K-Theory},
  volume    = {34},
  year      = {2005},
  pages     = {233--263}
}

Prerequisites

03.12.33
03.12.31
03.12.25
03.12.42

Tier anchors

beginner: Joyal 2008 *Notes on Quasi-categories* §1 informal opening on inner horns and composition; Riehl 2014 *Categorical Homotopy Theory* §15.1 for the intuition that composition is defined up to homotopy
intermediate: Joyal 2002 *Quasi-categories and Kan complexes* (J. Pure Appl. Algebra 175) §1-§4; Lurie 2009 *Higher Topos Theory* §1.1-§1.2; Boardman-Vogt 1973 *Homotopy Invariant Algebraic Structures* (LNM 347) §IV (weak Kan complexes)
master: Joyal 2002 *Quasi-categories and Kan complexes* (J. Pure Appl. Algebra 175); Joyal 2008 *The Theory of Quasi-categories and its Applications* (CRM lecture notes); Lurie 2009 *Higher Topos Theory* (Princeton Annals 170) §1.1-§2.2 (canonical reference); Boardman-Vogt 1973 *Homotopy Invariant Algebraic Structures on Topological Spaces* (LNM 347) (originator of weak Kan complexes); Cisinski 2019 *Higher Categories and Homotopical Algebra* (Cambridge Studies 180) §3

References

Joyal — Quasi-categories and Kan complexes · J. Pure Appl. Algebra 175 (2002) 207-222; introduces quasi-categories as the weak Kan complexes of Boardman-Vogt, characterises Kan complexes among them as the ones with all morphisms invertible, and develops the homotopy category of a quasi-category
Joyal — The Theory of Quasi-categories and its Applications · CRM Barcelona advanced course lecture notes (2008); constructs the Joyal model structure on sSet with cofibrations = monomorphisms and fibrant objects = quasi-categories, and proves the Quillen equivalence with the Bergner model structure on simplicial categories via the homotopy-coherent nerve
Lurie — Higher Topos Theory · Princeton Annals of Mathematics Studies 170 (2009); §1.1 the definition and basic theory of quasi-categories (Lurie's infinity-categories), §1.2 the homotopy category and mapping spaces, §2.2 the straightening/unstraightening equivalence and comparison of models
Boardman-Vogt — Homotopy Invariant Algebraic Structures · Springer Lecture Notes in Mathematics 347 (1973); §IV introduces 'restricted Kan complexes' (weak Kan complexes) satisfying inner-horn filling, as the homotopy-coherent nerves of topological categories
Cisinski — Higher Categories and Homotopical Algebra · Cambridge Studies in Advanced Mathematics 180 (2019); §3 the Joyal model structure via the theory of Cisinski model structures on presheaf categories, with categorical anodyne extensions generated by inner horns
Riehl — Categorical Homotopy Theory · Cambridge New Mathematical Monographs 24 (2014); §15 quasi-categories, the homotopy category, mapping spaces, and the comparison with simplicial categories

Estimated time

beginner: 20m
intermediate: 48m
master: 85m