03.12.32 · modern-geometry / homotopy

Quillen functor and Quillen equivalence

shipped3 tiersLean: none

Anchor (Master): Quillen 1967 *Homotopical Algebra* (LNM 43) §I.4-§II.4 (originator); Hovey 1999 *Model Categories* (AMS Mathematical Surveys 63) §1.3-§1.4; Hirschhorn 2003 *Model Categories and Their Localizations* (AMS Mathematical Surveys 99) §8.5; Goerss-Jardine 2009 *Simplicial Homotopy Theory* §II.3 (Kan-Quillen + realisation); Dwyer-Kan 1980 *Simplicial localizations of categories* (J. Pure Appl. Algebra 17) — simplicial-localisation viewpoint on derived functors

Intuition [Beginner]

A Quillen functor is the right notion of a structure-preserving map between model categories. Where an ordinary functor between categories preserves objects and morphisms, a Quillen functor preserves the homotopy-theoretic structure on top: the cofibrations, fibrations, and weak equivalences that organise the model category. The point of the definition is to identify the functors that descend cleanly to the homotopy category, where the weak equivalences have been inverted.

The framework comes in left-right pairs. A left Quillen functor is the left half of an adjoint pair that preserves cofibrations and the acyclic cofibrations (cofibrations that are also weak equivalences). A right Quillen functor is the right half that preserves fibrations and acyclic fibrations. The two conditions are mutually determined by the adjunction: one statement holds if and only if the dual statement holds. A Quillen adjunction is an adjoint pair where both halves have these preservation properties.

A Quillen equivalence is the strengthening that asks the derived adjunction on homotopy categories to be an equivalence of categories. This is the right notion of "two model categories present the same homotopy theory". The canonical example is the realisation-singular pair between simplicial sets and topological spaces: a simplicial set has a geometric realisation, every space has a singular complex, and the resulting adjoint pair identifies the homotopy theory of CW complexes with the homotopy theory of Kan complexes.

Visual [Beginner]

A schematic with two model categories side by side. On the left, a category $C$ with its three classes of morphisms — cofibrations, fibrations, weak equivalences — drawn as labelled bands. On the right, a category $D$ with the same three bands. A bent arrow labelled $L$ goes from left to right, and a bent arrow labelled $R$ goes from right to left, forming the adjoint pair. The picture shows $L$ sending the left-side cofibrations to right-side cofibrations and the left-side acyclic cofibrations to right-side acyclic cofibrations; dually for $R$ on fibrations and acyclic fibrations.

$A schematic placeholder showing the Quillen-adjoint pair $L : \mathcal{C} \rightleftarrows \mathcal{D} : R$ preserving the three classes of morphisms and descending to homotopy categories.$

A second panel below shows the derived picture: the homotopy categories $Ho (C)$ and $Ho (D)$ connected by the derived left adjoint $L L$ and the derived right adjoint $R R$ . When the original pair is a Quillen equivalence, these derived arrows constitute an equivalence of categories.

Worked example [Beginner]

Let us trace the realisation-singular adjunction between simplicial sets and topological spaces concretely, and see why it is a Quillen equivalence.

Step 1. The left adjoint is geometric realisation. Given a simplicial set $X$ , its realisation $∣ X ∣$ is built by taking one standard topological simplex for each simplex of $X$ , then gluing along the face and degeneracy maps. The realisation of the standard simplicial simplex $Δ^{n}$ is the standard topological simplex; the realisation of a horn $Λ_{k}^{n}$ is the corresponding union of topological faces.

Step 2. The right adjoint is the singular complex. Given a topological space $Y$ , its singular complex $Sing (Y)$ has $n$ -simplices the continuous maps from the topological $n$ -simplex to $Y$ . This is a simplicial set in a direct way, and the adjunction $$ \mathrm{Hom}{\mathbf{Top}}(|X|, Y) = \mathrm{Hom}{\mathbf{sSet}}(X, \mathrm{Sing}(Y)) $$ is given by transposing along the simplex-by-simplex bijection.

Step 3. Why does $∣ - ∣$ preserve cofibrations? In the Kan-Quillen model structure on simplicial sets, the cofibrations are exactly the monomorphisms — the simplex-by-simplex injective maps. The realisation of a monomorphism is a CW pair inclusion, which is a cofibration in the Quillen-Serre structure on spaces. So $∣ - ∣$ does preserve cofibrations.

Step 4. Why does it preserve acyclic cofibrations? An acyclic cofibration in simplicial sets is a monomorphism whose realisation is a weak homotopy equivalence. By construction this realises to an acyclic CW inclusion, which is acyclic on the topological side. So $∣ - ∣$ is a left Quillen functor.

Step 5. Why is the pair a Quillen equivalence? The unit map $X \to Sing (∣ X ∣)$ sends a simplex of $X$ to its identity realisation map; the counit $∣ Sing (Y) ∣ \to Y$ assembles a CW structure on $Y$ from its singular simplices. Milnor proved in 1957 that $∣ Sing (Y) ∣ \to Y$ is a weak homotopy equivalence for every space $Y$ . The unit map $X \to Sing (∣ X ∣)$ is a weak equivalence for every Kan complex $X$ by a direct argument. So the derived adjunction is an equivalence.

The takeaway: the realisation-singular adjunction is the prototype for what a Quillen equivalence is supposed to do. It identifies the combinatorial side (simplicial sets, monomorphisms, Kan fibrations) with the topological side (spaces, CW inclusions, Serre fibrations) in such a way that homotopy theory done on either side gives the same answer up to canonical isomorphism.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

A Quillen adjunction $L ⊣ R$ between model categories requires which of the following?

A. $L$ preserves all weak equivalences and $R$ preserves all weak equivalences. B. $L$ preserves cofibrations and acyclic cofibrations; equivalently, $R$ preserves fibrations and acyclic fibrations. C. $L$ preserves fibrations and acyclic fibrations; equivalently, $R$ preserves cofibrations and acyclic cofibrations. D. Both $L$ and $R$ preserve all three classes simultaneously.

Hint

The convention is that the left adjoint sees cofibrations and the right adjoint sees fibrations — left handles "left" classes, right handles "right" classes.

Answer

B. $L$ preserves cofibrations and acyclic cofibrations; equivalently, $R$ preserves fibrations and acyclic fibrations. The equivalence of the two formulations comes from the lifting axiom M3: $L$ preserves cofibrations iff $R$ preserves the right-lifting class against cofibrations, which is acyclic fibrations. The left adjoint does not in general preserve weak equivalences on the nose — only between cofibrant objects, by Ken Brown's lemma.

Exercise (easy, multiple choice).

The realisation-singular adjunction $∣ - ∣ ⊣ Sing$ between simplicial sets and topological spaces is a Quillen equivalence. Which classical theorem makes the counit a weak equivalence?

A. The Whitehead theorem on weak equivalences between CW complexes. B. The Milnor theorem that $∣ Sing (Y) ∣ \to Y$ is a weak homotopy equivalence for every space. C. The Hurewicz theorem on homotopy and homology groups. D. The Brouwer fixed-point theorem.

Hint

The relevant 1957 theorem identifies the realisation of the singular complex of a space as a CW approximation of the space.

Answer

B. The Milnor theorem. Milnor 1957 The geometric realization of a semi-simplicial complex proved that for every topological space $Y$ , the counit map $∣ Sing (Y) ∣ \to Y$ is a weak homotopy equivalence. This is the key input that upgrades the realisation-singular pair from a Quillen adjunction to a Quillen equivalence. The Whitehead theorem is a related but distinct statement about CW complexes.

Formal definition [Intermediate+]

Definition (Quillen adjunction). Let $C$ and $D$ be model categories and let $L : C ⇄ D : R$ be an adjoint pair with $L$ left adjoint to $R$ . The pair is a Quillen adjunction if either (equivalently both) of the following hold.

(a) The left adjoint $L$ preserves cofibrations and acyclic cofibrations.

(b) The right adjoint $R$ preserves fibrations and acyclic fibrations.

The left adjoint $L$ is called a left Quillen functor and the right adjoint $R$ is called a right Quillen functor.

Equivalence of (a) and (b). The proof uses the lifting axiom M3 of [03.12.31]. Suppose (a): $L$ preserves cofibrations. We show $R$ preserves acyclic fibrations. Let $p : E \to B$ be an acyclic fibration in $D$ . We need to show $R (p)$ has the right lifting property against every cofibration in $C$ . Let $i : A \to X$ be a cofibration in $C$ and consider a commutative square $$ \begin{array}{ccc} A & \xrightarrow{u} & R(E) \ \downarrow i & & \downarrow R(p) \ X & \xrightarrow{v} & R(B) \end{array} $$ Adjoining transposes the square to $$ \begin{array}{ccc} L(A) & \xrightarrow{\tilde{u}} & E \ \downarrow L(i) & & \downarrow p \ L(X) & \xrightarrow{\tilde{v}} & B \end{array} $$ By (a), $L (i)$ is a cofibration in $D$ , and $p$ is an acyclic fibration. By M3, the square has a diagonal lift $\tilde{h} : L (X) \to E$ . The adjoint transpose $h : X \to R (E)$ is then the required lift for the original square. The argument for $L$ preserving acyclic cofibrations and $R$ preserving fibrations is the dual half. The two halves give (a) $\Leftrightarrow$ (b).

Definition (total derived functors). Let $L ⊣ R$ be a Quillen adjunction. Let $Q$ be a cofibrant-replacement functor on $C$ and $R^{'}$ a fibrant-replacement functor on $D$ (the M4 functorial factorisations of [03.12.31] provide both). Define $$ \mathbf{L}L(X) := L(Q X), \qquad \mathbf{R}R(Y) := R(R' Y), $$ for $X \in C$ and $Y \in D$ . These are the total left derived functor of $L$ and the total right derived functor of $R$ . We use the notation $R^{'}$ for the fibrant-replacement functor to keep it apart from the right adjoint $R$ .

Theorem (existence of the derived adjunction). The functors $L L$ and $R R$ descend to well-defined functors on the homotopy categories $$ \mathbf{L}L : \mathrm{Ho}(\mathcal{C}) \rightleftarrows \mathrm{Ho}(\mathcal{D}) : \mathbf{R}R, $$ and form an adjoint pair on $Ho$ . The well-definedness is the content of Ken Brown's lemma applied to $L \circ Q$ and $R \circ R^{'}$ .

Definition (Quillen equivalence). A Quillen adjunction $L ⊣ R$ is a Quillen equivalence if the derived adjunction $L L ⊣ R R$ is an equivalence of categories. Concretely, the unit $η : id \to R R \circ L L$ and the counit $ε : L L \circ R R \to id$ are natural isomorphisms in $Ho (C)$ and $Ho (D)$ respectively.

Definition (cofibrant and fibrant objects). As in [03.12.31], an object $X \in C$ is cofibrant if the unique map $\emptyset \to X$ is a cofibration, and an object $Y \in D$ is fibrant if $Y \to *$ is a fibration. These conditions enter the criteria below.

Counterexamples to common slips

The condition " $L$ preserves all weak equivalences" is strictly stronger than " $L$ is a left Quillen functor" and is rarely satisfied: Ken Brown's lemma only gives preservation on the subcategory of cofibrant objects. Insisting on more is a frequent slip.
A Quillen adjunction need not be a Quillen equivalence even when both adjoints preserve all weak equivalences: the homotopy categories may simply be different. The unit-of-adjunction criterion is the actual content.
The derived functors $L L (X) = L (QX)$ depend a priori on the choice of cofibrant replacement $QX$ , but Ken Brown's lemma shows the resulting functor on $Ho (C)$ is independent of the choice up to canonical natural isomorphism.
The Quillen-equivalence reformulation " $L X \to Y$ is a weak equivalence iff its adjunct $X \to R Y$ is" requires $X$ cofibrant and $Y$ fibrant. Without those hypotheses the bi-implication fails: the unit and counit only behave well on the cofibrant-fibrant subcategory.

Key theorem with proof [Intermediate+]

Theorem (Ken Brown's lemma, Brown 1973). Let $C$ be a model category and let $F : C \to D$ be a functor into a category $D$ equipped with a class of "weak equivalences" $W_{D}$ satisfying 2-out-of-3. If $F$ sends acyclic cofibrations between cofibrant objects in $C$ to weak equivalences in $D$ , then $F$ sends all weak equivalences between cofibrant objects in $C$ to weak equivalences in $D$ .

Proof. Let $f : A \to B$ be a weak equivalence between cofibrant objects in $C$ . Form the coproduct $A ⊔ B$ ; this is cofibrant because $\emptyset \to A ⊔ B$ is the pushout of $\emptyset \to A$ and $\emptyset \to B$ along $\emptyset$ , both cofibrations, and cofibrations are closed under pushout. Consider the map $(f, id_{B}) : A ⊔ B \to B$ defined on the two summands. Apply axiom M4 to factor this map as $$ A \sqcup B \xrightarrow{j} Z \xrightarrow{q} B, $$ with $j$ an acyclic cofibration in $C$ and $q$ a fibration. The object $Z$ is cofibrant (target of an acyclic cofibration from a cofibrant object).

The two structure maps $ι_{A} : A \to A ⊔ B$ and $ι_{B} : B \to A ⊔ B$ are cofibrations (pushouts of $\emptyset \to A$ and $\emptyset \to B$ along $\emptyset$ ). The composites $j \circ ι_{A} : A \to Z$ and $j \circ ι_{B} : B \to Z$ are therefore acyclic cofibrations between cofibrant objects: each is a composite of a cofibration (the structure map $ι$ ) with an acyclic cofibration ( $j$ ), and we use that $ι_{A}$ and $ι_{B}$ are themselves weak equivalences after composition because of how the M4 factorisation interacts with the M1 two-out-of-three axiom (see the careful version below).

To make this precise without circularity, argue as follows. The composite $q \circ j \circ ι_{B} = (f, id_{B}) \circ ι_{B} = id_{B}$ is the identity. So $q \circ (j \circ ι_{B}) = id_{B}$ . The map $j \circ ι_{B}$ is a composite of two cofibrations, hence a cofibration; but more substantively, the identity $q \circ (j \circ ι_{B}) = id_{B}$ together with $j \in W$ and $q$ a fibration plus the M1 two-of-three says $j \circ ι_{B}$ is a weak equivalence (it composes with $q$ to the identity, which is a weak equivalence, and $j$ is a weak equivalence, so... we need an alternative tactic). The cleanest formulation: $j \circ ι_{B}$ is a section of $q$ , and $j$ is a weak equivalence; chasing through the two-of-three axiom on the triangle $A ⊔ B \to Z \to B$ in two ways gives $ι_{B} \in W$ and consequently $j \circ ι_{B} \in W \cap C = W \cap C$ , an acyclic cofibration. Similarly for $ι_{A}$ : we know $q \circ j \circ ι_{A} = f$ , and $f \in W$ by hypothesis; combined with $j \in W$ and M1, this gives $ι_{A} \in W$ , so $j \circ ι_{A} \in W \cap C$ is an acyclic cofibration between cofibrant objects.

By the hypothesis on $F$ , $F (j \circ ι_{A})$ and $F (j \circ ι_{B})$ are weak equivalences in $D$ . Now $q \circ (j \circ ι_{B}) = id_{B}$ gives $F (q) \circ F (j \circ ι_{B}) = F (id_{B}) = id$ , a weak equivalence; by M1 applied in $D$ , $F (q)$ is a weak equivalence. And $q \circ (j \circ ι_{A}) = f$ gives $F (f) = F (q) \circ F (j \circ ι_{A})$ , a composite of two weak equivalences. By M1 in $D$ , $F (f)$ is a weak equivalence. $□$

Bridge. Ken Brown's lemma is the foundational reason that left Quillen functors descend to homotopy categories, and it builds toward the entire derived-functor calculus of Quillen adjunctions. The central insight is that one only needs to check preservation on the smaller class of acyclic cofibrations between cofibrant objects — the M4 factorisation then propagates the preservation to all weak equivalences in the cofibrant subcategory. This is exactly the structure that makes the cofibrant-replacement functor $Q$ usable: $L \circ Q$ sends weak equivalences to weak equivalences because $Q$ takes values in cofibrant objects and $L$ preserves weak equivalences there. Putting these together with the dual statement for right Quillen functors and fibrant replacement, every Quillen adjunction induces a derived adjunction on homotopy categories with no further hypotheses. The bridge is between the bare axiomatic conditions (preserves cofibrations, preserves acyclic cofibrations) and the homotopical content (descends to $Ho$ , has a derived adjoint pair). The pattern generalises to the simplicial-localisation viewpoint of Dwyer-Kan: every functor of model categories that respects the appropriate structure on $W$ extends to a functor of simplicial localisations, and Quillen functors are precisely the case where this extension is the derived adjoint pair. The pattern appears again in the $\infty$ -categorical synthesis where every Quillen adjunction presents an adjunction of $(\infty, 1)$ -categories and every Quillen equivalence presents an equivalence. Ken Brown's lemma identifies the smallest hypothesis with the strongest conclusion in this framework.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Show that if $L : C ⇄ D : R$ is a Quillen adjunction and the adjoint pair is an equivalence of categories before passing to homotopy, then it is automatically a Quillen equivalence.

Hint

If $L ⊣ R$ is already an equivalence of categories, then $L$ and $R$ are inverse up to natural isomorphism. Show that this isomorphism preserves the three classes of morphisms and consequently descends to an equivalence on $Ho$ .

Answer

Suppose $L ⊣ R$ is both a Quillen adjunction and an equivalence of categories. Then the unit $η : id_{C} \to R L$ and counit $ε : L R \to id_{D}$ are natural isomorphisms, hence in particular weak equivalences (any isomorphism is a weak equivalence by M2 applied to the identity-as-retract; more directly, identities are weak equivalences in any model category, and natural isomorphisms are isomorphisms component-wise). The derived adjunction on $Ho$ inherits the unit and counit as natural isomorphisms after passing through cofibrant and fibrant replacement, which preserve weak equivalences and therefore isomorphisms in $Ho$ . So $L L ⊣ R R$ is an equivalence of categories, i.e. a Quillen equivalence. Rubric: full credit for identifying that the strict equivalence already passes through replacement cleanly.

Exercise 4 (medium, short-answer).

State the Quillen-equivalence criterion via the unit / counit: a Quillen adjunction $L ⊣ R$ between model categories is a Quillen equivalence if and only if which condition on the unit and counit?

Hint

The criterion involves the derived unit and counit, evaluated at cofibrant objects on the left and fibrant objects on the right.

Answer

The condition: for every cofibrant $X \in C$ , the composite $X η_{X} R L X \to R (R^{'} L X)$ (the unit followed by fibrant replacement on the right) is a weak equivalence in $C$ ; equivalently, for every fibrant $Y \in D$ , the composite $L (QR Y) \to L R Y ε_{Y} Y$ is a weak equivalence in $D$ . Even more cleanly: for every cofibrant $X \in C$ and fibrant $Y \in D$ , a morphism $L X \to Y$ in $D$ is a weak equivalence if and only if its adjoint transpose $X \to R Y$ is a weak equivalence in $C$ . Rubric: full credit for either the unit-counit form or the adjoint-transpose form.

Exercise 5 (medium, short-answer).

Verify that the geometric-realisation functor $∣ - ∣ : sSet \to Top$ preserves cofibrations in the Kan-Quillen and Quillen-Serre model structures.

Hint

Cofibrations in $sSet$ are monomorphisms; cofibrations in $Top$ include CW pair inclusions. Realise a monomorphism dimension by dimension.

Answer

Let $f : X ↪ Y$ be a monomorphism of simplicial sets. Its realisation $∣ f ∣ : ∣ X ∣ \to ∣ Y ∣$ is the induced map on geometric realisations. Working dimension by dimension: each $n$ -simplex of $X$ is sent to an $n$ -simplex of $Y$ , injectively. The realisation $∣ Y ∣$ is built by attaching topological simplices $Δ^{n}$ along their boundaries, and $∣ X ∣$ is built from the subset of simplices in the image of $f$ . So $∣ X ∣$ is a subcomplex of $∣ Y ∣$ in the CW structure inherited from the simplicial-set structure on $Y$ , and the inclusion $∣ X ∣ ↪ ∣ Y ∣$ is a CW pair inclusion. CW pair inclusions are cofibrations in the Quillen-Serre model structure on $Top$ — they are retracts of relative cell complexes (in fact, are relative cell complexes themselves). Hence $∣ f ∣$ is a cofibration. Rubric: full credit for the dimension-by-dimension argument plus identification of the result as a CW pair inclusion.

Exercise 6 (medium, symbolic).

Let $L ⊣ R$ be a Quillen adjunction. Show that for every cofibrant $X \in C$ and fibrant $Y \in D$ , the adjunction-induced bijection $$ \mathrm{Hom}\mathcal{D}(L X, Y) \cong \mathrm{Hom}\mathcal{C}(X, R Y) $$ descends to a bijection on homotopy classes $[L X, Y]_{D} ≅ [X, R Y]_{C}$ .

Hint

A left Quillen functor takes cylinder objects of cofibrant objects to cylinder objects, and dually for right Quillen functors and path objects. Use this to show the adjunction respects the left/right homotopy relations.

Answer

Recall that $[A, B]_{C}$ for $A$ cofibrant and $B$ fibrant in $C$ is $Hom_{C} (A, B) / ≃$ where $≃$ is left (equivalently right) homotopy. Given $f ≃^{ℓ} f^{'} : L X \to Y$ via a homotopy $H : Cyl (L X) \to Y$ , we want to show the adjoint transposes $\tilde{f}, \tilde{f}^{'} : X \to R Y$ are homotopic.

$L$ preserves the cylinder structure of cofibrant $X$ : $L (Cyl (X))$ is a cylinder object for $L X$ (because $L$ preserves coproducts as a left adjoint, $L$ preserves cofibrations between cofibrant objects, and by Ken Brown's lemma $L$ preserves weak equivalences between cofibrant objects). So a left homotopy $L (Cyl (X)) \to Y$ between $f$ and $f^{'}$ adjoins to a map $Cyl (X) \to R Y$ , which is a left homotopy between $\tilde{f}$ and $\tilde{f}^{'}$ . Therefore $≃^{ℓ}$ is preserved by the adjunction. Symmetrically, right homotopy is preserved using path objects on the right. The adjunction-bijection on Hom-sets thus descends to homotopy classes. Rubric: full credit for the cylinder/path argument and the explicit transpose.

Exercise 7 (hard, short-answer).

State the Dold-Kan correspondence as a Quillen equivalence and identify the relevant model structures and functors.

Hint

The correspondence relates simplicial $R$ -modules to non-negatively graded chain complexes of $R$ -modules. The Quillen-equivalence packaging adds a model structure on each side and identifies the equivalence as the derived adjunction.

Answer

Setup. Let $R$ be a ring. Let $sMod_{R}$ denote the category of simplicial $R$ -modules with the model structure where weak equivalences are the maps whose underlying simplicial-set map is a Kan-Quillen weak equivalence (equivalently, induces isomorphisms on the homotopy groups of the underlying simplicial sets), fibrations are the maps surjective on $n$ -simplices for $n \geq 0$ , and cofibrations are determined by left lifting. Let $Ch_{\geq 0} (R)$ denote chain complexes of $R$ -modules in non-negative degrees, with the projective model structure of [03.12.31]: weak equivalences are quasi-isomorphisms, fibrations are degree-wise surjections in positive degree, cofibrations are degree-wise injections with projective cokernel.

The functors. The normalised chain complex functor $N : sMod_{R} \to Ch_{\geq 0} (R)$ sends a simplicial module $A_{∙}$ to the chain complex $N (A)_{n} = ⋂_{i = 1}^{n} ker (d_{i} : A_{n} \to A_{n - 1})$ with differential $d_{0}$ . The Dold-Kan inverse $Γ : Ch_{\geq 0} (R) \to sMod_{R}$ sends a chain complex $C_{∙}$ to the simplicial module $Γ (C)_{n} = ⨁_{[n] ↠ [k]} C_{k}$ summed over surjections of the simplex category.

Theorem (Dold 1958, Kan 1958, upgraded by Quillen 1967 §II.4). The pair $Γ ⊣ N$ (with $Γ$ left adjoint and $N$ right adjoint, or in some conventions the other way around) is a Quillen equivalence. The induced equivalence on homotopy categories $$ \mathrm{Ho}(\mathbf{sMod}R) \simeq \mathrm{Ho}(\mathrm{Ch}{\geq 0}(R)) = D_{\geq 0}(R) $$ identifies the homotopy theory of simplicial $R$ -modules with the bounded-below derived category of $R$ . The classical Dold-Kan equivalence (an equivalence of categories before passing to homotopy) is a strict strengthening of the Quillen equivalence, but Quillen's framework is the right setting for the homotopical content.

Rubric: full credit for naming both functors, identifying both model structures, and stating the Quillen-equivalence conclusion.

Exercise 8 (hard, short-answer).

Prove that the realisation-singular adjunction $∣ - ∣ ⊣ Sing$ between $sSet$ (Kan-Quillen) and $Top$ (Quillen-Serre) is a Quillen adjunction, then identify which classical theorem upgrades it to a Quillen equivalence.

Hint

For Quillen-adjunction, check that $∣ - ∣$ preserves cofibrations (monomorphisms to CW pair inclusions) and acyclic cofibrations. For Quillen-equivalence, use Milnor 1957.

Answer

Quillen adjunction. Cofibrations in $sSet$ (Kan-Quillen) are monomorphisms; realisation sends a monomorphism to a CW pair inclusion (Exercise 5 above), which is a cofibration in $Top$ (Quillen-Serre). Acyclic cofibrations in $sSet$ are the monomorphisms whose realisation is a weak homotopy equivalence; their image under $∣ - ∣$ is therefore a weak homotopy equivalence and a CW pair inclusion, hence an acyclic cofibration in $Top$ . So $∣ - ∣$ preserves cofibrations and acyclic cofibrations: a left Quillen functor. The adjunction $Hom_{Top} (∣ X ∣, Y) = Hom_{sSet} (X, Sing (Y))$ exhibits the pair as a Quillen adjunction.

Quillen equivalence. The classical input is the Milnor theorem (Milnor 1957, The geometric realization of a semi-simplicial complex, Amer. J. Math. 79, 357-362): for every topological space $Y$ , the counit $∣ Sing (Y) ∣ \to Y$ is a weak homotopy equivalence. Combined with the fact that the unit $X \to Sing (∣ X ∣)$ is a weak equivalence for every Kan complex $X$ (by direct simplicial-set computation: $π_{n} (X)$ on a Kan complex matches $π_{n} (∣ X ∣)$ topologically), the derived unit and counit are isomorphisms in the appropriate homotopy categories. Hence $L ∣ - ∣ ⊣ R Sing$ is an equivalence of categories, i.e. $∣ - ∣ ⊣ Sing$ is a Quillen equivalence.

This is Quillen's Theorem II.3.1 in Homotopical Algebra (1967). Rubric: full credit for the Quillen-adjunction verification and the Milnor identification.

Lean formalization [Intermediate+]

Mathlib has the categorical infrastructure for adjunctions (Mathlib.CategoryTheory.Adjunction.Basic) and an early-stage Mathlib.AlgebraicTopology.ModelCategory.* namespace but does not yet ship the QuillenAdjunction or QuillenEquivalence API. The intended shape of the formalisation:

import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.AlgebraicTopology.ModelCategory.Basic

namespace CategoryTheory

variable {C D : Type*} [Category C] [Category D]
variable [ModelCategory C] [ModelCategory D]

/-- A Quillen adjunction between model categories: the left adjoint
preserves cofibrations and acyclic cofibrations. -/
structure QuillenAdjunction (L : C ⥤ D) (R : D ⥤ C) extends Adjunction L R where
  preserves_cofibrations :
    ∀ {X Y : C} (i : X ⟶ Y),
      (ModelCategory.structure : ModelStructure C).cofibrations i →
      (ModelCategory.structure : ModelStructure D).cofibrations (L.map i)
  preserves_acyclic_cofibrations :
    ∀ {X Y : C} (i : X ⟶ Y),
      ((ModelCategory.structure : ModelStructure C).cofibrations ⊓
       (ModelCategory.structure : ModelStructure C).weakEquivalences) i →
      ((ModelCategory.structure : ModelStructure D).cofibrations ⊓
       (ModelCategory.structure : ModelStructure D).weakEquivalences) (L.map i)

/-- The total left derived functor of a Quillen adjunction. -/
def QuillenAdjunction.derivedLeft {L : C ⥤ D} {R : D ⥤ C}
    (Q : QuillenAdjunction L R) :
    ModelCategory.homotopyCategory C ⥤ ModelCategory.homotopyCategory D :=
  sorry -- compose L with a cofibrant-replacement functor and descend via Ken Brown

/-- A Quillen equivalence is a Quillen adjunction whose derived adjunction
is an equivalence of categories. -/
class QuillenEquivalence {L : C ⥤ D} {R : D ⥤ C}
    (Q : QuillenAdjunction L R) : Prop where
  derived_is_equivalence : IsEquivalence Q.derivedLeft

end CategoryTheory

The pieces needed for a working instance: the ModelStructure C API with cofibrant and fibrant replacement functors (M4 functorial factorisation), the Ken Brown lemma as a verified theorem providing the well-definedness of the derived functor on $Ho (C)$ , and the construction of the derived adjunction including the verified unit and counit. With the Kan-Quillen structure on SSet and the Quillen-Serre structure on TopCat, the realisation-singular pair SSet.toTop ⊣ TopCat.toSSet would then be the canonical first verified Quillen-equivalence instance; the proof goes through Milnor's theorem (currently unformalised in Mathlib) on the counit being a weak homotopy equivalence.

Advanced results [Master]

Theorem (Quillen 1967, total derived functor characterisation). Let $L ⊣ R$ be a Quillen adjunction. The total left derived functor $L L : Ho (C) \to Ho (D)$ is characterised universally as the right Kan extension of $L \circ γ_{D}$ along the localisation $γ_{C} : C \to Ho (C)$ , restricted to cofibrant objects. Dually for $R R$ .

The characterisation makes derived functors a special case of Kan extension, identifying the model-categorical construction with the abstract categorical one. Riehl 2014 §10 develops this viewpoint in detail.

Theorem (Hovey 1999 §1.3.13, criterion for Quillen equivalence on cofibrant-fibrant subcategory). A Quillen adjunction $L ⊣ R$ is a Quillen equivalence if and only if the restricted adjunction on cofibrant-fibrant subcategories, $$ L : (\mathcal{C}{\mathrm{cf}})/!!\simeq \rightleftarrows (\mathcal{D}{\mathrm{cf}})/!!\simeq : R, $$ induces an equivalence of categories of cofibrant-fibrant objects modulo homotopy. This is equivalent to: for every cofibrant $X \in C$ and fibrant $Y \in D$ , a morphism $L X \to Y$ in $D$ is a weak equivalence if and only if its adjoint transpose $X \to R Y$ in $C$ is a weak equivalence.

This is the most computationally useful form: rather than verify an equivalence of full homotopy categories, one checks the bi-implication on a single pair of cofibrant-fibrant objects at a time.

Theorem (composition of Quillen adjunctions). If $L_{1} ⊣ R_{1} : C \to D$ and $L_{2} ⊣ R_{2} : D \to E$ are Quillen adjunctions between model categories, then the composite $L_{2} L_{1} ⊣ R_{1} R_{2} : C \to E$ is a Quillen adjunction. If both are Quillen equivalences, the composite is a Quillen equivalence.

Composition follows from the preservation conditions being preserved under composition. Hence Quillen equivalences form a well-behaved 2-categorical structure on the collection of model categories.

Theorem (substantive examples of Quillen equivalences). The following are Quillen equivalences:

(a) Realisation-singular: $∣ - ∣ ⊣ Sing : Top \to sSet$ between the Quillen-Serre structure on $Top$ and the Kan-Quillen structure on $sSet$ (Quillen 1967 Theorem II.3.1; the Milnor 1957 theorem on the counit is the key classical input).

(b) Dold-Kan: $N ⊣ Γ$ between $sMod_{R}$ and $Ch_{\geq 0} (R)$ , both with their standard model structures (Dold 1958, Kan 1958 — strict equivalence; Quillen 1967 §II.4 — upgrade to Quillen equivalence).

(c) Bar-cobar: $B ⊣ Ω$ between simplicial groups and reduced simplicial sets, the Quillen-equivalence underpinning the modern construction of classifying spaces of simplicial groups (Kan 1958; Quillen 1967 §I.4 in the model-category language).

(d) Hurewicz-Strøm versus Quillen-Serre on $Top$ : the identity functor $Top_{Strøm} \to Top_{Quillen}$ is a left Quillen functor but not a Quillen equivalence — the homotopy categories are genuinely distinct (one is the honest homotopy category, the other the weak-homotopy version).

(e) Different model structures on $sSet$ : Kan-Quillen and Joyal are not Quillen-equivalent; the former models $\infty$ -groupoids and the latter models $(\infty, 1)$ -categories, distinct homotopical content on the same underlying category.

Theorem (Hovey 1999 monoidal Quillen functors). If $C, D$ are monoidal model categories satisfying the pushout-product axiom and $L ⊣ R$ is a Quillen adjunction such that $L$ is strong monoidal and the unit map of the adjunction respects the monoidal structure, then $L L ⊣ R R$ is a monoidal adjunction on homotopy categories. If $L ⊣ R$ is a Quillen equivalence, the derived adjunction is a monoidal equivalence.

This is the framework for transporting monoidal structures across Quillen equivalences. Smash products on $sSet_{*}$ transport to topological spaces via the realisation-singular equivalence, and the derived smash product on the stable homotopy category lifts via the Hovey-Shipley-Smith 2000 Quillen equivalence between symmetric spectra and the topological-spectra of Bousfield-Friedlander.

Theorem (Dwyer-Kan simplicial-localisation viewpoint). Let $L ⊣ R$ be a Quillen adjunction. The simplicial localisations $L^{H} C$ and $L^{H} D$ at the weak-equivalence subcategories are connected by a Dwyer-Kan adjunction, and the model-categorical derived adjunction $L L ⊣ R R$ on $Ho (C) ⇄ Ho (D)$ agrees with the homotopy-category truncation of the simplicial-localisation adjunction. A Quillen equivalence presents a Dwyer-Kan equivalence of simplicial localisations.

This identifies the model-categorical derived adjunction with the abstract $\infty$ -categorical adjunction presented by the underlying $(\infty, 1)$ -categories.

Theorem (Lurie 2009 $(\infty, 1)$ -categorical presentation). Every Quillen adjunction presents an adjunction of $(\infty, 1)$ -categories via the nerve construction $N : ModCat \to (\infty, 1) Cat$ , and every Quillen equivalence presents an equivalence of $(\infty, 1)$ -categories. The model-categorical derived adjunction agrees with the $(\infty, 1)$ -categorical adjunction up to canonical equivalence.

This is the modern conceptual home of the Quillen-functor framework: a Quillen adjunction is a concrete presentation of an abstract $(\infty, 1)$ -categorical adjunction, and the framework's longevity comes from its computational accessibility relative to fully $(\infty, 1)$ -categorical methods.

Synthesis. The Quillen-functor and Quillen-equivalence framework is the foundational reason that abstract homotopy theory can be done category-theoretically rather than by tracking spaces, simplicial sets, and chain complexes separately. The central insight is that an adjoint pair between model categories, with the modest preservation conditions on cofibrations and fibrations encoded by the M3 lifting axiom of [03.12.31], automatically descends to a derived adjunction on homotopy categories — Ken Brown's lemma identifies the smallest hypothesis (preservation of acyclic cofibrations between cofibrant objects) with the strongest conclusion (descent to $Ho$ ). Putting these together with the unit-counit characterisation of equivalence, every Quillen equivalence presents an honest equivalence of homotopy theories with computational content on both sides. This is exactly the structure that identifies the simplicial side $sSet$ with the topological side $Top$ through the realisation-singular Quillen equivalence, and identifies the simplicial $R$ -module side with chain complexes through Dold-Kan; the bridge is between the concrete model-categorical presentation and the abstract $(\infty, 1)$ -categorical content. The pattern generalises to the full Hovey-Shipley-Smith calculus of stable homotopy theory, where Quillen equivalences between symmetric spectra and topological spectra preserve smash products as monoidal Quillen equivalences, and recurs in the modern derived-algebraic-geometry framework where Quillen equivalences between simplicial commutative rings and connective $E_{\infty}$ -rings present the underlying $(\infty, 1)$ -category of derived rings.

The framework builds toward the broader programme initiated by Dwyer-Kan and completed by Lurie: every $(\infty, 1)$ -category arises as the homotopy theory of a model category (under mild combinatorial hypotheses), and every functor between $(\infty, 1)$ -categories is presented by a Quillen adjunction up to homotopy. Quillen functors are the concrete computational tool; $(\infty, 1)$ -categorical functors are the abstract conceptual home; Quillen equivalences are the explicit equivalences that connect different presentations of the same homotopy theory. This pattern recurs through the modern algebraic-topology landscape: motivic homotopy theory (Morel-Voevodsky 1999) is presented by a Quillen-equivalence-equivalent family of model structures on simplicial presheaves; equivariant stable homotopy (Mandell-May 2002) by Quillen-equivalent symmetric-spectrum model structures; chromatic homotopy theory by Quillen-equivalent localisations at Morava $K$ -theory.

Full proof set [Master]

Proposition (Quillen adjunction: equivalence of left and right characterisations). Let $L : C ⇄ D : R$ be an adjoint pair between model categories. The following are equivalent:

(i) $L$ preserves cofibrations and acyclic cofibrations.

(ii) $R$ preserves fibrations and acyclic fibrations.

(iii) $L$ preserves cofibrations and $R$ preserves fibrations.

(iv) $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations.

Proof. We show (i) $\Leftrightarrow$ (iii) $\Leftrightarrow$ (ii); the equivalence with (iv) follows by combining (i) and (ii) and noting the redundancy.

(i) $\Rightarrow$ (iii). Suppose $L$ preserves cofibrations and acyclic cofibrations. We show $R$ preserves fibrations. Let $p : E \to B$ be a fibration in $D$ . To show $R (p)$ is a fibration in $C$ , by the M3 dual characterisation (Proposition in [03.12.31]) it suffices to show $R (p)$ has the right lifting property against every acyclic cofibration in $C$ . Let $j : A \to X$ be an acyclic cofibration in $C$ . Given a commutative square $$ \begin{array}{ccc} A & \xrightarrow{u} & R(E) \ \downarrow j & & \downarrow R(p) \ X & \xrightarrow{v} & R(B) \end{array} $$ adjoin transposes to $$ \begin{array}{ccc} L(A) & \xrightarrow{\tilde{u}} & E \ \downarrow L(j) & & \downarrow p \ L(X) & \xrightarrow{\tilde{v}} & B \end{array} $$ By hypothesis (i), $L (j)$ is an acyclic cofibration in $D$ . By M3, the square has a diagonal lift $\tilde{h} : L (X) \to E$ . The adjoint transpose $h : X \to R (E)$ is the required lift, exhibiting $R (p)$ as a fibration.

(iii) $\Rightarrow$ (ii). Suppose $L$ preserves cofibrations and $R$ preserves fibrations. We show $R$ preserves acyclic fibrations. Let $p : E \to B$ be an acyclic fibration in $D$ . To show $R (p)$ is an acyclic fibration in $C$ , use the M3 dual characterisation: $R (p)$ is an acyclic fibration if it has the right lifting property against every cofibration. Let $i : A \to X$ be a cofibration in $C$ . The adjoint-transpose argument identical to the (i) $\Rightarrow$ (iii) step, with $L (i)$ a cofibration (by hypothesis) and $p$ an acyclic fibration: M3 lifts in $D$ , the transpose lifts in $C$ .

(ii) $\Rightarrow$ (i). Symmetric to (i) $\Rightarrow$ (ii): suppose $R$ preserves fibrations and acyclic fibrations. The same adjunction-transpose argument with the squares reversed shows $L$ preserves cofibrations and acyclic cofibrations.

The four conditions are therefore equivalent. The standard convention is to use (i) as the definition; condition (ii) is the dual; condition (iv) is the "minimal" form that suffices to give both (i) and (ii) by combining with (iii). $□$

Proposition (well-definedness of the derived adjunction). Let $L ⊣ R$ be a Quillen adjunction. Let $Q$ and $R^{'}$ be cofibrant and fibrant replacement functors on $C$ and $D$ respectively. Then $L L = L \circ Q$ and $R R = R \circ R^{'}$ descend to functors on the homotopy categories, the descended functors are well-defined up to canonical natural isomorphism, and they form an adjoint pair on $Ho (C) ⇄ Ho (D)$ .

Proof. Three steps: descent of $L \circ Q$ , descent of $R \circ R^{'}$ , and the adjunction relation.

Descent of $L \circ Q$ . The cofibrant replacement functor $Q$ takes every object to a cofibrant object, and sends every weak equivalence to a weak equivalence between cofibrant objects (by the cofibrant-replacement functoriality lemma in [03.12.31]). Ken Brown's lemma applied to $L$ — which by hypothesis sends acyclic cofibrations between cofibrant objects to weak equivalences — shows $L$ sends all weak equivalences between cofibrant objects to weak equivalences in $D$ . The composite $L \circ Q$ therefore sends weak equivalences in $C$ to weak equivalences in $D$ , hence descends to a functor $L L : Ho (C) \to Ho (D)$ . Independence of the choice of $Q$ : any two cofibrant-replacement functors are connected by a natural homotopy equivalence (cofibrant-replacement functoriality), and $L$ preserves homotopy equivalence between cofibrant objects, so the resulting functors on $Ho (C)$ agree up to canonical natural isomorphism.

Descent of $R \circ R^{'}$ . Symmetric: fibrant replacement is functorial up to homotopy, $R$ preserves weak equivalences between fibrant objects by the dual of Ken Brown's lemma, and $R \circ R^{'}$ therefore descends to $R R : Ho (D) \to Ho (C)$ independent of the choice of $R^{'}$ .

Adjunction relation. For any cofibrant $X \in C$ and fibrant $Y \in D$ , the adjunction bijection $$ \mathrm{Hom}\mathcal{D}(L X, Y) \cong \mathrm{Hom}\mathcal{C}(X, R Y) $$ descends to homotopy classes by Exercise 6 (cylinder/path-object preservation across the adjunction). For arbitrary objects, replace $X$ by $QX$ (cofibrant) and $Y$ by $R^{'} Y$ (fibrant), and use that $X \to QX$ and $Y \to R^{'} Y$ are weak equivalences (becoming isomorphisms in $Ho$ ). The bijection becomes $$ \mathrm{Hom}{\mathrm{Ho}(\mathcal{D})}(\mathbf{L}L X, Y) \cong \mathrm{Hom}{\mathrm{Ho}(\mathcal{C})}(X, \mathbf{R}R Y), $$ exhibiting $L L ⊣ R R$ on homotopy categories. $□$

Proposition (Quillen-equivalence characterisations). Let $L ⊣ R$ be a Quillen adjunction. The following are equivalent.

(i) $L L ⊣ R R$ is an equivalence of categories.

(ii) For every cofibrant $X \in C$ and fibrant $Y \in D$ , a morphism $f : L X \to Y$ in $D$ is a weak equivalence if and only if its adjoint transpose $\tilde{f} : X \to R Y$ is a weak equivalence in $C$ .

(iii) For every cofibrant $X \in C$ , the composite $X η_{X} R L X R (ρ_{L X}) R (R^{'} L X)$ — the unit followed by the fibrant-replacement map of $L X$ — is a weak equivalence in $C$ . Equivalently for the counit of fibrant $Y$ .

Proof. (i) $\Rightarrow$ (ii): if $L L ⊣ R R$ is an equivalence on $Ho$ , then the derived unit and counit are isomorphisms in $Ho$ , hence weak equivalences when computed on the cofibrant-fibrant model. The bi-implication of (ii) follows by chasing the adjoint-transpose construction: $f$ a weak equivalence in $D$ becomes an isomorphism in $Ho (D)$ ; the adjunction-bijection on $Ho$ sends it to the corresponding $\tilde{f}$ in $Ho (C)$ , which is an isomorphism iff $\tilde{f}$ is a weak equivalence.

(ii) $\Rightarrow$ (iii): apply (ii) with $f$ the fibrant-replacement map $L X \to R^{'} L X$ , which is a weak equivalence by construction. Its adjoint transpose is the composite in (iii), hence a weak equivalence by (ii).

(iii) $\Rightarrow$ (i): the unit isomorphism on $Ho (C)$ at $X$ is, computed concretely, the composite $X \to R L X \to R (R^{'} L X) = R R L L X$ . By (iii) this composite is a weak equivalence, hence an isomorphism in $Ho (C)$ . Symmetrically the counit is an isomorphism. So $L L ⊣ R R$ is an equivalence on $Ho$ . $□$

Proposition (realisation-singular Quillen equivalence). The adjunction $∣ - ∣ ⊣ Sing : Top \to sSet$ is a Quillen equivalence between the Quillen-Serre model structure on $Top$ and the Kan-Quillen model structure on $sSet$ .

Proof sketch (Quillen 1967 Theorem II.3.1). Three pieces: $∣ - ∣$ is a left Quillen functor, the counit is a weak equivalence, the unit is a weak equivalence on cofibrant objects.

Left Quillen functor. Cofibrations in $sSet$ are monomorphisms; realisation sends a monomorphism to a CW pair inclusion (Exercise 5), which is a cofibration in $Top$ . Acyclic cofibrations are monomorphisms with weak-equivalence realisation; their image is an acyclic cofibration in $Top$ tautologically.

Counit weak equivalence. The Milnor 1957 theorem: for every topological space $Y$ , the map $∣ Sing (Y) ∣ \to Y$ obtained by assembling all singular simplices into a CW structure is a weak homotopy equivalence. The proof uses that $Sing (Y)$ is a Kan complex (horns extend by the homotopy lifting property of $Sing$ ), and the realisation of a Kan complex matches the simplicial-set homotopy groups dimension by dimension.

Unit weak equivalence on cofibrant objects. Every simplicial set is cofibrant in Kan-Quillen (monomorphisms include $\emptyset \to X$ for every $X$ ). The unit $X \to Sing (∣ X ∣)$ on a Kan complex $X$ is a weak equivalence by the same dimension-by-dimension homotopy-group argument as the counit. On a non-Kan simplicial set, factor $X \to R^{'} (X)$ through Kan replacement and use that $∣ X ∣ = ∣ R^{'} (X) ∣$ up to homotopy on the topological side.

Combining the three pieces with the criterion of the previous proposition (the unit/counit characterisation), the pair $∣ - ∣ ⊣ Sing$ is a Quillen equivalence. $□$

Proposition (Dold-Kan Quillen equivalence). Let $R$ be a ring. The normalised-chain-complex adjunction $Γ ⊣ N : sMod_{R} \to Ch_{\geq 0} (R)$ (with $Γ$ left adjoint, $N$ right adjoint) is a Quillen equivalence between the standard model structure on $sMod_{R}$ and the projective model structure on $Ch_{\geq 0} (R)$ .

Proof sketch (Quillen 1967 §II.4; modern: Goerss-Jardine 2009 §III.2). The classical Dold-Kan correspondence is the statement that $N$ and $Γ$ are mutually inverse equivalences of ordinary categories — a strict equivalence on the underlying categories. Quillen's contribution is the upgrade to the model-categorical level. The two model structures are designed so that $N$ matches the simplicial-module weak equivalences with the chain-complex quasi-isomorphisms: $N$ is the normalised chain complex, and the simplicial homotopy groups of a simplicial $R$ -module $A_{∙}$ match the homology groups of $N (A)$ by the Dold-Kan-Eilenberg-Zilber theorem.

Since $N$ and $Γ$ are already inverse equivalences before passing to homotopy, the derived adjunction $L Γ ⊣ R N$ is automatically an equivalence on $Ho$ , by the strict-equivalence-implies-Quillen-equivalence observation (Exercise 3). The model-categorical content is that the categorical equivalence respects the chosen weak-equivalence classes, which is the content of the classical Eilenberg-Zilber theorem. $□$

Proposition (Quillen equivalence is preserved under composition). Let $L_{1} ⊣ R_{1} : C \to D$ and $L_{2} ⊣ R_{2} : D \to E$ be Quillen equivalences. The composite $L_{2} L_{1} ⊣ R_{1} R_{2} : C \to E$ is a Quillen equivalence.

Proof. Composition is a Quillen adjunction by direct preservation: $L_{2} L_{1}$ preserves cofibrations and acyclic cofibrations because each factor does. The derived composite $L (L_{2} L_{1}) = L L_{2} \circ L L_{1}$ is the composite of two equivalences on $Ho$ , hence an equivalence. Concretely, we need that the derived composite agrees with $L L_{2} \circ L L_{1}$ ; this holds because cofibrant replacement is functorial up to homotopy and $L_{1}$ preserves cofibrant objects up to weak equivalence, so $L_{2} (Q (L_{1} (QX)))$ agrees with $L_{2} L_{1} (QX)$ in $Ho (E)$ . $□$

Connections [Master]

Quillen model category 03.12.31. The direct prerequisite: Quillen functors are morphisms between model categories, and the entire derived-functor calculus depends on the M3 lifting and M4 factorisation axioms of [03.12.31]. Ken Brown's lemma is the bridge that lets the model-category structure descend to homotopy categories under the modest preservation conditions on a left Quillen functor.
Simplicial sets and geometric realization 03.12.25. The realisation-singular adjunction is the canonical example of a Quillen equivalence and the foundational compatibility result identifying the simplicial side with the topological side. The Kan-Quillen model structure on $sSet$ from [03.12.25] provides the simplicial model category; the geometric realisation functor from the same unit is the left adjoint of the Quillen equivalence.
Delta-complex / semi-simplicial set 03.12.22. The semi-simplicial framework provides the face-only data underlying the full simplicial-set construction. Quillen functors between semi-simplicial sets and simplicial sets — the inclusion-realisation-degeneracy adjunctions — sit one level beneath the Quillen-equivalence framework, presenting the equivalence of homotopy theories between $Δ$ -complexes and CW complexes that classical algebraic topology used to verify by hand.
Homotopy 03.12.01. Classical homotopy of continuous maps and homotopy of simplicial-set maps are matched across the realisation-singular Quillen equivalence: the homotopy classes $[X, Y]_{sSet}$ between Kan complexes correspond bijectively to the classical $[∣ X ∣, ∣ Y ∣]_{Top}$ via $∣ - ∣$ . The Quillen-equivalence framework is what makes this correspondence functorial and computable.
CW complex 03.12.10. The cofibrant objects in the Quillen-Serre structure on $Top$ are exactly the retracts of CW complexes. The realisation $∣ X ∣$ of a simplicial set is canonically a CW complex, and the realisation-singular Quillen equivalence identifies the CW homotopy theory in $Top$ with the Kan-complex homotopy theory in $sSet$ . The Whitehead theorem (every weak equivalence between CW complexes is a homotopy equivalence) is recovered as the cofibrant-fibrant restriction of the Quillen equivalence.
Eilenberg-MacLane spaces 03.12.05. The Dold-Kan Quillen equivalence between simplicial abelian groups and non-negatively graded chain complexes identifies $K (π, n)$ on the topological side with the chain complex $π [n]$ concentrated in degree $n$ on the algebraic side. The cohomology classification $H^{n} (X; π) = [X, K (π, n)]$ is then a corollary of the Quillen-equivalence-transferred adjunction.
Sullivan minimal models 03.12.06. Rational homotopy theory is presented by the Bousfield-Gugenheim Quillen equivalence between cdga's over $Q$ and rational simplicial sets (or rational topological spaces). The Sullivan minimal model is the cofibrant replacement of a space in the cdga model category, and rational homotopy equivalence transports across the Quillen equivalence.
Singular homology 03.12.11. The homotopy-invariance of singular homology is a corollary of the realisation-singular Quillen equivalence combined with the Dold-Kan Quillen equivalence: $H_{*} (X; Z)$ is the composite of the singular complex functor (right Quillen) with the normalised-chains functor (right Quillen for Dold-Kan) and then homology, and homology takes quasi-isomorphisms to isomorphisms.

Historical & philosophical context [Master]

Daniel Quillen introduced the model-category framework and the Quillen-adjunction-equivalence calculus in his 1967 Lecture Notes in Mathematics monograph Homotopical Algebra (Springer LNM 43) ^{[Quillen 1967]}. The defining conditions on left and right Quillen functors appear in §I.4, where Quillen showed that an adjoint pair preserving cofibrations and fibrations descends to an adjunction on homotopy categories via the cofibrant and fibrant replacement constructions. The Quillen-equivalence concept appears in the same chapter, and §II.3-§II.4 of the same monograph proved the two foundational instances: the realisation-singular pair $∣ - ∣ ⊣ Sing$ between simplicial sets and topological spaces, and the Dold-Kan pair between simplicial $R$ -modules and chain complexes. Both Quillen equivalences were the substantive compatibility results connecting the combinatorial and topological (or algebraic) sides of the framework.

Ken Brown's lemma — the result that a left Quillen functor preserves weak equivalences between cofibrant objects — appears explicitly in Brown's 1973 Transactions of the American Mathematical Society paper Abstract homotopy theory and generalized sheaf cohomology (TAMS 186, 419-458) ^{[Brown 1973]}, as Proposition 1.3 in the abstract-homotopy-theory section. Brown's framework was a slight variant of Quillen's, designed for sheaf-theoretic applications; the lemma was the key technical input that made Quillen functors into a workable calculus.

The framework's modern reformulation came in Mark Hovey's 1999 monograph Model Categories (AMS Mathematical Surveys 63) ^{[Hovey 1999]}, which gave the first systematic treatment of Quillen adjunctions and equivalences as a calculus, including the functorial-factorisation refinement of Quillen's original construction and the unit-counit characterisation of equivalence. Philip Hirschhorn's 2003 Model Categories and Their Localizations (AMS Mathematical Surveys 99) ^{[Hirschhorn 2003]} developed the theory in further generality, with the small-object argument and Bousfield localisation as the main applications. Paul Goerss and Rick Jardine's Simplicial Homotopy Theory ^{[Goerss-Jardine 2009]} gave the canonical modern treatment of the realisation-singular and Dold-Kan Quillen equivalences in the Kan-Quillen model structure on simplicial sets.

The simplicial-localisation viewpoint of William Dwyer and Daniel Kan, developed in their 1980 Journal of Pure and Applied Algebra paper Simplicial localizations of categories (J. Pure Appl. Algebra 17, 267-284) ^{[Dwyer-Kan 1980]}, gave an alternative presentation of derived functors via the simplicial localisation $L^{H} C$ at a class of weak equivalences. Dwyer-Kan showed that the model-categorical derived functor of a Quillen adjunction agrees with the simplicial-localisation construction; both are presentations of the underlying $(\infty, 1)$ -categorical adjunction. The $(\infty, 1)$ -categorical synthesis was completed by André Joyal (2002 paper on quasi-categories) and Jacob Lurie in Higher Topos Theory (2009) ^{[Lurie 2009]}, where Quillen adjunctions and equivalences are identified as concrete presentations of abstract $(\infty, 1)$ -categorical adjunctions and equivalences.

Applications of the Quillen-functor calculus extend across the modern landscape of algebraic topology and derived algebra: motivic homotopy theory (Morel-Voevodsky 1999) is built on Quillen-equivalent simplicial-presheaf model structures; symmetric spectra (Hovey-Shipley-Smith 2000) are Quillen-equivalent to topological spectra and provide a monoidal model for the stable homotopy category; derived algebraic geometry (Toën-Vezzosi 2008, Lurie 2009-) uses Quillen-equivalent simplicial-commutative-ring model categories to present derived rings; and chromatic homotopy theory uses Quillen-equivalent localisations at Morava $K$ -theory.

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

Prerequisites

03.12.01
03.12.10
03.12.22
03.12.25
03.12.31

Tier anchors

beginner: Dwyer-Spalinski 1995 *Homotopy theories and model categories* §9-§10, the canonical informal introduction to Quillen functors and equivalences
intermediate: Quillen 1967 *Homotopical Algebra* (LNM 43) §I.4 left/right derived functors, §II.3 the realisation-singular adjunction; Hovey 1999 *Model Categories* §1.3.1-§1.3.10; Goerss-Jardine 2009 *Simplicial Homotopy Theory* §II.3
master: Quillen 1967 *Homotopical Algebra* (LNM 43) §I.4-§II.4 (originator); Hovey 1999 *Model Categories* (AMS Mathematical Surveys 63) §1.3-§1.4; Hirschhorn 2003 *Model Categories and Their Localizations* (AMS Mathematical Surveys 99) §8.5; Goerss-Jardine 2009 *Simplicial Homotopy Theory* §II.3 (Kan-Quillen + realisation); Dwyer-Kan 1980 *Simplicial localizations of categories* (J. Pure Appl. Algebra 17) — simplicial-localisation viewpoint on derived functors

References

TODO_REF
Quillen — Homotopical Algebra · Springer Lecture Notes in Mathematics 43 (1967); §I.4 derived functors and Quillen adjunctions, §I.5 Ken Brown's lemma, §II.3 the Quillen equivalence $|{-}| \dashv \mathrm{Sing}$ between simplicial sets and topological spaces, §II.4 Dold-Kan as a Quillen equivalence
TODO_REF
Hovey — Model Categories · AMS Mathematical Surveys and Monographs 63 (1999); §1.3.1 Quillen adjunctions, §1.3.4 Ken Brown's lemma, §1.3.10 Quillen equivalences and the homotopy-category characterisation
TODO_REF
Dwyer-Spalinski — Homotopy theories and model categories · Handbook of Algebraic Topology (Elsevier 1995) Ch. 2 pp. 73-126; §9 Quillen functors and equivalences with the realisation example worked in detail
TODO_REF
Hirschhorn — Model Categories and Their Localizations · AMS Mathematical Surveys and Monographs 99 (2003); §8.5 Quillen functors, total derived functors via cofibrant and fibrant approximations, §8.5.23 Quillen equivalence criterion
TODO_REF
Goerss-Jardine — Simplicial Homotopy Theory · Birkhäuser Progress in Mathematics 174 (1999, repr. 2009); §II.3 the realisation-singular Quillen equivalence, §III.2 simplicial $R$-modules and the Dold-Kan Quillen equivalence
TODO_REF
Brown — Abstract homotopy theory and generalized sheaf cohomology · Transactions of the American Mathematical Society 186 (1973) 419-458; originator of Ken Brown's lemma (Proposition 1.3 there) — a left Quillen functor preserves weak equivalences between cofibrant objects
TODO_REF
Dwyer-Kan — Simplicial localizations of categories · Journal of Pure and Applied Algebra 17 (1980) 267-284; alternative presentation of derived functors via the simplicial localisation $L^H \mathcal{C}$ — equivalent to the model-category derived functor when both presentations apply
TODO_REF
Dold-Kan correspondence — Dold 1958, Kan 1958 · Dold 1958 *Homology of symmetric products and other functors of complexes* Ann. Math. 68; Kan 1958 *Functors involving css complexes* Trans. AMS 87; the original correspondence between simplicial abelian groups and non-negatively graded chain complexes, upgraded by Quillen 1967 §II.4 to a Quillen equivalence
TODO_REF
Milnor — The geometric realization of a semi-simplicial complex · American Journal of Mathematics 79 (1957) 357-362; the foundational classical theorem that $|\mathrm{Sing}(X)| \to X$ is a weak homotopy equivalence for every space $X$, underpinning the realisation-singular Quillen equivalence
TODO_REF
Riehl — Categorical Homotopy Theory · Cambridge 2014; §10 derived functors via deformations, §11 Quillen adjunctions and equivalences from the categorical viewpoint

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 90m