SOME SIMPLICIAL HOMOTOPY THEORY 1. Motivation and Basic

SOME SIMPLICIAL HOMOTOPY THEORY

1. Motivation and basic definitions Fabian proved in Topology II that the canonical map |Sing(X)| → X is a weak equivalence for every topological space X, so we “know” that simplicial sets are sufficient to model (weak) homotopy types. We would like to try to ex- pand on this idea by showing that it is possible to “do homotopy theory” in the category of simplicial sets just as one usually does in topological spaces. One of the key outcomes of this discussion will be the observation that there is essentially no difference between simplicial sets and topological spaces as far as homotopy theory is concerned. Of course, the translation between these two categories will be performed through the familiar adjunction between geometric realisation and the singular set functor. However, our goal will be to develop a homotopy theory for simplicial sets independently of topological spaces and only make the comparison at the very end. To get things off the ground, let us attempt to find simplicial analogues of the key notions that allow us to do homotopy theory in topological spaces: these are fibrations, cofibrations and weak equivalences. Let us first find an appropriate notion of fibration. The following terminology is not only useful for this purpose, but will reappear throughout the entire lecture course. 1.1. Definition. Let C be a category, let f : X → Y be a morphism in C and let Σ be a class of morphisms in C. (1) The map f has the right lifiting property (or RLP for short), with respect to Σ if for every commutative square S x X s f y T Y with s ∈ Σ there exists a commutative diagram of the following form: S x X y s f y T Y (2) The map f has the left lifting property (or LLP) with respect to Σ if for every commutative square X x S f s y Y T with s ∈ Σ there exists a commutative diagram of the following form: X x S y f s y Y T 1 2

1.2. Notation. Let C be a category with ﬁnite products and pushouts. For i: A → B and f : X → Y morphisms in C, let

i f :(A × Y ) ∪A×X (B × X) → B × Y be the canonical arrow induced by the universal property of the pushout. Recall that a map of topological spaces p: E → B is a (Serre) fibration if it has the RLP for the set of inclusions n−1 n (S ,→ D ) ({0} ,→ [0, 1]) | n ≥ 1 . There is an evident analogue of these maps in simplicial sets, namely n n 1 (∂∆ ,→ ∆ ) ({0} ,→ ∆ ). However, note that there is a homeomorphism n−1 n ∼ n−1 n (S ,→ D ) ({0} ,→ [0, 1]) = (S ,→ D ) ({1} ,→ [0, 1]) which has no simplicial analogue. This is the reason why it will also be important to consider the maps n n 1 (∂∆ ,→ ∆ ) ({1} ,→ ∆ ). 1.3. Definition. (1) A map of simplicial sets is a left fibration if it has the RLP with respect to the set n n 1 LC := (∂∆ ,→ ∆ ) ({0} ,→ ∆ ) | n ≥ 1 . (2) A map of simplicial sets is a right fibration if it has the RLP with respect to the set n n 1 RC := (∂∆ ,→ ∆ ) ({1} ,→ ∆ ) | n ≥ 1 . (3) A map of simplicial sets is a Kan fibration if it is both a left and a right fibration, ie if it has the RLP with respect to the set C := LC ∪ RC. 1.4. Remark. With this definition, it is not clear that a simplicial set X is Kan if and only if X → ∆0 is a Kan fibration. This will be shown in Corollary 2.15. Similarly, cofibrations in topological spaces were defined in terms of the homotopy extension property. We could try to mimic this definition directly, but this turns out to be a bad idea. For example, we would certainly expect the inclusion n n 1 (∂∆ ,→ ∆ ) ({0} ,→ ∆ ) to be a cofibration. However, if we took “cofibration” to mean “has the homotopy extension property with respect to all simplicial sets”, then this map would have to have a left inverse. It is easy to check in the case n = 1 that no such retraction exists (nor does it for any other n). On the other hand, we can think of every (left) homotopy extension problem as a commutative diagram

1 A × ∆ ∪A×{0} B × {0} X

B × ∆1 ∆0 In particular, for A = ∂∆n and B = ∆n we have already deﬁned this problem to have a solution if X → ∆0 is a left ﬁbration.

1 1.5. Definition. A morphism i: A → B is a left cofibration if i ({0} ,→ ∆ ) has the LLP with respect to all Kan fibrations X → ∆0. SOME SIMPLICIAL HOMOTOPY THEORY 3

1.6. Remark. i) It would actually be more natural to require X → ∆0 to be a left Kan fibration in the definition of a left cofibration. We will see later that every left fibration over ∆0 is a Kan fibration. ii) One could equally well consider right cofibrations and cofibrations, and develop a parallel theory. We will show soon that this is not a parallel theory, but the same theory. 1.7. Example. (1) By definition, the inclusion ∂∆n → ∆n is a left cofibration for all n ≥ 0 if we set ∂∆0 := ∅. (2) The map ∅ → B is a left cofibration for every simplicial set B. Finally, recall that the following statements are equivalent for a map f : X → Y of CW-complexes: (1) f is a homotopy equivalence; (2) f induces a bijection on π0 and an isomorphism on all homotopy groups with respect to every choice of base point; (3) for every space Z, the map f induces a weak homotopy equivalence ∗ f : HomTop(Y,Z) → HomTop(X,Z); (4) for every space Z, the map f induces a bijection ∗ f : π0 HomTop(Y,Z) → π0 HomTop(X,Z). When defining cofibrations, we already observed that homotopies in the category of simplicial sets are more rigid than homotopies in topological spaces, so it is unlikely that copying condition (1) will give a sensible notion. For example, our geometric n n 1 intuition tells us that (∂∆ ,→ ∆ ) ({0} ,→ ∆ ) should be an equivalence, but it is certainly not a homotopy equivalence. Conditions (2) and (3) would require us to first develop a sensible notion of homotopy groups for simplicial sets. So we will try to adopt (4) as our definition. Note that π0 Map(Y,Z) will correspond to homotopy classes of maps Y → Z. If we were to require this condition for every simplicial set Z, we would run into the same problems we faced when defining cofibrations. Hence, we only require condition (4) to hold when Z is a Kan complex (see again Remark 1.6 ii). 1.8. Definition. A map of simplicial sets f : X → Y is a weak equivalence if it induces a bijection ∗ f : π0 F(Y,K) → π0 F(X,K) for every left Kan complex K. 1.9. Definition. A morphism f : X → Y is a retract of a morphism f 0 : X0 → Y 0 if it is a retract in Fun([1], C), ie there exists a commutative diagram p X i X0 X

f f 0 f j q Y Y 0 Y such that pi = idX and qj = idY . 1.10. Deﬁnition. Let C be a category and let Σ be a class of morphisms in C. (1) The class Σ satisﬁes the two-out-of-six-property if the following holds: Given three composable morphisms

f1 f2 f3 X0 −→ X1 −→ X2 −→ X3 4

such that f2f1 and f3f2 belong to Σ, then f1, f2, f3 and f3f2f1 also belong to Σ. (2) The class Σ is closed under retracts every morphisms which is a retract of a member of Σ also belongs to Σ. 1.11. Proposition. The class of weak equivalences satisfies the two-out-of-six property and is closed under retracts. Proof. This follows directly since bijections satisfy two-out-of-six and are closed under retracts. 2. Fibrations via horn-filling As already promised, there is an easier criterion to check whether a map of simplicial sets is a left or right fibration: 2.1. Theorem. Let f : X → Y be a map of simplicial sets. (1) f is a left fibration if and only if it has the RLP with respect to the set of left horn inclusions n n LH := Λi → ∆ | n ≥ 1, 0 ≤ i < n}. (2) f is a right fibration if and only if it has the RLP with respect to the set of right horn inclusions n n {Λi → ∆ | n ≥ 1, 0 < i ≤ n}. (3) f is a Kan fibration if and only if it has the RLP with respect to the set of horn inclusions H := LH ∪ RH.1 The proof of Theorem 2.1 relies on the observation that we may enlarge the class of morphisms against which we test the RLP without changing the class of morphisms which possess the RLP. 2.2. Lemma. Let C be a category with countable colimits and let f : X → Y be a morphism in C. (1) Suppose that f has the RLP with respect to s: S → T and let g : S → Z be arbitrary. Then f has the RLP with respect to the induced map Z → Z∪S T . (2) Suppose that f has the RLP with respect to s0 and that s is a retract of s0. Then f has the RLP with respect to s. (3) If {si : Si → Ti}i∈I is an arbitrary family of morphisms and f has the RLP with respect to every si, then f has the RLP with respect to the coproduct ` ` ` i∈I si : i∈I Si → i∈I Ti. s1 s2 (4) Let S0 −→ S1 −→ ... be a sequence of composable arrows. If f has the RLP with respect to every sn, then f has the RLP with respect to the canonical map S0 → colimn Sn.

Proof. Draw the appropriate diagrams, see eg [Lan, Lemma 3.8]. 2.3. Remark. Note that f has the RLP with respect to s if and only if s has the LLP with respect to f. Lemma 2.2 can therefore also be read as a statement about preservation of the LLP. 2.4. Deﬁnition. Let C be a category and let Σ be a class of morphisms in C. Then Σ is saturated if it is closed under the four constructions appearing in Lemma 2.2: (1) S is closed under pushouts: If s: S → T is in Σ and g : S → Z is an arbitrary morphism, then the induced morphism Z → Z ∪S T is also in Σ.

1These are the standard deﬁnitions one will ﬁnd in the literature. SOME SIMPLICIAL HOMOTOPY THEORY 5

(2) S is closed under retracts: If s0 is a retract of a morphism s ∈ Σ, then s0 ∈ Σ. (3) S is closed under coproducts: If {si : Si → Ti}i∈I is a family of morphisms ` ` ` in Σ, then i∈I si : i∈I Si → i∈I Ti is in Σ. s1 s2 (4) S is closed under countable compositions: If S0 −→ S1 −→ ... is a sequence of composable arrows in S, then S0 → colimn Sn is in Σ. Every class of morphisms Σ in a category C is contained in a smallest saturated class of morphisms sat(Σ), which we call the saturation of Σ. Since any intersection of saturated classes is saturated, \ sat(Σ) = S. S⊆mor(C) saturated Σ⊆S Note that the indexing set of the intersection is non-empty since mor(C) certainly is saturated. With this terminology, Lemma 2.2 can be rephrased as saying that a morphism has the RLP with respect to a class Σ if and only if it has the RLP with respect to sat(Σ). 2.5. Remark. Note that the saturation of a class of injective maps in sSet contains only injections. There are numerous classes of morphisms we have considered already and whose saturation we might be interested in. 2.6. Deﬁnition. (1) The class B of boundary inclusions is given by B := {∂∆n ,→ ∆n | n ≥ 1} . (2) The class I of injections is given by I := {i: X → Y | i injective} . Observe that I is a saturated class. As a little warm-up, we consider the classes B and I. 2.7. Proposition. The following classes of morphisms coincide: (1) sat(B); (2) I.

Proof. Exercise. 2.8. Corollary. If a map of simplicial sets has the RLP with respect to B, then it is a Kan ﬁbration and a homotopy equivalence. Proof. Let f : X → Y have the RLP with respect to B. By Proposition 2.7, it has the RLP with respect to all inclusions, so it is a Kan ﬁbration. Since ∅ → Y is an injection, there exists a section s: Y → X. Since fsf = f, we have a commutative diagram sftid X × {0, 1} X X

1 f (∅→X)({0,1}→∆ ) X × ∆1 Y where the map X × ∆1 → Y is the constant homotopy on f. Since f has the RLP with respect to all injections, there exists a homotopy X × ∆1 → X witnessing sf ' idX . 6

Obviously, LC ⊆ C, RC ⊆ C, LH ⊆ H and RH ⊆ H. The analogous inclusions hold for the saturations of these classes. 2.9. Proposition. Let LCg denote the class of morphisms 1 (X → Y ) ({0} ,→ ∆ ) | X → Y injective . Then the following holds: (1) LCg ⊆ sat(LC); (2) LC ⊆ sat(LH); (3) LH ⊆ sat(LC); Proof. For (1), we use again that for every injection j : X → Y there is a ﬁltration X = Y (−1) ⊆ Y (0) ⊆ Y (1) ⊆ Y (2) ⊆ ...Y such that G Y (n) =∼ Y (n−1) ∪ ∆n, F ∂∆n i∈In i∈In (n) ∼ (n) colimn Y = Y and j becomes identiﬁed with the canonical map X → colimn Y . Since pushouts commute with each other and −×∆1 commutes with pushouts, there exists for every n ≥ 0 a pushout

F n n 1 (n−1) 1 i∈I (∆ × {0}) ∪ (∂∆ × ∆ ) (Y × {0}) ∪ (Y × ∆ ) n ∂∆n×{0} Y (n−1)×{0}

F n 1 (n) 1 i∈I (∆ × ∆ ) (Y × {0}) ∪ (Y × ∆ ) n Y (n)×{0} Consequently, the sequence of maps ... → (Y × {0}) ∪ (Y (n−1) × ∆1) → (Y × {0}) ∪ (Y (n) × ∆1) → ... Y (n−1)×{0} Y (n)×{0} lies in sat(LC). It follows that (Y × {0}) ∪ (X × ∆1) → colim (Y × {0}) ∪ (Y (n) × ∆1) X×{0} n Y (n)×{0} =∼ Y × ∆1 also lies in sat(LC). n n 1 For (2), it suffices to show that (∂∆ → ∆ ) ({0} → ∆ ) can be obtained by filling left horns in the domain. Note that ∆n × ∆1 =∼ N([n] × [1]), so simplices correspond to order-preserving maps α:[k] → [n] × [1]. Such a simplex is non- degenerate if and only if α is injective. For k < n, α factors as [k] → [n − 1] × [1] −−−→di×id [n] × [1] for some i. So every k-simplex is contained in ∂∆n × ∆1 for k < n. Let p:[n] × [1] → [n] and q :[n] × [1] → [1] be the two projection maps. An n n 1 n-simplex α is not contained in (∆ × {0}) ∪∂∆n×{0} (∂∆ × ∆ ) if and only if n n 1 pα = id[n] and img(qα) 6= {0}. Moreover, (∆ × {0}) ∪∂∆n×{0} (∂∆ × ∆ ) does not contain any non-degenerate (n + 1)-simplices. An (n+1)-simplex α is uniquely determined by the number j(α) ∈ [n] satisfying qα(j(α)) = 0 and qα(j(α)+1) = 1. Hence, there are precisely n+1 non-degenerate (n + 1)-simplices. These come with a canonical ordering α0, . . . , αn by requiring that j(αi) = i. n Consider αn. Then there is exactly one face which does not lie in (∆ × n 1 {0}) ∪∂∆n×{0} (∂∆ × ∆ ), namely dnαn. The remaining faces of αn define a map n+1 n n 1 Λn → (∆ × {0}) ∪∂∆n×{0} (∂∆ × ∆ ), SOME SIMPLICIAL HOMOTOPY THEORY 7 and filling this inner horn adds precisely αn and dnαn to give a simplicial subset n 1 Xn ⊆ ∆ × ∆ . n 1 For αn−1, observe that dn+1αn−1 lies in ∂∆ × ∆ , dnαn−1 = dnαn and dkαn−1 n 1 also lies in ∂∆ × ∆ for k < n − 1. So all faces of αn−1 except dn−1αn−1 define n+1 a map Λn−1 → Xn. Filling this horn adds αn−1 and dn−1αn−1 to Xn to obtain Xn−1. Proceeding like this inductively, one ends up with α0. All faces of α0 except d0α0 n+1 have been added in previous steps. Filling the corresponding (left!) horn Λ0 → n 1 X1 adds the last two missing simplices α0 and d0α0 to X1 to give X0 = ∆ × ∆ . There are closed formulas for all the face relations which are not difficult to figure out. If you want to look them up, see [GJ99, Proposition 4.2]. This shows (2). n n Finally, consider the inclusion of a left horn Λk → ∆ , ie k < n. Then there is a retract diagram

j|Λn r | Λn k (∆n × {0}) ∪ (Λn × ∆1) k ... Λn k n k k Λk ×{0}

j ∆n ∆n × ∆1 rk ∆n n ∼ n n 1 in which j is given by the inclusion ∆ = ∆ × {1} → ∆ × ∆ and rk is induced by the map of posets ( l i = 1 or l < k, rk :[n] × [1] → [n], (l, i) 7→ k else.

Note that rkj = id, and rk does restrict as indicated in the diagram: we only need to check that the inclusion [n] \{k} ⊆ img(rkα) is impossible. If α:[p] → [n] × {0} is a simplex, then rkα is a map [p] → [k]. Since k < n, img(rkα) cannot contain n 1 n. Suppose that α:[p] → [n] × [1] is a simplex in Λk × ∆ . Then img(rkα) ⊆ img(qα) ∪ {k}, where q :[n] × [1] → [n] is the projection. Since [n] \{k} is not a subset of img(qα), it cannot be a subset of img(rkα), either. This proves (3). 2.10. Corollary. sat(LH) = sat(LC) = sat(LCg)

Proof. Note that LC ⊆ LCg. By Proposition 2.9, we have sat(LH) ⊆ sat(LC) ⊆ sat(LCg) ⊆ sat(LC) ⊆ sat(LH). Analogous statements hold for the right cylinders and right horn inclusions. Instead of saying that all proofs work identically, we employ a little trick. 2.11. Deﬁnition. Deﬁne an automorphism ι: ∆ → ∆ by setting ι([n]) := [n] and op op ι(α:[m] → [n]) := ιn α ιm, where op ιn :[n] → [n] , i 7→ n − i. Denote the induced isomorphism of categories − ◦ ιop : sSet → sSet by (−)op and call the image Xop of a simplicial set under this functor the opposite of X. 2.12. Example. ι(di) = dn−i 2.13. Remark. (1) For a category C, there is a natural isomorphism N(C)op =∼ N(Cop). (2) The functor ι preserves inner horn inclusions, so Xop is a quasi-category if X is a quasi-category. 8

2.14. Corollary. The following classes of morphisms coincide: (1) sat(RC); (2) sat(RH); (3) the saturation of 1 RCg := (X → Y ) ({1} ,→ ∆ ) | Y → X injective . Proof. Note that (−)op induces bijections LC ∼ RC, L ∼ R and LCg ∼ RCg. As an isomorphism of categories, (−)op also preserves saturations, so the corollary follows from Corollary 2.10. Combining Corollary 2.10 and Corollary 2.14, we immediately obtain the following. 2.15. Corollary. The following classes of morphisms coincide: (1) sat(C); (2) sat(H); (3) the saturation of 1 1 Ce := X × {i} ∪ Y × ∆ ,→ X × ∆ | Y → X injective, i = 0, 1 . Proof. By Proposition 2.9 (and its “opposite”), we have the following inclusions: (1) LH ∪ RH ⊆ sat(LC) ∪ sat(RC) ⊆ sat(LC ∪ RC); (2) LCg ∪ RCg ⊆ sat(LH) ∪ sat(RH) ⊆ sat(LH ∪ RH); (3) LC ∪ RC ⊆ LCg ∪ RCg. Hence, we have sat(C) ⊆ sat(LCg ∪ RCg) ⊆ sat(H) ⊆ sat(C). 2.16. Deﬁnition. The class of morphisms described in Corollary 2.15 is called the class of anodyne extensions.2 Similarly, Corollaries 2.10 and 2.14 describe the classes of left anodyne extensions and right anodyne extensions, respectively.

Proof of Theorem 2.1. Immediate from Lemma 2.2 and Corollary 2.15. 2.17. Remark. In particular, this shows that X is a Kan complex if and only if 0 X → ∆ is a Kan fibration. Since all inclusions F n (0, n) ⊆ F n (0, n) lie in C(Λk ) C(∆ ) Ce by Lemma IV.34, the proof of Theorem IV.31 is now complete. 2.18. Corollary. A map of simplicial sets is (1) a left fibration if and only if it has the RLP with respect to all left anodyne extensions. (2) a right fibration if and only if it has the RLP with respect to all right anodyne extensions. (3) a Kan fibration if and only if it has the RLP with respect to all anodyne extensions. 2.19. Corollary. For every simplicial set X there exists an anodyne extension X → X, where X is a Kan complex. Proof. For any simplicial set X, define X0 by the pushout ` n n Λ X f :Λi →X i

` n 0 n ∆ X f :Λi →X

2The terminology is due to Gabriel–Zisman. “Anodyne” is Greek and translates to “painless”. SOME SIMPLICIAL HOMOTOPY THEORY 9 where the coproduct runs over all maps from all possible horns to X. Note that X → X0 is an anodyne extension. Given a fixed simplicial set X, let X0 := X and define Xn+1 := Xn. Then the Yoneda lemma implies that every horn in X := colimn Xn can be filled. Moreover, the canonical map X → X is a left anodyne extension. Using Remark 2.3, Lemma 2.2 is enough to show that cofibrations admit a much easier description than our original definition suggests. 2.20. Theorem. A map of simplicial sets is a left cofibration if and only if it is injective.3 Proof. Exercise: Fix an injection j. Let Σ be a saturated class of morphisms. Then the class of morphisms S := {i | i j ∈ Σ} is saturated. Consequently, the class of left cofibrations is saturated. Since ∂∆n → ∆n is a left cofibration for all n ≥ 0, Corollary 2.10 implies that every injection is a left cofibration. Let i: A → B be a cofibration. By Corollary 2.19, there is an anodyne extension 1 B × {0} ∪A×{0} A × ∆ → K for some Kan complex K. Since i is a left cofibration, this injection extends to a 1 1 1 map B × ∆ → K. Consequently, B × {0} ∪A×{0} A × ∆ → B × ∆ is injective. Since i can be recovered from this map via the inclusion of A × {1}, it follows that i is injective. 2.21. Corollary. The classes of left cofibrations, right cofibrations and injections coincide. From now on, we will only (and intercheangeably) use the terms “injection” and “cofibration”. 2.22. Proposition. Let i: A → B be a left anodyne extension and let j : X → Y be a cofibration. Then

(A × Y ) ∪A×X (B × X) → B × Y is left anodyne. In particular, i × X : A × X → B × X is a left anodyne extension for every simplicial set X. The analogous statement holds for right anodyne and anodyne maps. Proof. For fixed j, the class of morphisms S := {k : A → B | k j left anodyne} is saturated by the exercise set in the proof of Theorem 2.20. 1 By Corollary 2.10, it suffices to check that l ({0} → ∆ ) lies in S for all cofibrations l : A → B. Since − × X and − × Y commute with pushouts, the morphism (B×{0}) ∪ (A×∆1)×Y ∪ (B×∆1)×X → (B×∆1)×Y A×{0} (B×{0}) ∪ (A×∆1)×X A×{0} can be identified with the morphism (B×Y ×{0}) ∪ (A×Y ×∆1) ∪ (B×X×∆1) → (B×Y )×∆1, A×Y ×{0} (B×X×{0}) ∪ (A×X×∆1) A×X×{0}

3As in the case of fibrations, this is usually the definition of a cofibration. 10 which is left anodyne by Corollary 2.10. 2.23. Corollary. If i: A → B is a cofibration and p: X → Y is a left/right/Kan fibration, then the induced map F(B,X) → F(A, X) × F(B,Y ) F(A,Y ) is a left/right/Kan fibration. In particular, the following holds: (1) If B is any simplicial set and p: X → Y is a left/right/Kan fibration, then p∗ : F(B,X) → F(B,Y ) is a left/right/Kan fibration. (2) If i: A → B is a cofibration and X is a Kan complex, then i∗ : F(B,X) → F(A, X) is a Kan fibration of Kan complexes. Proof. Every lifting problem

n Λk F(B,X)

∆n F(A, X) × F (B,Y ) F(A,Y ) corresponds to a commutative diagram (B × Λn) ∪ (A × ∆n) X k n A×Λk p

B × ∆n Y The left vertical map is an anodyne extension by Proposition 2.22, and hence has the LLP with respect to p. The resulting lift provides the required lift of the original lifting problem. 2.24. Corollary. Let K be a Kan complex and let X be a simplicial set. Then F(X,K) is a Kan complex. 0 Proof. This is Corollary 2.23 (1) for p: K → ∆ . Corollary 2.24 shows that the simplicially enriched category Kan of Kan complexes is enriched in Kan complexes. So the following definition is justified. 2.25. Definition. Define the quasi-category of anima to be An := Nc(Kan). Note that the name An is abusive at this point in time. We have defined “anima” in such a way that it is a theorem that “anima” is synonymous to “Kan complex”. Our notation anticipates that we will prove this eventually. Also, let us emphasise that at the time of writing the name “anima” is very much non-standard. Most people prefer the term “spaces” and write S or Spc instead of An. 2.26. Corollary. Let i: A → B be a left/right/∅ anodyne extension and let p: X → Y be a left/right/Kan fibration. Then the induced map F(B,X) → F(A, X) × F(B,Y ) F(A,Y ) has the RLP with respect to I. In particular, it is a Kan fibration and a homotopy equivalence (see Corollary 2.8). SOME SIMPLICIAL HOMOTOPY THEORY 11

Proof. Since sat(B) = I, it is enough to consider B. Any lifting problem ∂∆n F(B,X)

∆n F(A, X) × F(B,Y ) F(A,Y ) corresponds to a lifting problem (A × ∆n) ∪ (B × ∂∆n) X A×∂∆n p

B × ∆n Y The left vertical map is left/right/∅ anodyne by Proposition 2.22, so there exists a solution. 2.27. Corollary. Every anodyne extension is a cofibration and a weak equivalence. Proof. We only need to show that every anodyne extension is a weak equivalence. 0 This follows from Corollary 2.26 for Y = ∆ by taking π0. 2.28. Definition. (1) A category I is filtered if for every diagram F : J → I where J has only finitely many objects and morphisms there exists an object i ∈ I and a natural transformation F ⇒ consti. (2)A filtered colimit is a colimit of a diagram indexed by a filtered category. (3) Let C be a category with all filtered colimits. Then c ∈ C is compact if HomC(c, −) commutes with all filtered colimits. (4) Let C be a cocomplete category. Then c ∈ C is tiny if HomC(c, −) commutes with all colimits. 2.29. Example. (1) Every directed poset is a filtered category, eg N. (2) A set X is compact in Set if and only if X is finite. (3) A simplicial set X is compact in sSet if and only if X is finite. (4) A module M over some ring R is compact in Mod(R) if and only if M is finitely presented. The following statement is often referred to as the “small object argument”. 2.30. Proposition. Let C be a cocomplete category. Let S be a set of morphisms such that the domain of each morphism in S is compact. Then every morphism f : X → Y of simplicial sets can be factored as f = pi, where i lies in sat(S) and p has the RLP with respect to S. Proof. For an arbitrary map g : V → W , let LP(S, g) := {(a: A → V, b: B → W, i: A → B) | i ∈ S, ga = bi}. Define V 0 as the pushout ` P a (a,b,i)∈LP(S,g) A V

` i g P ` b 0 (a,b,i)∈LP(S,g) B V Then there is a canonically induced map V 0 → W , and the map V → V 0 lies in the saturation of S. 12

0 n+1 n 0 n Now define X := X and X := (X ) . Set E := colimn X . By definition, the induced map i: X → E is in sat(S). Moreover, the induced map p: E → Y satisfies pi = f. Consider a lifting problem A a V j f B b W with j ∈ S. Since A is compact by assumption, a factors as a: A → Xn → colim Xn = E n for some n ∈ N. By construction, the induced lifting problem

0 A a Xn+1 j f B b Y has a solution, and this also provides a solution to the original lifting problem. Note that this contains Corollary 2.19 as a special case (take S = H and f to be the projection map X → ∆0). We can now prove a converse to Corollary 2.18. 2.31. Corollary. Let C be a cocomplete category and S a set of maps such that the domain of every morphism in S is compact. Then sat(S) = {f : X → Y | f has the LLP with respect to the class of morphisms which have the RLP with respect to S.} =: S

In particular, a morphism is a left/right/∅ anodyne extension if and only if it has the LLP with respect to every left/right/Kan ﬁbration. Proof. Since S is a saturated class by Lemma 2.2 and contains S, we have sat(S) ⊆ S. Let f : X → Y be in S. By Proposition 2.30, factor f into a map i: X → E in sat(S) followed by a map p: E → Y which has the RLP with respect to S. The lifting problem X i E f s p Y id Y has a solution since f has the LLP with respect to p. These maps deﬁne a retract diagram X id X id X

f i f p Y s E Y So f is in sat(S).

3. Simplicial homotopy groups 3.1. Proposition. Let K be a Kan complex. Then πX is a groupoid. In particular, 0 homotopy is an equivalence relation on HomsSet(∆ ,K). SOME SIMPLICIAL HOMOTOPY THEORY 13

Proof. Let x, y ∈ X and let e be an edge such that d0(e) = y and d1(e) = x. 2 Consider e and s0(x) as a map Λ0 → X. For any filler F , the edge d0(F ) provides a morphism y → x in πX such that d0F ◦e = idx. Similarly, e and s0(x) also define a 2 map Λ2 → X, and d2 of any filler provides a right inverse to e. Since isomorphisms satisfy two-out-of-six, e is an isomorphism in πX. 3.2. Corollary. Let K be a Kan complex, let X be a simplicial set, and let A be a simplicial subset of X. Then homotopy relative A is an equivalence relation on HomsSet(X,K). Proof. Let f : A → K be given and consider the pullback

F(X,K)f F(X,K)

f ∆0 F(A, K)

Then F (X,K)f is a Kan complex by Corollary 2.23. Homotopy of 0-simplices in F (X,K)f corresponds precisely to homotopy relative A of maps X → K which restrict to the given map f on A. So the corollary follows from Proposition 3.1. 3.3. Corollary. Every weak equivalence between Kan complexes is a homotopy equivalence. Proof. Let f : K → L be a weak equivalence between Kan complexes. Since ∗ f : π0(L, K) → π0(K,K) is a bijection, there exists by Corollary 3.2 a map g : L → ∗ K such that gf ' idK . Since fgf ' f, the injectivity of f implies that fg ' idL as well. 3.4. Deﬁnition. Let A ⊆ X and B ⊆ Y be inclusions of simplicial subsets. Deﬁne F((X,A), (Y,B)) as the pullback F((X,A), (Y,B)) F(X,Y )

F(A, B) F(A, Y )

Note that F((X,A), (Y,B)) is a Kan complex if B and Y are Kan complexes (by Corollaries 2.23 and 2.24). 3.5. Deﬁnition. For n ≥ 0, let n 1 n ∼ n := (∆ ) = N([1] ) be the n-cube. The simplicial subset n n ∂ := {α:[k] → [1] | ∃ 1 ≤ i ≤ n: pri α constant} 0 is called the boundary of the n-cube. We agree that ∂ = ∅.

3.6. Definition. Let X be a Kan complex and x ∈ X0. Define n n πn(X, x) := π0(F(( , ∂ ), (X, x))). Let Y be a Kan complex and y ∈ Y . Define

π1(Y, y) := EndπY (y). Unwinding deﬁnitions and using Remark II.30, there is a natural isomorphism ∼ n−1 n−1 πn(X, x) = π1(F(( , ∂ ), (X, x)), cx) for all n ≥ 1, where cx denotes the constant map with value x. So the composition law in the homotopy category equips πn(X, x) with the structure of a monoid. 14

Conjugating the identiﬁcation ∼ n−1 n−1 πn(X, x) = π1(F(( , ∂ ), (X, x)), cx) 1 n by the transposition τ1,i acting on the factors of (∆ ) , we in fact obtain n monoid structures {·i}1≤i≤n on πn(X, x) which have the same neutral element (namely the class represented by the constant map).

3.7. Proposition. Let X be a Kan complex and x ∈ X. Then πn(X, x) is a group for all n ≥ 1. For n ≥ 2, the compositions ·i all agree and are commutative. n−1 n−1 Proof. Since F(( , ∂ ), (X, x)) is a Kan complex for all n ≥ 1, the ﬁrst assertion follows from the fact that πY is a groupoid for every Kan complex Y . n n Let α, β, γ, δ :( , ∂ ) → (X, x) represent elements in πn(X, x) for n ≥ 2. We claim that

([α] ·1 [β]) ·2 ([γ] ·1 [δ]) = ([α] ·2 [γ]) ·1 ([β] ·2 [δ]).

If this holds, the Eckmann–Hilton argument shows that ·1 = ·2 and that this composition is abelian. 2 2 n−2 The four given elements induce a map Λ1 × Λ1 × → X. Decoding the deﬁnition of the composition laws, the left hand composition amounts to extending this map along the anodyne extensions 2 2 n−2 2 2 n−2 2 2 n−2 Λ1 × Λ1 × → ∆ × Λ1 × → ∆ × ∆ × , while the right hand side corresponds to extending along the anodyne extensions 2 2 n−2 2 2 n−2 2 2 n−2 Λ1 × Λ1 × → Λ1 × ∆1 × → ∆ × ∆ × . Note that we are ignoring various boundary conditions for legibility. Both extensions provide an extension along the anodyne extension 2 2 n−2 2 2 n−2 Λ1 × Λ1 × → ∆ × ∆ × . Since such an extension is unique up to homotopy (Corollary 2.26), this proves the claim. For arbitrary ·i and ·j, the same proof works after appropriate permutation of the factors.

Of course, one can construct the group structure on πn(X, x) directly and observe that there are n diﬀerent compositions on πn(X, x). As soon as there are at least two, the Eckmann–Hilton argument applies. Each πn deﬁnes a functor

πn : Kan∗ → Grp which is invariant under pointed simplicial homotopies.

3.8. Theorem. Let p: X → Y be a Kan ﬁbration of Kan complexes and x ∈ X0. Let F denote the ﬁbre of p over p(x). Then there exists a natural long exact sequence

p∗ ∂ i∗ p∗ ... → πn+1(X, x) −→ πn+1(Y, p(x)) −→ πn(F, x) −→ πn(X, x) −→ πn(Y, p(x)) → ... In low degrees, exactness means the following:

(1) There is an action π1(Y, p(x)) × π0(F, x) → π0(F, x) such that • the image of p∗ : π1(X, x) → π1(Y, y) is precisely the stabiliser of [x] ∈ π0(F ); • two elements in π0(F, x) map to the same element in π0(X, x) if and only if they lie in the same orbit of this action. (2) An element in π0(X, x) maps to the basepoint in π0(Y, p(x)) if and only if they lie in the image of π0(F, x) → π0(X, x). SOME SIMPLICIAL HOMOTOPY THEORY 15

Proof. Since we are using cubes to model homotopy groups, the proof is more or less identical to the proof for topological spaces. Set y := p(x). n+1 n+1 We begin by deﬁning the boundary map. Let α:( , ∂ ) → (Y, y) repre- n n 1 sent any element in πn+1(Y, y). Since (∂ → ) ({0} → ∆ ) is anodyne, the lifting problem (∆1 × ∂ n) ∪ ({0} × n) constx X n {0}×∂ α p

1 n α ∆ × Y has a solution. Set

∂[a] := [α| n ]. {1}× n+1 1 ∼ 1 n 1 This is well-defined: let h: ×∆ = ∆ × ×∆ → Y be a homotopy relative n+1 0 0 1 n ∂ from α to α . Any two lifts α and α as above define a map (∆ × × ∂∆1) → X. This map glues with the constant map on 1 n n 1 ((∆ × ∂ ) ∪ n ({0} × )) × ∆ {0}×∂ 1 n n to define a partial lift of h. Since ({0} → ∆ ) (∂ → ) is anodyne, so is 1 n n 1 1 ({0} → ∆ ) (∂ → ) (∂∆ → ∆ ). 1 n 1 0 Any lift h: ∆ × × ∆ → X witnesses that α and α are homotopic relative n ∂ . So ∂ is well-defined. If n ≥ 1, we claim that ∂ is a group homomorphism. Two elements [α], [β] define a map 1 2 n−1 ∆ × Λ1 × → Y. 1 2 n−1 Similarly, the lifts α and β glue to a map on ∆ ×Λ1 × which has an extension 1 2 n−1 to a map γ : ∆ × ∆ × → X. Then γ| 2 n−1 defines ∂[a] · ∂[β]. On {1}×d1∆ × the other hand, p ◦ γ is an extension of the original map, and (p ◦ γ)| 1 2 n−1 ∆ ×d1∆ × defines [α] ·2 [β], so ∂([α] ·2 [β]) = ∂[α] · ∂[β].

Since ·2 = ·, this proves that ∂ is a homomorphism. Let e ∈ F0 and [α] ∈ π1(Y, y). Solve the lifting problem ∆0 e X α p ∆1 α Y to define [α][e] := [α(1)]. It is easy to check that this is well-defined and defines a group action. We recover the definition of the connecting map by letting π1(Y, y) act on [x]. We begin by showing exactness in low degrees. Exactness at π0(X): The composition π0(F ) → π0(X) → π0(Y ) obviously 0 0 only hits the basepoint. Suppose that x ∈ X such that [p(x )] = [y] ∈ π0(Y ). Then the lifting problem 0 ∆0 x X h p p(x0)'y ∆1 Y 0 has a solution, and h(1) is a point in F such that [h(1)] = [x ] ∈ π0(X). 0 0 Exactness at π0(F ): Suppose that [e], [e ] ∈ π0(F ) satisfy [α][e] = [e ] for some 1 [α] ∈ π1(Y, y). Then, by definition, there exists α: ∆ → X such that α(0) = e and α(1) is homotopic to [e0] in F . This implies that e and e0 are homotopic in X. 16

0 Conversely, suppose that [e] = [e ] ∈ π0(X). Then there exists a homotopy 1 0 h: ∆ → X from e to e . The map ph defines an element in π1(Y, y) since p(e) = 0 0 y = p(e ). By definition, we have [ph][e] = [e ]. So the fibres of π0(F ) → π0(X) correspond precisely to the orbits of the π1(Y, y)-action on π0(F ). Exactness at π1(Y, y): Let [α] ∈ π1(X, x). Then α itself is a solution to the lifting problem ∆0 x X p pα ∆1 Y so [pα][x] = [x]. Conversely, suppose that [α] ∈ π1(Y, y) satisfies [α][x] = [x]. Then α lifts by definition of the action to a loop in X. So the image of p∗ : π1(X, x) → π1(Y, y) is precisely the stabiliser of [x] ∈ π0(F ). Exactness at πn(X, x) for n ≥ 1: The composition p∗i∗ is obviously trivial. n 1 Suppose that p∗[α] = 1 ∈ πn(Y, y). Then there exists a homotopy h: × ∆ → Y n relative ∂ from pα to the constant map with value y. Since p is a Kan fibration, the lifting problem

( n × {0}) ∪ (∂ n × ∆1) α∪constx X n ∂ ×{0} p

n 1 h × ∆ Y has a solution. The endpoint of this solution provides a preimage of [α] under i∗. Exactness at πn(F, x) for n ≥ 1: Let [α] ∈ πn+1(Y, y). Then ∂[α] is by n construction represented by a map which is homotopic to constx relative ∂ in X. Consequently, i∗∂ is trivial. Conversely, suppose that [iα] is trivial in πn(X, x) for some [α] ∈ πn(F, x). Then there exists a homotopy (note we are free to choose the direction of the homotopy) 1 n n h: ∆ × → X relative ∂ from constx to iα. Then, by definition, h represents an element in πn+1(Y, y) with ∂[h] = [α]. Exactness at πn+1(Y, y) for n ≥ 1: Let [α] ∈ πn+1(X, x). Then α solves the n lifting problem used to define ∂[pα]. Since α is constant on ∂ , this proves that ∂p∗ is trivial. Conversely, suppose that ∂[α] = [α| n ] = 1 for some [α] ∈ π (Y, y). Then {1}× n+1 1 n n there exists a homotopy h: ∆ × → F relative ∂ from α| n to const . {1}× x Since p is a Kan fibration, the lifting problem

(α∪h)∪const (Λ2 × n) ∪ (∆2 × ∂ n) x X 1 2 n Λ1×∂ p

2 n α(s1×id) ∆ × Y has a solution h, where s1 denotes the appropriate degeneration map. Then h| 2 n deﬁnes an element in π (X, x) such that [ph| 2 n ] = [α]. d1∆ × n+1 d1∆ × 3.9. Lemma. Let f :(X, x) → (Y, y) be a pointed map of pointed Kan complexes. Then f is a simplicial homotopy equivalence if and only if it is a pointed simplicial homotopy equivalence. Proof. The basepoint inclusions induce Kan ﬁbrations

evx : F(X,Z) → Z, evy : F(Y,Z) → Z SOME SIMPLICIAL HOMOTOPY THEORY 17 for every Kan complex Z. For z ∈ Z0, the ﬁbres of evx and evy over z are precisely F((X, x), (Z, z)) and F((Y, y), (Z, z)), respectively. Consider the induced map on long exact sequences in low degrees:

∂Y π1(Z, z) π0 F((Y, y), (Z, z)) π0 F(Y,Z) π0(Z)

∗ ∗ id fp ∼ f id

∂X π1(Z, z) π0 F((X, x), (Z, z)) π0 F(X,Z) π0(Z)

∗ By a diagram chase, fp is surjective. Inserting (Z, z) = (X, x), there exists a pointed map g :(Y, y) → (Z, z) such that gf is pointed homotopic to idX . In particular, g is an unpointed homotopy equivalence. Repeating the argument with g in place of f, we obtain a pointed map f 0 :(X, x) → (Y, y) such that f 0g is pointed homotopic to idY . By two-out-of-six, it follows that f is a pointed homotopy equivalence. 3.10. Corollary. Let f :(X, x) → (Y, y) be a map of pointed Kan complexes such that f : X → Y is a homotopy equivalence. Then f induces isomorphisms ∼= f∗ : πn(X, x) −→ πn(Y, y) for all n. Our next goal is to prove a converse to Corollary 3.10. 3.11. Theorem. A map f : X → Y between Kan complexes is a homotopy equivalence if and only if f induces a bijection on π0 and an isomorphism πn(X, x) → πn(Y, f(x)) for all x ∈ X0 and n ≥ 1. We begin by characterising contractible Kan complexes by their homotopy groups. The proof relies on a description of the homotopy groups in terms of simplices.

3.12. Deﬁnition. For X a Kan complex, x ∈ X0 and n ≥ 1, deﬁne 4 n n πn (X, x) := π0(F((∆ , ∂∆ ), (X, x))). 4 The comparison of πn with πn rests on the following lemma.

3.13. Lemma. Let A1 ⊆ A2 ⊆ X be simplicial sets and suppose that A1 ⊆ A2 is an anodyne extension. Then the restriction map

F((X,A2), (Y,B)) → F((X,A1), (Y,B)) induces a bijection on π0. Proof. The map under consideration comes from a map of pullback squares

F((X,A2), (Y,B)) F(X,Y )

F((X,A1), (Y,B)) F(X,Y )

F(A2,B) F(A2,Y )

F(A1,B) F(A1,Y )

The restriction maps F(A2,B) → F(A1,B) and F(A2,Y ) → F(A1,Y ) have the RLP with respect to I by Corollary 2.26. Let a: A1 → A2, i1 : A1 → X, i2 : A2 → X and j : B → Y denote the various inclusion maps. 18

We ﬁrst prove surjectivity on π0. Let f :(X,A1) → (Y,B) be a vertex in

F((X,A1), (Y,B)). Let f|A1 : A1 → B be the induced map. Then f|A1 has a lift to map f2 : A2 → B. By assumption, jf2a = jf|A1 = fi1 : A1 → Y . Since F(A2,Y ) → F(A)1,Y ) has the RLP with respect to I, there exists a homotopy fi2 ' jf2 relative A1. Since fi2 extends to X and F(X,Y ) → F(A2,Y ) is a Kan ﬁbration, we obtain 0 0 a homotopy X × ∆ → Y relative A1 from f to a map f satisfying f i2 = jf2. By 0 construction, this homotopy gives a 1-simplex in F((X,A1), (Y,B)) from f to f , 0 and f is a map (X,A2) → (Y,B). For injectivity, let f0, f1 :(X,A2) → (Y,B) be two maps such that there exists a 1 homotopy h: X × ∆ → Y from f0 to f1 which restricts to a homotopy h|A1 : A1 × 1 ∆ → B from f0|A1 to f1|A1 . Since F(A2,B) → F(A1,B) has the RLP with 1 respect to I, the homotopy h|A1 extends to a homotopy k : A2 × ∆ → B from f0|A2 ' f1|A2 . Consider the map

1 1 1 A2 × (∆ × {0} ∪ ∂∆ ) × ∆ → B ∂∆1×{0}

1 which is given by h|A2 on A2 × ∆ × {0} and by the constant homotopies on fe on 1 A2 × {e} × ∆ . This map glues with k to give a map

1 1 1 1 A2 × (∆ × ∂∆ ∪ ∂∆ × ∆ ) → B, ∂∆1×{0}

1 1 which extends to a map A2 × ∆ × ∆ . Since h|A2 extends to a homotopy on X and F(X,Y ) → F(A2,Y ) is a Kan ﬁbration, there exists a map h: X × ∆1 × ∆1 → Y such that h|X×∆1×{0} = h and h|X×{e}×∆1 is the constant homotopy on fe. The

1 endpoint h|X×∆ ×{1} is a homotopy f0 ' f1 which restricts to a homotopy f0|A2 ' f1|A2 .

3.14. Proposition. There is a natural bijection

4 πn (X, x) ∼ πn(X, x). Proof. By deﬁnition,

n n πn(X, x) = π0(F(( , ∂ ), (X, x))). n n Let C ⊆ denote the simplicial subset obtained by removing the simplex σn :[n] → [1]n, i 7→ 1 ... 1 0 ... 0 . | {z } | {z } i times n−i times

n n Claim: The inclusion ∂ → C is anodyne. For someone who is good at ﬁnite combinatorics, this can probably be proven directly by successively ﬁlling horns. We will derive it later from results that are independent of the present discussion, but would lead us on a tangent right now. Consequently, the restriction map induces by Lemma 3.13 a bijection

n n ∼ n n π0(F(( ,C ), (X, x))) −→ π0(F(( , ∂ ), (X, x))). By inspection, we have for every pair of simplicial sets A ⊆ B a natural isomorphism F((B,A), (X, x)) =∼ F ((B/A, ∗), (X, x)). SOME SIMPLICIAL HOMOTOPY THEORY 19

Consequently, we have natural bijections n n n n π0(F(( ,C ), (X, x))) ∼ π0(F(( /C , ∗), (X, x))) n n ∼ π0(F((∆ /∂∆ , ∗), (X, x))) n n ∼ π0(F((∆ , ∂∆ ), (X, x))). 4 3.15. Remark. The bijection πn (X, x) → πn(X, x) from Proposition 3.14 sends a n n n map α: (∆, ∂∆ ) → (X, x) to the map ( , ∂ ) → (X, x) which is given by α on the simplex σn appearing in the proof and is constant on all other simplices.

3.16. Lemma. Let (X, x) be a pointed Kan complex such that πn(X, x) is trivial for all n ≥ 0. Let f : B → X be a map, let A ⊆ B and let h: A × ∆1 → X be a homotopy from f|A to the constant map with value x. Then there exists an extension of h to a homotopy B × ∆1 → X from f to the constant map with value x. Proof. The extension may be constructed simplex by simplex, so it suﬃces to n n n n 1 consider B = ∆ and A = ∂∆ . Since (∂∆ → ∆ ) ({0} → ∆ ) is ano- 0 n 1 0 dyne, there exists a homotopy h : ∆ × ∆ → X such that h |∆n×{0} = f and 0 0 0 4 h |∂∆n×{1} = constx. Therefore, f := h |∆n×{1} represents an element in πn (X, x). Since πn(X, x) is trivial, Proposition 3.14 guarantees the existence of a homotopy h00 relative ∂∆n from f 0 to the constant map with value x. The two homotopies h0 and h00 glue with the map n ∂∆n × ∆2 −−−−−→∂∆ ×s1 ∂∆n × ∆1 −→h X to give a map eh: (∆n × Λ2) ∪ (∂∆n × ∆2) → X. 1 n 2 ∂∆ ×Λ1 n n 2 2 Since (∂∆ → ∆ ) (Λ1 → ∆ ) is anodyne, there exists an extension of eh to n 2 n 2 ∆ × ∆ . Restricting to ∆ × d1∆ yields the desired extension of h. 3.17. Corollary. Let X be a Kan complex. The following are equivalent: (1) The map X → ∆0 has the RLP with respect to I. (2) The map X → ∆0 is a homotopy equivalence. (3) X is non-empty and πn(X, x) is trivial for all x ∈ X0 and for every n ≥ 0. (4) π0(X) is a singleton and there exists some x ∈ X0 such that πn(X, x) is trivial for all n ≥ 1. Proof. (1) implies (2) by Corollary 2.8.(2) implies (3) by Corollary 3.10.(3) obviously implies (4). Suppose (4) holds. Let f ∂∆n X

∆n ∆0 be a lifting problem. By Lemma 3.16, there exists a homotopy from f to the n n 1 constant map with value x. Since (∂∆ → ∆ ) ({1} → ∆ ) is anodyne, we obtain a homotopy ∆n × ∆1 → X whose restriction to ∆n × {0} is the desired lift. 3.18. Corollary. Let p: X → Y be a Kan ﬁbration. The following are equivalent: (1) p has the RLP with respect to I. (2) For every y ∈ Y , the ﬁbre p−1(y) is contractible. −1 (3) For every y ∈ Y , the set π0(p (y)) is a singleton and there exists some −1 −1 x ∈ p (y) such that πn(p (y), x) is trivial for all n ≥ 1. 20

Proof. Assume (1). Then, being a pullback, p−1(y) → ∆0 has the RLP with respect to I. Then (2) and (3) follow by Corollary 3.17. Note that (2) and (3) are equivalent, also by Corollary 3.17. Suppose (2) holds and that we are given a lifting problem

∂∆n ∂σ X p ∆n σ Y

Let h: ∆n × ∆1 → ∆n be a simplicial homotopy from id to the constant map with value n (this exists eg by Exercise 2 on Sheet 4). Then σ ◦ h is a homotopy from σ to the constant map with value σ(n). Then (σ ◦ h)|∂∆n×∆1 admits a lift to a homotopy k in X whose endpoint is a map ∂τ : ∂∆n → p−1(σ(n)). Since p−1(σ(n)) is assumed to be contractible, Corollary 3.17 implies that ∂τ extends to a map τ : ∆n → p−1(σ(n)). Since (∂∆n → n 1 ∆ ) ({1} → ∆ ) is anodyne, there exists a homotopy k extending k such that k|∆n×{1} = τ and k|∂∆n×{0} = ∂σ. So σ := k|∆n×{0} is the desired lift.

Proof of Theorem 3.11. One direction is provided by Corollary 3.10. For the converse, assume that f induces a bijection on π0 and an isomorphism πn(X, x) → πn(Y, f(x)) for all x ∈ X0 and n ≥ 1. By Proposition 2.30, factor f into an anodyne extension i: X → X followed by a Kan fibration p: X → Y . Since i is a weak equivalence of Kan complexes by Corollary 2.27, it is a homotopy equivalence. So we only need to show that p is an equivalence. We show that all fibres of p are contractible. Exercise: Show that the fibres over any two vertices in the same path component are homotopy equivalent. Since f is surjective on π0, we may therefore only consider y ∈ Y0 such that there exists x ∈ X0 with f(x) = y. By Corollary 3.17 and the long exact sequence of a fibration (Theorem 3.8), it suffices to show that p∗ : πn(X, i(x)) → πn(Y, y) is a bijection for all n ≥ 0. Since

i∗ πn(X, x) πn(X, i(x))

f∗ p∗ πn(Y, y) commutes, f∗ is assumed to be a bijection and i∗ is a bijection by Corollary 3.10, the theorem follows.

3.19. Remark. It should in principle be possible to prove Theorem 3.11 without factoring the morphism, working with relative homotopy groups instead. However, this would force us to consider homotopy classes of maps of triples, which would certainly not reduce the amount of bookkeeping we would have to do.

4. Ex∞ The discussion in Section 3 provides us with a good understanding of Kan fibrations which are homotopy equivalences. However, it remains unclear how these relate to Kan fibrations which are weak equivalences. For example, if p: X → Y is any Kan fibration, we could apply the small object argument (Proposition 2.30) twice to replace Y by a Kan complex j : Y → Y and then factor the map jp into SOME SIMPLICIAL HOMOTOPY THEORY 21 an anodyne map i: X → X followed by a Kan fibration p: X → Y :

X i X p p j Y Y However, we are unable to control the effect of this construction on the fibres. This is remedied by a more sophisticated construction called Ex∞. 4.1. Theorem. There exist a functor Ex∞ : sSet → sSet and a natural transforma- ∞ ∞ tion ρ : idsSet → Ex such that the following holds: (1) Ex∞ ∆0 =∼ ∆0; (2) Ex∞ X is a Kan complex for every simplicial set X; (3) ρ∞ : X → Ex∞ X is a weak equivalence for every simplicial set X; (4) Ex∞ preserves finite limits; (5) Ex∞ preserves Kan fibrations. Before we start with the proof of this theorem, we establish some important consequences. 4.2. Corollary. Ex∞ preserves and detects weak equivalences. Proof. If f : X → Y is a map, it follows from the commutative square

ρ∞ X Ex∞ X

f Ex∞ f ρ∞ Y Ex∞ Y and Theorem 4.1 that f is a weak equivalence if and only if Ex∞ f is a weak equivalence. 4.3. Corollary. Let p: X → Y be a Kan ﬁbration and a weak equivalence. Then p has the RLP with respect to I. Proof. By Theorem 4.1, we have a commutative square

ρ∞ X Ex∞ X p Ex∞ p ρ∞ Y Ex∞ Y Since p is a weak equivalence, Ex∞ p is a homotopy equivalence. Consequently, ∞ all fibres of Ex p are contractible. For any y ∈ Y0, we have a weak equivalence ρ∞ : p−1(y) → (Ex∞ p)−1(ρ∞(y)). Since Ex∞ preserves finite limits, the target of this map is the fibre of Ex∞ p over ρ∞(y). Consequently, p−1(y) is contractible. It follows that p has the RLP with respect to I. 4.4. Corollary. If f : X → Y is a cofibration and a weak equivalence, then it is anodyne. Proof. Factor f into an anodyne map i: X → X followed by a Kan fibration p: X → Y . Any lifting problem against a Kan fibration q : V → W gives rise to a commutative diagram X V

i q p X Y W 22

There exists a lift l : X → V . Since f is a weak equivalence, so is p. By Corollary 4.3, p has the RLP with respect to I. In particular, there exists by Corollary 2.8 a section s. Then l ◦ s is the desired lift. 4.5. Corollary (HELP). Let i: A → B be a coﬁbration and let p: X → Y be a Kan ﬁbration. If i or p is a weak equivalence, then the canonical map F(B,X) → F(A, X) × F(B,Y ) F(A,Y ) has the RLP with respect to I. Proof. If i is a weak equivalence, it is an anodyne extension by Corollary 4.4. Then the corollary follows from Corollary 2.23. As usual, any right lifting problem with respect to ∂∆n → ∆n corresponds to a lifting problem ∆n × A ∪ ∂∆n × B X ∂∆n×A

n n p (∂∆ →∆ )i ∆n × B Y If p is a weak equivalence, it has the RLP with respect to I by Corollary 4.3, so this lifting problem has a solution. 4.6. Remark. Informally, Corollary 4.5 tells us that “the space of solutions to any lifting problem A X

i p B Y is contractible”. The proof of Theorem 4.1 occupies the remainder of this section. We begin by defining all the relevant constructions. 4.7. Definition. Let P be a poset. Define chains(P ) as the poset whose elements are finite, linearly ordered subsets of P such that a chain c is smaller than a chain c0 if c is a suborder of c0. This defines an endofunctor on the category of posets. 4.8. Definition. Define sd ∆• : ∆ → sSet as the functor sending [n] to N(chains[n]). The (barycentric) subdivision sd: sSet → sSet is the unique colimit-preserving extension of sd ∆. By Exercise 2 on Sheet 3, sd has a right adjoint Ex: sSet → sSet. 4.9. Definition. The last vertex map is the natural transformation

λ: sd → idsSet induced by the natural map chains[n] → [n], c 7→ max c. Denote by ρ the adjoint transformation

ρ: idsSet → Ex . 4.10. Definition. Define Ex∞ : sSet → sSet as the colimit of 2 ρ ρ Ex 2 ρ Ex 3 idsSet −→ Ex −−−→ Ex −−−→ Ex → ... ∞ ∞ Let ρ : idsSet → Ex be the induced natural transformation. 4.11. Corollary. Ex∞ commutes with finite limits. SOME SIMPLICIAL HOMOTOPY THEORY 23

Proof. Since Ex is a right adjoint, it commutes with arbitrary limits. Filtered colimits commute with ﬁnite limits in Set (by inspection), so the same statement holds in sSet. This implies the corollary. Since Ex arises as the right adjoint of sd, everything we want to know about Ex will be deduced from appropriate statements about the subdivision functor. 4.12. Proposition. Let X be a simplicial set. Then every horn in Ex X has a ﬁller in Ex2 X.

n n f ρEx X Proof. For every map f :Λk → Ex X, we have to show that Λk −→ Ex X −−−→ 2 n Ex X admits a ﬁller. Let g : sd Λk → X be the map determined by f. The map adjoint to ρ ◦f is given by λ ◦sd f, which is the same as f ◦λ n by naturality Ex X Ex X Λk of λ. The adjoint of this map is

sd λΛn 2 n k n g sd Λk −−−−→ sd Λk −→ X. Consequently, we have to show that the dotted arrow in the diagram

2 n sd λ n sd Λk sd Λk

sd2 ∆n X exists and makes the resulting square commute. Note that it suﬃces to ﬁnd the dotted arrow in the case g = id. n Observe that sd Λk is the nerve of the poset P := chains[n] \{[n], [n] \{k}}, so we have to specify a map of posets

chains chains[n] → P, (c0 ⊆ ... ⊆ cr) 7→ {max(g c0),..., max(g cr)}, where max0 is deﬁned by ( k c = [n] or c = [n] \{k}, maxc := g max c else. It is straightforward to check that this is well-deﬁned, and the map evidently extends sd λ. 4.13. Corollary. Ex∞ X is a Kan complex for every simplicial set X.

n n ∞ n Proof. Since Λk is compact, every map Λk → Ex factors through some Ex X. n n n ρ Ex n+1 n Assume n ≥ 1. Then Λk → Ex X −−−→ Ex X extends to a map ∆ → n+1 ∞ Ex X by Proposition 4.12, and this provides also a ﬁller in Ex X. We consider the question whether Ex preserves Kan ﬁbrations next. By the adjunction between sd and Ex, this reduces to the question whether sd preserves anodyne extensions.

4.14. Lemma. Let J be a non-empty, ﬁnite set. The inclusion LJ := P(J)\{J} → P(J) induces a left anodyne map on nerves.

Proof. Pick some element j ∈ J. Then the inclusion map N(LJ ) → N(P(J)) is isomorphic to the map 1 1 (N(LJ\{j}) × ∆ ) ∪ (N(P(J \{j})) × {0}) → N(P(J \{j}) × ∆ . N(LJ\{j})×{0} This map is left andoyne by Proposition 2.22. 4.15. Proposition. The subdivision functor preserves anodyne extensions. 24

Proof. Since sd is a colimit-preserving functor, the class of morphisms Σ := {i | sd i is anodyne} is saturated. Hence it suﬃces to show that H ⊆ Σ. Recall that sd ∆n is the nerve n of the poset chains[n], and that Λk is the nerve of the subposet P := chains[n] \ {[n], [n]\{k}}. In this proof, we identify chains[n] with the set P∗([n]) of non-empty subsets of [n]. Set [n]k := [n] \{k}. The map

P([n]k) → P∗([n]),S 7→ S ∪ {k} is injective. Let X ⊆ sd ∆n denote the image of the induced map on nerves. Then n ∼ X ∩ sd Λk = N(L[n]k ), and there is a pushout square

∼ n n N(L[n]k ) = X ∩ sd Λk sd Λk

∼ N(P([n]k) = X Y

n Consequently, sd Λk → X is left andoyne by Lemma 4.14. Consider now the injective map ( S r = 0, P([n]k) \{∅} × [1] → P∗([n]), (S, r) 7→ c(S) = S ∪ {k} r = 1, and denote its image by Z. Then ∼ Z∩Y = N((P([n]k)\{∅})×{1}) ∪ N((P([n]k)\{∅, [n]k})×[1]) N((P([n]k)\{∅,[n]k})×{1}) and there is a pushout square

Z ∩ Y Y

Z sd ∆n The left vertical map is right anodyne by Proposition 2.22, so X → sd ∆n is also n n right anodyne. Consequently, the composition sd Λk → X → sd ∆ is anodyne.

n 4.16. Corollary. Ex preserves Kan fibrations for every n ∈ N ∪ {∞}. Proof. Suppose by induction that Exn preserves Kan fibrations. Any lifting problem for Exn+1 p, where p: X → Y is a Kan fibration, corresponds to a lifting problem n n sd Λk Ex X Exn p

sd ∆n Exn Y n n Since sd Λk → sd ∆ is anodyne by Proposition 4.15, a solution to the lifting problem exists. n n The statement for n = ∞ follows since Λk and ∆ are compact objects in sSet. Since Ex∞ ∆0 is obviously isomorphic to ∆0, Corollaries 4.11, 4.13 and 4.16 take care of Theorem 4.1 with the exception of (3). SOME SIMPLICIAL HOMOTOPY THEORY 25

4.17. Proposition. Let X,Y : N → sSet be diagrams and let f : X → Y be a transformation. Suppose that x(n): X(n) → X(n + 1) and y(n): Y (n) → Y (n + 1) are coﬁbrations for all n, and that f(n): X(n) → Y (n) is a weak equivalence for every n.

Then the induced map colimN f : colimN X → colimN Y is a weak equivalence. Proof. Let g : V → W be an arbitrary transformation of diagrams N → sSet. We claim that g can be factored as g = pi, where i(n) is anodyne for every n and p(n) is a Kan fibration for every n. This factorisation can be built by induction. In degree 0, apply Proposition 2.30 to factor g(0) into an anodyne i(0): V (0) → V (0) and a Kan fibration p(0): V (0) → W (0). Then the induced map j(1): V (1) → V (1)∪V (0)V (0) is anodyne and the map V (0) → V (1) ∪V (0) V (0) is a cofibration. Moreover, g(1) induces a map V (1) ∪V (0) 0 V (0) → W (0). Factor this map into an andoyne map j (1): V (1) ∪V (0) V (0)V (1) and a Kan fibration p(1): V (1) → W (1). Set i(1) := j0(1) ◦ j(1). Proceeding like this inductively produces the desired factorisation. Apply the claim to the unique transformation Y → ∆0 to obtain a transformation j : Y → Y such that j(n) is anodyne for all n and Y (n) is a Kan complex for every n. Apply the claim a second time to jf : X → Y , producing a pointwise anodyne transformation i: X → X and a pointwise Kan fibration p: X → Y . Since Y (n) is a Kan complex and p(n) is a Kan fibration, X(n) is a Kan complex for every n. Moreover, p(n) is a weak equivalence, so it has the RLP with respect to I by Corollary 3.18 and Theorems 3.8 and 3.11. n n Since ∂∆ and ∆ are compact, it follows directly that colimN p has the RLP with respect to I. In particular, colimN p is a weak equivalence. Therefore, it suffices to show the proposition in the case when all components of the transformation are anodyne: then both colimN j and colimN i are weak equivalences, which readily implies that colimN f is a weak equivalence. Hence, assume that f(n) is anodyne for all n. Let K be a Kan complex. We have the show that the induced map ∗ (colim f) : π0 F(colim Y,K) → π0 F(colim X,K) N N N is a bijection. This map fits into a commutative square

(colim f)∗ N π0 F(colimN Y,K) π0 F(colimN X,K) ∼= ∼= ∗ (lim op f) N op op π0(limN F(Y,K)) π0(limN F(X,K)) Let us show directly that the lower horizontal map is a bijection. We claim in slightly greater generality that if we have a transformation between towers of Kan ﬁbrations of Kan complexes

u(2) u(1) ... U(2) U(1) U(0)

q(2) q(1) q(0) v(2) v(1) ... V (2) V (1) V (0) such that q(n) has the RLP with respect to I for all n, then the induced map

op op π0(limN U) → π0(limN V ) is a bijection. 0 0 op Q Given (bn)n ∈ limN V , there exists a sequence (an)n U(n) such that q(n)(an) = 0 0 N bn. Set a0 := a0. Since q(0)(u(1)(a1) = b0, the constant homotopy on b0 lifts to 0 a homotopy u(1)(a1) ' a0. Since u(1) is a Kan ﬁbration, this homotopy lifts to 1 0 a homotopy h1 : ∆ → U(1) from a1 to some vertex a1. Note that u(1)(a1) = a0. 26

Moreover, q(1)◦h1 provides a homotopy b1 ' q(1)(a1) such that v(1)◦q(1)◦h1 is the op constant homotopy on b0. Continuing inductively, we find a vertex (an)n ∈ limN U 1 op and a homotopy ∆ → limN V from (bn)n to (q(n)(an))n. So the comparison map is surjective. 0 Suppose that (an)n,(an)n are two vertices such that there exists a homo- 1 0 op topy ∆ → limN V from (q(n)(an))n to (q(n)(an))n, ie a sequence of homo- 1 0 topies (hn : ∆ → V (n))n such that hn is a homotopy q(n)(an) ' q(n)(an) and v(n + 1) ◦ hn+1 = hn. Since each q(n) has the RLP with respect to I, there exists 0 1 0 0 a sequence of homotopies (kn : ∆ → U(n))n such that kn is a homotopy an ' an. 0 Set k0 := k0. 2 0 Consider the map H0 : ∂ → U(0) which is given by u(1)k1 and k0 on two 0 opposite faces, and by the constant homotopies on a0 and a0 on the two other 2 pr 1 h0 faces, respectively. Then q(0) ◦ H0 extends to the map −→ ∆ −→ V (0). As 2 q(0) has the RLP with respect to I, the map H0 extends to a map K0 : → U(0) lifting the given map. Since u(1) is a Kan fibration, K0 lifts further to a homotopy 0 k1 ' k1 (relative endpoints). Again, one continues this procedure inductively to 1 0 op obtain a homotopy ∆ → limN U from (an)n to (an)n. This proves injectivity, and thus finishes the proof of the proposition.

4.18. Proposition. Let

f X X0

p p0 g Y Y 0 be a commutative diagram such that p and p0 are Kan ﬁbrations of Kan complexes and g is a weak equivalence. Then the following are equivalent: (1) f is a weak equivalence. (2) The induced map p−1(y) → (p0)−1(g(y)) is a homotopy equivalence for every y ∈ Y0.

Proof. Exercise.

4.19. Proposition. The last vertex map λ: sd X → X is a weak equivalence for every simplicial set X.

Proof. Since sd preserves colimits and coﬁbrations, it follows from Proposition 4.17 that it suﬃces to show that λ: sd skn(X) → skn(X) is a weak equivalence for every n. We do this by induction on n. For n = 0, one sees directly that λsk0(X) is an isomorphism. Let K be a Kan complex. The vertical maps in the commutative square

λ∗ skn+1(X) F(skn+1(X),K) F(sd skn+1(X),K)

λ∗ skn(X) F(skn(X),K) F(sd skn(X),K) are Kan fibrations of Kan complexes, and λ∗ is a weak equivalence by induction skn(X) hypothesis. By Proposition 4.18, it suffices to show that the induced maps on vertical fibres are weak equivalences. SOME SIMPLICIAL HOMOTOPY THEORY 27

Consider the pullback square

F(sk X,K) F(F ∂∆n+1,K) n+1 In+1

F(sk X,K) F(F ∆n+1,K) n In+1 arising from the pushout describing the attachment of (n + 1)-simplices. Since sd preserves colimits, there is an analogous square in which all sources of func- tion complexes have been replaced by their subdivisions. Consequently, for f ∈ F(skn(X),K) the induced map on fibres can be identified with G G {f 0} × F( ∆n+1,K) → {f 0} × F( sd ∆n+1,K), F n+1 F n+1 F( I ∂∆ ,K) F( I sd ∂∆ ,K) n+1 In+1 n+1 In+1 where f 0 : F ∂∆n+1 → K is the map induced by f. In+1 Again by Proposition 4.18, it suffices to show that the two horizontal maps in the diagram

∗ F(F ∆n+1,K) λ F(F sd ∆n+1,K) In+1 In+1

∗ F(F ∂∆n+1,K) λ F(F sd ∂∆n+1,K) In+1 In+1 are homotopy equivalences. The lower horizontal map is again a homotopy equivalence by induction hypothesis. A product of homotopy equivalenes is a homotopy equivalence, so it suﬃces to see n+1 n+1 n+1 that λ∆n+1 : sd ∆ → ∆ is a homotopy equivalence. Since both sd ∆ and ∆n+1 are nerves of posets with a largest element, this is immediate from Exercise 2 on Sheet 4. 4.20. Lemma. Let X be a simplicial set and let K be a Kan complex. Then the ∼ bijection HomsSet(sd X,K) = HomsSet(X, Ex K) induces a bijection ∼ π0 F(sd X,K) = π0 F(X, Ex K). Proof. Suppose that h: sd X ×∆1 → K is a homotopy. Then the adjoint morphism to sd X×λ sd(X × ∆1) → sd X × sd ∆1 −−−−−−→∆1 sd X × ∆1 −→h K provides a homotopy between the morphisms adjoint to h0 and h1. Suppose that X × ∆1 → Ex K is a homotopy. Let h: sd(X × ∆1) → K be the adjoint morphism. Let k denote the composition sd(X × ∆1) → sd X −→h0 K. Since X × {0} → X × ∆1 is anodyne, the restriction map F(sd(X × ∆1),K) → F(sd(X × {0},K) has the RLP with respect to I by Proposition 4.15 and Corollary 2.26. Since h and k restrict to the same map sd(X ×{0}) → K, there exists a homotopy sd(X ×∆1)× ∆1 → K between h and k. The restriction of this homotopy to sd(X × {1}) × ∆1 provides the required homotopy between h0 and h1. 4.21. Corollary. The transformation ρ: K → Ex K induces a bijection ∼ π0 F(X,K) −→ π0 F(X, Ex K) for every Kan complex K and every simplicial set X. 28

Proof. By Lemma 4.20, there is a commutative triangle

π0(X, Ex K) ρ∗

∼ π0 F(X,K) = λ∗

π0 F(sd X,K)

The lower diagonal arrow is a bijection by Proposition 4.19. 4.22. Lemma. Let X be a simplicial set. Then there is a simplicial homotopy ρEx X ' Ex ρX . Proof. Deﬁne the map of posets h: chains(chains[n])) × [1] → chains[n] ( max c0 ≤ ... ≤ max ck i = 0, (c0 ⊆ ... ⊆ ck, i) 7→ ck i = 1.

Note that max c0 ≤ ... ≤ max ck is a suborder of ck, so this is well-deﬁned. Taking nerves, we obtain a homotopy from sd λ∆n to λsd ∆n . This is natural in n, so we obtain a natural transformation sd2(−) × ∆1 ⇒ sd (using that − × ∆1 commutes with arbitrary colimits). Since there is also a natural transformation sd2(− × ∆1) ⇒ sd2(−) × ∆1, we obtain a natural transformation sd2(− × ∆1) ⇒ sd. Using the solution to Exercise 1 on Sheet 4, we obtain a natural transformation Ex ⇒ F(∆1, Ex2(−)) which provides (by “currying”) the required homotopy ρ Ex ' Ex ρ. 4.23. Proposition. The transformation ρ: X → Ex X is a weak equivalence for every simplicial set X. Proof. Let K be a Kan complex. Consider the diagram

∗ ρX π0 F(Ex X,K) π0 F(X,K)

(ρK )∗ (ρK )∗ ∗ ρX π0 F(Ex X, Ex K) π0 F(X, Ex K) The dotted diagonal map is induced by the functoriality of Ex: note that Ex preserves simplicial homotopy since a homotopy X ×∆1 → K gives rise to a homotopy Ex X ×∆1 → Ex X ×Ex ∆1 =∼ Ex(X ×∆1) → K. Both vertical maps are bijections by Corollary 4.21. Then the lower triangle commutes by the naturality of ρ. For f : Ex X → K, we have by Lemma 4.22

Ex(f ◦ ρX ) = Ex f ◦ Ex ρX ' Ex f ◦ ρEx X = ρK ◦ f. So the upper triangle commutes as well. It follows that all maps in the diagram are bijections. 4.24. Remark. Note that ρ is injective. Since λ is evidently surjective, this follows from the commutativity of

HomsSet(Y, Ex X) (ρX )∗

Hom (Y,X) ∼= sSet ∗ λY

HomsSet(sd Y,X) SOME SIMPLICIAL HOMOTOPY THEORY 29

Proof of Theorem 4.1. (1) is obvious. (2), (4) and (5) hold by Corollaries 4.11, 4.13 and 4.16, respectively. So we are left with showing that ρ∞ : X → Ex∞ X is a weak equivalence for every simplicial set X. By Proposition 4.23, each composite X → Exn X is a weak equivalence for every n. By Proposition 4.17, it follows that ∞ ρ is also a weak equivalence. 5. The comparison with the homotopy theory of CW-complexes To ﬁnish our discussion of simplicial homtopy theory, we outline the relation to the homotopy theory of CW-complexes. We will assume some facts about the geometric realisation functor. This is summarised in the following theorem: 5.1. Theorem. Let Kan denote the simplicially enriched category of Kan complexes and let CW denote the simplicially enriched category of CW-complexes (by taking Sing of the spaces of all continuous maps). Then both |−|: Kan → CW and Sing: CW → Kan are Dwyer–Kan equivalences, ie • they induced essentially surjective functors on homotopy categories and • they induce weak equivalences on all mapping spaces. The remainder of this section is occupied by the proof of Theorem 5.1 5.2. Proposition. Geometric realisation |−|: sSet → kTop commutes with ﬁnite products.

Proof. [GZ67, Section 3.3] 5.3. Corollary. Both Sing: kTop → sSet and |−|: sSet → kTop deﬁne enriched functors which are adjoint. More precisely, the adjunction (|−|, Sing) induces natural bijections ∼ op FkTop(|−|, −) = FsSet(−, Sing −): sSet × kTop → sSet. Proof. Let f : X × ∆nY . By Proposition 5.2, the map |f| may be regarded as a map |∆n| × |X| =∼ |∆n × X| → |Y |, and therefore deﬁnes an n-simplex in FkTop(|X|, |Y |). n An n-simplex in FkTop(T,U) is given by a map f : |∆ | × T → U which induces a map

u n ×Sing T Sing f ∆n × Sing T −−−−−−−−→∆ Sing|∆n| × Sing T =∼ Sing(|∆n| × T ) −−−−→ Sing U. It is straightforward to check that these rules do deﬁne enriched functors. n For the second part, observe that an n-simplex ∆ → FkTop(|X|,T ) induces by Proposition 5.2 a unique map |X × ∆n| =∼ |X| × |∆n| → T, n which by adjunction is the same as a map ∆ → F(X, Sing T ). 5.4. Lemma. There are natural isomorphisms ∼ π0(−) −→ π0(Sing(−)): kTop → Set and ∼ πn(−) −→ πn(Sing(−)): kTop∗ → Grp for every n ≥ 0. 30

Proof. Exercise.

5.5. Corollary. Both |−| and Sing preserve weak equivalences.

Proof. Exercise.

5.6. Theorem. The following are equivalent: (1) Simplicial approximation holds for maps from ﬁnite simplicial sets: Let A ⊆ X be ﬁnite simplicial sets and let Y be a simplicial set. Suppose that f0 : A → Y and g : |X| → |Y | are maps such that g||A| = |f0|. Then there exist for some n a map f : sdn X → Y such that

n f sdn A λ A 0 Y

f sdn X commutes and a homotopy |f| ' g|λn| relative |sdn A|. (2) Simplicial approximation holds for maps into Kan complexes: Let A ⊆ X be simplicial sets and let K be a Kan complex. Suppose that f0 : A → K and g : |X| → |K| are maps such that g||A| = |f0|. Then there exist a map f : X → K and a homotopy |f| ' g relative |A|. (3) The unit morphism X → Sing|X| is a homotopy equivalence for every Kan complex. (4) The unit morphism X → Sing|X| is a weak equivalence for every simplicial set. (5) Geometric realisation induces an equivalence F(X,K) −→' F(|X|, |K|) for every simplicial set X and every Kan complex K.

Proof. We only sketch why (1) is equivalent to (2). If (1) holds, one constructs an approximation simplex by simplex. This reduces the problem to the case A = ∂∆n and X = ∆n. In this case, the simplicial approximation whose existence is guaranteed by (1) yields a commutative square

f ∂∆n 0 K ρn f ∆n Exn K Since ρn is an anodyne map between Kan complexes, it has a retraction. This yields the desired extension. Conversely, we may replace Y by Ex∞ Y and apply (2). Since X is assumed to be ﬁnite, the resulting map X → Ex∞ Y factors through some Exn Y and gives thus rise to a map sdn X → Y with the desired properties. We now show (2) ⇒ (3). By one of the triangle identities, the composition

|u | c |X| −−−→|X Sing|X|| −−→||X| X| is the identity. By (2), there exists a map k : Sing|X| → X extending idX such that |k| is homotopic to c|X| relative |X|. Let h: |Sing|X||×[0, 1] → |X| be such a homotopy. Since |∆1| =∼ [0, 1] and |−| commutes with ﬁnite products, this corresponds to a map |Sing|X| × ∆1| → |X| whose adjoint provides a deformation retraction of Sing|X| onto |X|. SOME SIMPLICIAL HOMOTOPY THEORY 31

For (3) ⇒ (4), consider the commutative square

X uX Sing|X|

ρ∞ Sing|ρ∞|

u ∞ Ex∞ X Ex X Sing|Ex∞ X| The vertical maps are weak equivalences by Theorem 4.1 and Corollary 5.5, and the lower horizontal map is a weak equivalence by assumption. So uX is also a weak equivalence. For (4) ⇒ (5), consider the commutative diagram

FCW(|X|, |K|) |−|

∼ FsSet(X,K) = (uK )∗

FsSet(X, Sing|K|)

Since uK is a homotopy equivalence by assumption, (5) follows. Finally, we show that (5) ⇒ (2). There is a map of Kan ﬁbrations of Kan complexes |−| FsSet(X,K) FCW(|X|, |K|)

|−| FsSet(A, K) FCW(|A|, |K|) and by assumption both horizontal maps are weak equivalences. It follows from Proposition 4.18 that the induced map on ﬁbres over f0 and |f0| is a homotopy equivalence. Picking a preimage of g yields the desired simplicial approximation. Standing assumption: Let us assume for a moment that one of the simplicial approximation statements holds. 5.7. Corollary. The counit |Sing T | → T is a weak equivalence for every T ∈ Top.

Proof. By Lemma 5.4, it suﬃces to check that Sing cT : Sing|Sing T | → Sing T induces bijections on all πn. By one of the triangle identities, the identity on Sing T factors as u Sing c Sing T −−−−→Sing T Sing|Sing T | −−−−→T Sing T.

Since uSing T is a homotopy equivalence, Sing cT is also a homotopy equivalence. So the corollary follows from Corollary 3.10. 5.8. Corollary. The geometric realisation detects weak equivalences, ie if f : X → Y is a map of simplicial sets such that |f| is a weak equivalence, then f is a weak equivalence. Proof. By Corollary 5.5, Sing|f| is a weak equivalence. Then it follows from the commutative diagram f X Y

uX uY Sing|f| Sing|X| Sing|Y | and Theorem 5.6 that f is a weak equivalence. 32

Proof of Theorem 5.1. Theorem 5.6 deals almost with the entire statement since it already tells us that Sing is essentially surjective on homotopy categories and that |−| induces weak equivalences on mapping spaces. By Corollary 5.7, the functor |−| is also essentially surjective on homotopy categories. Using Corollary 5.7 a second time, the commutative diagram

FsSet(Sing T, Sing U) Sing

∼= FCW(T,U) ∗ cT

FCW(|Sing T |,U) shows that Sing also induces weak equivalences on mapping spaces. In the remainder of this section, we outline why simplicial approximation holds for simplicial sets. Our discussion follows Jardine [Jar04], but many of the properties in which we are interested (plus many other, more subtle properties) are also studied in [WJR13, Section 2]. We assume the classical simplicial approximation theorem for simplicial complexes. To formulate it, we need to recall the barycentric subdivision of a simplicial complex: Given an (abstract) simplicial complex (V,S), we deﬁne sd(V,S) as the abstract simplicial complex whose set of vertices is given by S, and whose n-simplices are given by strictly ascending chains s0 ⊆ s1 ⊆ ... ⊆ sn of length n. 5.9. Remark. Recall that any locally ordered simplicial complex gives rise to a simplicial set by formally adjoining degeneracies (see Exercise 3 on Sheet 3). Denote this functor by (−)+ : SClo → sSet. Canonical examples of locally ordered simplicial complexes are those arising from posets: For a poset P , let V (P ) be the simplicial complex with vertex set P and n-simplices strictly ascending chains p0 < . . . < pn. The simplicial complex V (P ) is often called the nerve of the poset P , but we will avoid that terminology because the nerve of a poset is a simplicial set for us. However, note that there is a natural isomorphism NP =∼ V (P )+, so the two notions are not unrelated. Moreover, it is straightforward to check that (sd V (P ))+ =∼ sd V (P )+ for any poset P .

5.10. Proposition. Let U ⊆ V and W be simplicial complexes. Let f0 : U → W be a simplicial map and let g : |V | → |W | be a map such that g||U| = |f0|. Suppose that V is ﬁnite. Then there exist some n a map f : sdn V → W such that

n f sdn U λ U 0 W

f sdn V commutes and a homotopy |f| ' g|λn| relative |sdn U|.

Proof. Exercise, possibly. If not, I will insert a textbook reference here later. Our line of approach is to deduce a simplicial approximation theorem for simplicial sets by turning the simplicial sets in question into simplicial complexes. Here is a particularly brutal way of building a simplicial complex from a simplicial set: SOME SIMPLICIAL HOMOTOPY THEORY 33

5.11. Definition. Let X be a simplicial set. Let nd X denote the set of non- degenerate simplices of X. Denoting by hxi the smallest simplicial subset generated by a non-degenerate simplex x, equip nd X with the partial order given by x ≤ y :⇔ hxi ⊆ hyi. The Barrat nerve B X of X is defined as B X := N(nd X). Any map f : X → Y of simplicial sets induces a map of posets nd f : nd X → nd Y (since every simplex is the unique, possibly trivial, degeneration of a unique non- degenerate simplex), and thus induces a map B f :B Xto B Y . Therefore, we obtain a functor B: sSet → sSet. 5.12. Remark. In general, the Barratt nerve of a simplicial set has little to do with the original simplicial set. For example, B(∆1/∂∆1) is the nerve of the poset [1] and therefore simplicially contractible. On the other hand, B(V,S)+ =∼ sd(V,S)+ for any locally ordered simplicial complex (V,S). 5.13. Lemma. For every simplicial set X, the canonical maps colim sdhxi → sd X and colim Bhxi → B X x∈nd X x∈nd X are isomorphisms. ∼ Proof. Since colimx∈nd X hxi = X, the first part follows because sd is colimit- preserving. Let ξ = x0 ≤ ... ≤ xn be an n-simplex in B X. Then ξ is also an n-simplex in hxni. So colimx∈nd X Bhxi → B X is surjective. Suppose that ξ = x0 ≤ ... ≤ xk ∈ Bhxi and η = y0 ≤ ... ≤ yl ∈ Bhyi map to the same simplex in B X. Then ξ and η define the same sequence in nd X. In particular, yl = xk. Call this simplex z. Then ξ and η both lie in Bhzi and define the same element there. So ξ = η in colimx∈nd X Bhxi. By Remark 5.9, the restrictions of sd and B along the Yoneda embedding ∆ → sSet agree. By the universal property of the left Kan extension, there is an induced natural transformation π : sd → B . 5.14. Lemma. Let X be a simplicial set. The map π : sd X → B X is (1) surjective in all degrees; (2) bijective on vertices. In particular, two n-simplices in sd X map to the same simplex in B X if and only if they have the same set of vertices.

Proof. By definition, an n-simplex in B X is a sequence x0 ≤ ... ≤ xn in nd X. ∗ Choose injective maps δi such that xi = δi (xn). Then each δi can be considered as |xn| |xn| a simplex in ∆ , and thus δ0 ≤ ... ≤ δn defines an n-simplex in sd ∆ . The pair (δ0 ≤ ... ≤ δn, xn) defines an n-simplex in sd X which maps to the original n-simplex in B X under π. For any two vertices in sd X, choose k, l minimal such that they can be represented as sd y ∆0 −→v sd ∆k −−→sd x sd X and ∆0 −→w sd ∆l −−→ sd X 34 for non-degenerate simplices x and y of X. By the minimality of k and l, v = [k] and w = [l]. If πX ◦ sd x ◦ v = πX = sd y ◦ w, then use (B X)0 = nd X and B ∆k =∼ sd ∆k to see that

x = B x(v) = πX (sd x(v)) = πX (sd y(w)) = B y(w) = y.

5.15. Deﬁnition. A non-degenerate simplex x ∈ Xn is regular if

n−1 dn(x) ∆ hdn(x)i

dn ∆n x hxi is a pushout square. A simplicial set is regular if all its non-degenerate simplices are regular.

5.16. Example. If the characteristic map of every non-degenerate simplex in X is injective (ie X is non-singular), then X is regular. In particular, the nerve of every poset is regular.

5.17. Proposition. The subdivision sd X is regular for every simplicial set X.

Proof. Let c: ∆n → sd X be a non-degenerate simplex. Choose k minimal such that c factors as n κ k sd x c: ∆ −→ sd ∆ −−→ sd skk(X) for some non-degenerate k-simplex x of X and some non-degenerate n-simplex κ of sd ∆k (if κ was degenerate, c would also be degenerate; if x was degenerate, k would not have been minimal). We may assume k > 0. Since k is minimal, k κ(n) = [k]. In particular, dn(κ) is a simplex in sd ∂∆ . Therefore, there is an induced commutative diagram

d (κ) ∆n−1 n sd ∂∆k sd ∂hxi

dn ∆n κ sd ∆k sdhxi where ∂hxi denotes the simplicial set hxi with the unique top-dimensional simplex x removed. The right hand square is a pushout. Consider the outer square

d (c) ∆n−1 n sd ∂hxi

dn ∆n c sdhxi

n n If y is a simplex in ∆ which does not lie in dn∆ , its image contains [n]. Hence c(y) ∈/ sd ∂hxi. Consequently, this square is a pullback. Since this square ﬁts into the commutative diagram

n−1 dn(κ) ∆ hdn(c)i sd ∂hxi

dn ∆n κ hci sdhxi SOME SIMPLICIAL HOMOTOPY THEORY 35 and all arrows in the right hand square are monomorphisms as indicated, the square

n−1 dn(c) ∆ hdn(c)i

dn ∆n c hci is also a pullback. Therefore, it suﬃces to show that the induced map n n ∆ \ dn∆ → hci \ hdn(c)i is a degreewise bijection. This map is surjective since c: ∆n → hci is surjective. n n Suppose that σ, τ are in ∆ \ dn∆ such that c(σ) = c(τ) ∈ hci ⊆ sdhxi. Then c(σ) and c(τ) factor through sd ∆k, so κ(σ) = κ(τ). Since sd ∆k is non-singular and κ is non-degenerate, σ = τ. This proves the proposition. 5.18. Deﬁnition. A simplicial set X is op-regular if Xop is regular, ie for every n-simplex x the diagram

n−1 d0(x) ∆ hd0(x)i

d0 ∆n x hxi is a pushout. Let sSetopreg denote the full subcategory of sSet spanned by the op-regular simplicial sets. 5.19. Proposition. There exists a natural transformation

γ :B |sSetopreg ⇒ id such that λ|sSetopreg = γ ◦ π|sSetopreg .

Proof. If such a factorisation γX exists for every op-regular simplicial set X in opreg some full subcategory C of sSet , then the collection (γX )X∈C is automatically a natural transformation on C: for every f : X → Y we have

πX ◦ B f ◦ γY = γY ◦ πY ◦ sd f = λY ◦ sd f = f ◦ λX = f ◦ γX ◦ πX . Since π is an epimorphism by Lemma 5.14, naturality of γ follows. So we only need opreg to show that γX exists for every X ∈ sSet . By Lemma 5.13 and the preliminary observation, the existence of γX follows if we can show that for every non-degenerate simplex x ∈ X there exists a commutative triangle π sdhxi hxi Bhxi

γ λhxi x hxi

We proceed by induction on the dimension of x. If x is a 0-simplex, sdophxi,Bophxi 0 and hxi are all isomorphic to ∆ , and the existence of γx follows trivially. Suppose that the dimension of x is n > 0. By induction hypothesis, there exists a commutative triangle

πhd0(x)i sdhdn(x)i Bhdn(x)i

γd (x) λhd0(x)i 0 hdn(x)i 36

Note the slight abuse of notation: d0(x) is not necessarily non-degenerate. However, it is the unique degeneration d0(x) = s(y) of a non-degenerate simplex y, and evidently hyi = hd0(x)i. Since X is op-regular, λhxi ﬁts into a map of pushout squares (the front and back face of the cube):

n−1 sd ∆ sdhd0(x)i λ λ n−1 ∆ d0 hdn(x)i

n d0 sd ∆ sdhxi λ λ ∆n x hxi

Consequently, we only need to factor the composite map

λ sd ∆n −−→sd x sdhxi −−−→hhxi xi in a way which is compatible with the given factorisation on sd ∆n−1. k n Therefore, suppose that σ, τ : ∆ → sd ∆ are k-simplices such that πhxi ◦sd(x)◦ σ = πhxi ◦ sd(x) ◦ τ. Write σ = σ0 ≤ ... ≤ σk and τ = τ0 ≤ ... ≤ τk. By Lemma 5.14, this means that

{x ◦ σ0, . . . , x ◦ σk} = {x ◦ τ0, . . . , x ◦ τk}. We have to show that

λhxi ◦ sd(x) ◦ σ = λhxi ◦ sd(x) ◦ τ.

Let i be the smallest index such that 0 lies in the image of σi and τi. Then σj = τj for all j ≥ i since x is injective on that part of ∆n. Set i := k + 1 if no such index exists. n If i = k + 1, then σ and τ both map to sd d0∆ . This case is covered by the induction hypothesis. So suppose i < k + 1. If i = 0, then σ = τ, so obviously λ ◦ sd(x) ◦ σ = λhxi ◦ sd(x) ◦ τ. Suppose that 0 < i < k + 1. Then λ ◦ σj = λ ◦ d0(σj) and λ ◦ τj = λ ◦ d0(τj) for all j ≥ i. Consequently, the simplices 0 σ := σ0 ≤ ... ≤ σi−1 ≤ d0(σi) ≤ ... ≤ d0(σk) and 0 τ := τ0 ≤ ... ≤ τi−1 ≤ d0(τi) ≤ ... ≤ d0(τk) n both lie in sd d0∆ and map to the same simplex as σ and τ under λ ◦ sd(x), respectively. Since x ◦ d0(σj)) = x ◦ σj ◦ d0 = x ◦ τj ◦ d0 = x ◦ d0(τj), the simplices σ0 and τ 0 map to the same simplex under π, and therefore satisfy λ ◦ sd(x) ◦ σ0 = 0 λ ◦ sd(x) ◦ τ by induction. This ﬁnishes the proof. For the following discussion, abbreviate the composite functor (−)op ◦ sd: sSet → sSet by sdop. By Proposition 5.17, sdop takes values in the full subcategory of op-regular simplicial sets. It comes equipped with a natural transformation

ϕ: sdop → idsSet, the ﬁrst vertex map, induced by the transformation (chains[n])op → [n], c 7→ min c. SOME SIMPLICIAL HOMOTOPY THEORY 37

5.20. Lemma. There is a natural isomorphism Φ: sd =∼ sd ◦(−)op 2 ∼ Let Ψ: sd = sd sdop denote the induced isomorphism. Then there exists a simplicial homotopy sd2 X × ∆1 → X from ϕλΨ to λ2. Proof. It suﬃces to show that the lemma holds on the full subcategory of standard simplices. For any poset P , Φ is given by op ΦP : chains P → chains(P ), c 7→ c with reversed order. The transformation λ2 is induced by

chains(chains[n]) → [n], (c0 ⊆ ... ⊆ cr) 7→ max cr, and ϕλΨ is induced by

chains(chains[n]) → [n], (c0 ⊆ ... ⊆ cr) 7→ min c0.

Since min c0 ≤ max cr, these two maps combined deﬁned a natural map chains(chains[n]) × [1] → [n] which induces the required homotopy. 5.21. Theorem (Simplicial approximation for simplicial sets). Let A ⊆ X be ﬁnite simplicial sets and let Y be a simplicial set. Suppose that f0 : A → Y and g : |X| → |Y | are maps such that g||A| = |f0|. k Then there exist for some k a map f : sd sdop sd sdop X → Y such that

k k ϕλϕλ f0 sd sdop sd sdop A A Y

f k sd sdop sd sdop X

|sd sdop i| |ϕλ| g|ϕλ| |sd sdop X| |Y |

Since sd sdop i is a coﬁbration (even the inclusion of a sub-CW complex) and |ϕλ| is a homotopy equivalence by Lemma 5.20 and Proposition 4.19, there exists a map

g : |sd sdop X| → |sd sdop Y | such that g ◦ |sd sdop i| = |sd sdop f0| and |ϕλ| ◦ g ' g|ϕλ| relative |sd sdop A|. Using Propositions 5.17 and 5.19, we can consider the maps

B sdop sd sdop f0 B sdop sd sdop A B sdop Y

B sdop sd sdop i

B sdop sd sdop X and |ϕγ| g π ge: |B sdop sd sdop X| −−→|sd sdop X| −→|sd sdop Y | −→|B sdop Y |. 38

Proposition 5.10 applies to these to yield a commutative diagram

n λn B sdop sd sdop f0 sd B sdop sd sdop A B sdop sd sdop A B sdop Y

f n sd B sdop sd sdop X

k k ϕλϕλ f0 sd sdop sd sdop A A Y

f 0 k sd sdop sd sdop X Compose this approximation with the canonical map ρ∞ : K → Ex∞ Y . Since A → X is a cofibration and Ex∞ Y is Kan, there exists by Lemma 5.20 a map f : sdk+3 X → Ex∞ Y ∞ k+3 ∼ k ρ ◦f ∞ such that f is simplicially homotopic to sd X = sd sdop sd sdop X −−−→ Ex Y ∞ k+3 and extends ρ f0λ . As X is assumed finite, sdk+3 X is also finite, so the approximation factors through some Exn Y . Adjoining to a map sdk+n+3 X → Y yields the desired approximation. References [GJ99] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. BirkhäuserVerlag, Basel, 1999. [GZ67] P. Gabriel and M Zisman. Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc., New York, 1967. [Jar04] J. F. Jardine. Simplicial approximation. Theory Appl. Categ., 12:No. 2, 34–72, 2004. [Lan] Markus Land. Introduction to infinity-categories. Lecture notes. [WJR13] Friedhelm Waldhausen, Bjørn Jahren, and John Rognes. Spaces of PL manifolds and categories of simple maps, volume 186 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2013.