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Homotopy in Model Categories

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Homotopy in Model Categories Homotopy in Model Categories 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 follow- ing 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 par- ticular, 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 follow- ing 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 2 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.
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