Homotopy in Model Categories

Home , Homotopy category, Model category

Daniel Fuentes-Keuthan

1 Left Homotopy

The homotopy category of a model category is defined as the localization at the class of weak equivalences. In the case of topological spaces (and more gen- erally as we will see much later), one can form a much more naive construction by taking morphisms instead to be homotopy classes of maps between spaces. The cellular approximation theorem, along with Whitehead’s theorem, imply that one can view the homotopy category as the full subcategory of this naive homotopy category restricted to the CW (cofibrant) spaces. In this section we discuss homotopy in model categories and prove a more general version of this result. Throughout we let M be a model category, with categories of cofibrant, fibrant, and cofibrant-fibrant objects Mc, Mf , and Mcf = Mfc. To give a homotopy between continuous maps f, g : A → X amounts to giving a continuous map from the cylinder A ∧ I → X which restricts to f and g on opposite ends of the cylinder. The cylinder itself is a factorization of the codiagonal of A, A q A,→ A ∧ I −→∼ A, and the homotopy is a map which makes the following commute

A A ∧ I A

H f g X We generalize this notion with the following definitions Definition 1.1. A cylinder object for A is a factorization of the codiagonal q A q A −→i Cyl(A) −→ A. We call the cylinder object good if i is a cofibration, ∼ and very good if in addition q is a fibration. Definition 1.2. Two maps f, g : A → X are left homotopic if there is a cylinder object A ∧ I and a morphism H : A ∧ I → X making the following commute.

A A ∧ I A

H f g X

The left homotopy is good (very good) precisely when the cyclinder object is.

1 Remark 1.3. Our notion of homotopy here requires the choosing of a cylinder object in M, whereas in the category of topological spaces we had a natural choice of cylinder object. We also do not require our cylinder objects to be very good, whereas the natural cylinder object in topological spaces always has this property. However this is not as big of a problem as it seems, as we can often upgrade cylinder objects. Lemma 1.4. Every left homotopy can be upgraded to a good homotopy. If in addition X is fibrant, then every homotopy can be upgraded to a very good homotopy. Proof. For the first part, we factor the map A → A ∧ I further into A,→ σ A ∧ I −−− A ∧ I and take Hσ to be our new homotopy. For the second, ∼ we choose a good cyclinder object as we have just done, then use our other ∼ 0 factorization to factor A ∧ I into A ∧ I ,−→ A ∧ I A. Because X is fibrant, the diagram

A ∧ I H X H0 ∼ A ∧ I0 ∗ Has a lift which deﬁnes a very good homotopy, as desired.

Since we hope to form equivalence classes with respect to homotopy, we must check that this relation deﬁnes an equivalence relation. This is not the case in general, but nevertheless.

Lemma 1.5. When A is cofibrant, left homotopy is an equivalence relation on M(A, X). Proof. Left homotopy is clearly symmetric, and the constant homotopy f q f : A q A → X gives a left homotopy between f and f, where A q A acts as the cylinder object. Hence we must prove transitivity. Let f ∼ g and g ∼ h using 0 00 0 good cylinder objects A∧I and A∧I . Then the pushout A∧I = A∧I qA A∧I defines a good cylinder object and a homotopy f ∼ h. Definition 1.6. The set of left homotopy equivalence classes of M(A, X) will be denoted πl(A, X). Lemma 1.7. If A is cofibrant and A∧I is a cylinder object for A, the morphisms i0, i1 : A → A ∧ I are cofibrations Proof. The two maps are weak equivalences by the two out of three property. If A is cofibrant, i0, i1 are compositions of cofibrations, hence are cofibrations, as seen in the diagram below where the square is a pushout

2 ∅ A i1

A A q A A ∧ I

While our axioms for a model category do not require unique lifts, the following lemma presents a whitehead theorem which states that cofibration/trivial- fibration lifting diagrams have a unique lift up to homotopy when the cofibration is a map ∅ ,→ A from the initial object of M, ie when A is cofibrant. In particular, given A in M, there is a model structure on the slice category A ↓ M where the cofibrant objects are precisely the cofibrations out of A. The following lemma implies then that lifts in cofibration/trivial-fibration lifting diagrams have a unique lift up to left homotopy over A. ∼ Lemma 1.8. Let A be cofibrant. Any trivial fibration p: X −−− Y induces ∼ a bijection on the set of homotopy equivalence classes of maps πl(A, Y ) −→= πl(A, X). Proof. The first thing to note is that this map is well-defined as any homotopy H : f 7→ g gives a homotopy pH : pf 7→ pg. Now given any f ∈ M(A, X), there is a lift ∅ Y f 0 p A X f hence pf 0 = f, and in particular [pf 0] = [f], proving surjectivity. Now suppose pf ∼ pg so that we have a diagram A A ∧ I A f g Y H p X This gives a square

fqg A q A Y H0 p A ∧ I X H and a lift H0 : A ∧ I → Y which provides a homotopy between f and g. This next lemma will help establish a composition on homotopy classes of maps, which appears as the next proposition 1.10.

3 Lemma 1.9. Suppose X is ﬁbrant, f, g : A → X are left homotopic, and h: A0 → A is any morphism, then fh ∼ gh. Proof. By 1.4, we can chose a very good cylinder object A ∧ I and homotopy H for A, and a good A0 ∧ I for A0. Any lift in the diagram below gives a homotopy fh ∼ gh upon precomposition with H.

A0 q A0 hqh A q A A ∧ I 0 H ∼ A0 ∧ I A0 A ∼ h

Proposition 1.10. If X is fibrant the composition map πl(A, X)×πl(A0,A) → πl(A0,X) given by ([f], [h]) 7→ [hf] is well defined. Proof. First note A is not cofibrant, so we cannot appeal to transitivity of left homotopy. We must check that if h ∼ k : A0 → A and f ∼ g : A → X, then [fh] = [gk]. For this it is enough to check that fh ∼ gh and gh ∼ gk. The first foloows from 1.10, and the second from composing the homotopy giving h ∼ k with g.

2 Right Homotopies and the Relation to Left Ho- motopy

Up until this point we have speciﬁcally made reference to left homotopies. Of course by adjunction we can deﬁne homotopies in topological spaces in terms of path objects. To distinguish between the two notions of homotopy, we will denote them from now on as ∼l and ∼r. All of our results above dualize to this case, and we list them quickly for reference.

Definition 2.1. A path object for X is a factorization of the diagonal X −→i ∼ p XI −→ X × X. We call the path object good if p is a fibration, and very good if in addition i is a cofibration. Definition 2.2. Two maps f, g : A → X are right homotopic if there is a path object XI and a morphism H : A → XI making the following commute.

X XI Xs

H f g A

The right homotopy is good (very good) precisely when the path object is. Lemma 2.3. If X is ﬁbrant and XI is a path object for X, the morphisms I p0, p1 : X → X are trivial ﬁbrations

4 Lemma 2.4. Every right homotopy can be upgraded to a good homotopy. If in addition X is fibrant, then every homotopy can be upgraded to a very good homotopy. Lemma 2.5. If X is fibrant then ∼r is an equivalence relation of M(A, X). We denote the set of equivalence classes of morphisms as πr(A, X). ∼ Lemma 2.6. If X is fibrant, any trivial cofibration i: A ,−→ B induces a bijec- ∼ tion πr(B,X) −→= πr(A, X). Lemma 2.7. Suppose A is cofibrant, f, g : A → X are right homotopic, and h: X → X0 is any morphism, then fh ∼r gh. Lemma 2.8. If A is cofibrant, composition induces a map πr(X,X0)×πr(A, X) → πr(A, X0). By now the topologically minded reader may be somewhat concerned over the a priori different notions of right and left homotopy given above. The reason for this distinction is the number of results that require either the domain or codomain of our morphisms to be either cofibrant or fibrant. In the category of topological spaces all objects are fibrant, and so the two notions of homotopy collapse due to the following lemma. Lemma 2.9. Let f, g : A → X be morphisms. If A is cofibrant and f ∼l g, then f ∼r g. Dually, if X is fibrant, f ∼r g implies f ∼l g.

j Proof. We prove the ﬁrst statement. Fix a good homotopy (AqA −−−→i0qi1 A∧I −→ ∼ p q I A, H), and a good path object X −→ X −−− X × X. By 1.7 i0 is a trivial ∼ coﬁbration, so there is a lift σ in the diagram

qf A XI σ i0 p A ∧ I X × X (fj,H)

We check p0σi1 = fji1 = f and p1σi1 = Hi1 = g, so that σi1 deﬁnes a right homotopy f ∼r g. Deﬁnition 2.10. Given two maps f, g : A → X, we say f ∼ g if both f ∼l g and f ∼r g.

3 The Subcategory of Coﬁbrant-Fibrant Objects

We now restrict down to the category Mcf of cofibrant-fibrant objects. Work- ing in this subcategory provides many benefits, for example by 1.5 and 2.5 we see that homotopy is an equivalence relation on homsets, and by 1.10 and 2.8 it plays nice with composition. In particular there is a well defined notion of homotopy equivalence between two objects.

5 Deﬁnition 3.1. Objects A, X ∈ Mcf are homotopy equivalent if there are morphisms f : A → X and g : X → A so that fg ∼ 1X and gf ∼ 1A. The maps f, g are called homotopy equivalences. As a further example, we give a generalization of Whitehead’s Theorem.

Theorem 3.2 (Whitehead). A morphism f : A → X between coﬁbrant-ﬁbrant objects is a weak equivalence if and only if it is a homotopy equivalence.

q p Proof. First suppose f is a weak equivalence. Factor f as A ,−→ C −−− X. By ∼ the two out of three property, p is also a weak equivalence. Since q is a trivial cofibration and A is fibrant, there is a left inverse rq = 1, r : C → A given as a lifting in A A q,∼ r C ∗ By 1.8, q induces a bijection π(C,C) → π(A, C). Under this map, [qr] 7→ [qrq] 7→ [q], but also [1] 7→ [q], hence [qr] = [1]. This shows q is a homotopy equivalence, and by a dual argument p is as well, hence f is a homotopy equivalence. q p Now suppose f has a homotopy inverse g. Factor f again as A ,−→ C −−− ∼ X. We must show p is a weak equivalence to complete the argument. Let l H : X ∧ I → X be a homotopy fg ∼ 1X . We have a lift qg X C H0 ∼ p X ∧ I H X 0 Let s = H i1 so that ps = 1X . Now q is a weak equivalence, so by the first part of the proof q has a homotopy inverse r. Since pq = f, we have a homotopy p ∼ fr. Since s ∼ qg using H0, we have a chain of homotopies sp ∼ qgp ∼ qgfr ∼ qr ∼ 1C , hence sp is a weak equivalence (being homotopic to a weak equivalence makes you a weak equivalence). Finally, p is a retract of sp as in the diagram below, which implies p is a weak equivalence. C C C p sp p

X s C p X

Before proceeding, we make a note that our development of the proof of Whitehead’s theorem was enitrely formal and devoid of the technical machinery used in the proof of the topological case. However, this does not come for free, as

6 the proof of existence of the model structure on topological spaces reintroduces much of the technical machinery. We close this section by examining the relationship between the naive homotopy category and the homotopy category of M. Homotopy is a well deﬁned equivalence relation on Mcf , so we may form the category Mcf / ∼. Whitehead’s Theorem 3.2 implies that the quotient functor δ : Mcf → Mcf / ∼ sends weak equivalences to isomorphisms, hence factors through the category Ho(Mcf ).

δ Mcf Mcf / ∼

γ σ

Ho(Mcf )

We also have an inclusion morphism Mcf → M which induces a morphism τ Ho(Mcf ) −→ Ho(M). Our main theorem will be that both of these morphisms are isomorphisms, proving the generalization of the classical result mentioned in the begining of this section. τ σ Theorem 3.3. The morphisms Ho(Mcf ) −→ Ho(M) and Ho(Mcf ) −→ Mcf / ∼ ' together give an equivalence Ho(M) −→ Mcf / ∼.

Proof. We first prove that τ is an equivalence. The inclusion i: Mc → M preserves weak euivalences so induces a map Hoi: Ho(Mc) → Ho(M). Let Q: M → Mc be a cofibrant replacement functor for M. Q is a equipped with a natural weak equivalence q : Q ⇒ 1M , hence preserves weak equivlaences by the two out of three property and induces a map HoQ: Ho(M) → Ho(Mc). The transformation q acts as a natural weak equivalence i◦Q ⇒ 1M and Q◦i ⇒ 1Mc , and so is a natural equivalence on the level of homotopy categories. A repetition of this argument using an inclusion Mcf → Mc and a fibrant replacement functor (R, r) finished this half of the proof. To show that σ is an isomorphism, we show that Mcf has the universal property of Ho(Mcf ). Let F : Mcf → D be a functor which takes weak equivalences to isomorphisms. We must show that F takes (left) homotopic maps to the same map. Let f, g : A → X be homotopic morphisms using a cylinder object A ∧ I and homotopy H : A ∧ I → X. The i0qi1 s factorization A q A −−−→ A ∧ I −−− A gives si0 = si1 = 1A, and since ∼ s is a weak equivalence, so are i0, i1. Hence F (i0) = F (i1). Then F (f) = F (H)F (i0) = F (H)F (i1) = F (g), as desired. Finally as a corollary, the above proof shines some light on the structure of the homotopy category.

Corollary 3.4. There are natural isomorphisms

M(QRX, QRY )/ ∼=∼ M(QX, RY )/ ∼=∼ Ho(M)(X,Y ) and the localization M → Ho(M) inverts homotopy equivalences.

7 Proof. The second ismomorphism of the first statement as well as the second statement follow from the proof of 3.3. The first isomorphism of the first statement follows from 1.8 and its dual 2.6.