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The homotopy category is a homotopy category. By Arne Str¢m In [4] Quillen defines the concept of a category of (a model category for short). models for a homotopy theory A model category is a category K together with three distinguished classes of morphisms in C ("cofibrations"), and K: F ("fibrations"), W ("weak equivalences"). These MO-M 5 of (4]. classes are required to satisfy axioms A closed model category is a model category satisfying the extra axiom M 6 (see [4] for the statement of the axioms To each model category HoK M0- M6 ). K one can associate a category called the homotopy category of K. is obtained by turning the morphisms in Essentially, W into HoK isorn~rphisrns. It is shown in [4] that the category of topological spaces is a closed model category if one puts fibrations} and F = {Serre W = {weak homotopy equivalences}, and takes C to be the class of all maps having a certain lifting property. From an aesthetical point of view, however, it would be nicer to work with ordinary (Hurewicz) fibrations, cofibrations and homotopy equivalences. The corresponding homotopy category - 2 - would then be the ordinary homotopy category of topological spaces, i.e. the objects would be all topological spaces and the morphisms would be all homotopy classes of continuous maps. In the first section of this paper we prove that this is indeed feasible, and in the last section we consider the case of spaces with base points. \ - 3 - 1. The model category structure of To2. Let Top be the category of topological spaces and continuous maps. By fibrations (cofibrations) we shall mean maps having the homotopy lifting (extension) property with respect to all spaces. F = {fibrations}, Let C = {closed cofibrations}, and W = {homotopy equivalences}. If i: A + X and shall say that i respect to and that (RLP) p, p: E B are morphisms in + has the left lifting property with respect to p Top, we (LLP) with has the right lifting property i, if every commutative square of the form X in Top admits a diagonal Proposition 1. F, C and W. (a) (b) (c) (d) ----> X + B E. The following relations hold between p E F <=> p has the RLP with respect to all i E Q flW. i E C <=> i has the LLP with respect to all p E F n W. pE FnW <=> p has the RLP with respect to all i€ C. 1€ C ():W <=> i has the LLP with respect to all p E F. Proof. (a) follows from [61, Theorem 8, and the definition of fibrations. =>in (b) follows from (6], Theorem 9. To prove <= we first note that if i has the LLP with respect to all pEFnW, then i is a cofibration. It remains to show that i is closed. We may - 4 assume that i is an inclusion, =A E and define I X p: E ~ u X X by X (0 ,1) i: Ac X. c X X I' p(x,t) = x. p Let is a homotopy equivalence, and it is also a fibration, for given and G: Y lifting I x X with + G: Y x I + E g: Y ~ E G = pg, we can construct a suitable 0 by letting G(y,t) = (G(y,t), t+ (1-t)prig(y)), where pri: E + I is the projection map. (This construction works for any inclusion map.) Hence has the i j(a) = (a,O). by j : A + E LLP there is a map f: X A = f- 1 pr! 1 (O) is closed in + E and i' Then and consequently pj = lxi, f =pi= p'i', Now define It follows that j • x. [6], Theorems 8 and 9, and (b). Every continuous map Proposition 2. i p. extending (c) and (d) follow from factored as with respect to where p f: X and are closed cofibrations, and p' i ~ Y can be are fibrations, and p' are homotopy equivalences. f = pi: Proof. 5.27) that f It is well knO\'ln (see for instance r1), can be factored f = nj, X as a strong deformation retract of fibration. = (x,O), retract of W, and j: XC W imbeds n: W + Y is a As in the proof of Proposition 1 (b) let E = X x I u W x ( 0, 1) i(x) where E and define 7r'(w,t) = w. and Then i (X) = pri-1 ( 0 ) • i: X + E, TI': E ~ W by i(X) is a strong deformation It follows that 1 is a - 6 - MO and M5 are obviously true and above, \'Thile 2. M6 M2 is just Proposition 2 follows from Proposition 1, r112 and M5. Some lemmas. The following lemmas will be useful in the next section. Lemma 4. If i: A c X is a cofibration and i#: Aye XY compact space, then the map Y is a induced by i is also a cof'ibration (with respect to the compact-open topology). Proof: functions H, ~ are as in (6], Lemma 4, then corresponding If H: xY X H(f,t)(y) l(f) I + xY xY ¢: and I + are given by = H(f(y),t), = sup ¢f(y). y€ y Lemma 5. i and ij Proof. If j: B +A i: A+ X are maps such that are cofibrations, then We can assume that There exists a halo retraction and r: U + U around A. Since is also a cofibration. j i A in and j are inclusion maps. X together with a U is also a halo around B c U is a cofibration ([2], Satz 2, Korollar). Now consider a commutative diagram (1) _.....;;..f_> y B in X, - 7 where ~ 0 (w) = w(O). The diagram B (\ (2) u is also commutative, and since G: U admits a diagonal It follows that + Y1 • B c: U is a cofibration, ( 2) GIA is then a diagonal in (1). is a cofibration. j Recall that a well-pointed space is a space * with a base point ~ X together X such that the inclusion maP {*} c X is a closed cofibration. Lemma 6. Consider a pullback diagram f' E' - - - > E lp 1 f B' - - - > B where p is a fibration. pointed and that f and Suppose that p E, B, B' are well- respect base points. Then (with the obvious base point) is also well-pointed. the fiber of Proof. p Consider the sequence j * i f' F C E'-> E is the base point of base point of In particular, is well-pointed. pq.c where E' B'. By [6], E' and F is the fiber over the Theorem 12, i and f'i are 8 - = cofibrations. fibration; ij The pointed Let E is well-pointed, hence, by Lemma 5 above, follows that 3. Since Top * Top All base points will be denoted by Top * are defined exactly as except that all maps and homotopies are required to From now on homotopies, fibrations, etc. will be referred to as free homotopies, fibrations, etc. It is clear that if a map i: A-+ X in Top * cofibration (that is, when considered as a map in is a cofibration in Top * It be the category of pointed spaces and continuous respect the base points. in is a cofibration. case~ Fibrations and cofibrations in Top, is a co- is a cofibration. base point preserving maps. in j (f'i)j Top * • is a free Top), then it On the other hand, a fibration in is also a free fibration. Just as in the free case one can prove that all cofibrations in Top * are imbeddings. closed, an inclusion only if Also, if the base point of A c X in (X x 0 u A x I) I* xI deformation retract) of Top * X is is a cofibration if and is a retract (and hence a strong X x I/* x I The arguments are similar to those in (A need not be closed). [5] and [6). We shall need the following result, analogous to (6], Lemma 4. Proposition 7. Let i: AC X be an inclusion map in and suppose that there exists a continuous function w: X -+ I ~ Top· , - 9 1jJ -1( 0) such that = {*}. Then i is a cofibration if and only if there exist a continuous function and a homotopy H: X H(x,t) whenever A (f. If such that cj> Proof. Since i with H0 AC = ~X cj> - l (0), and H exist they can be chosen in such a way x EX. Suppose first that ~ X X 0 u A p: X x 0 u A x I p(x,t) I X rel A such that + for all K = { (X' t) and define + Min(t,l)J(x)) > cj>(x). and cj>(x) < ljJ(x) I x cj>: X + K i X is a cofibration. II Let t < l)J(x)} by = (x,Min(t,l)J(x))). is a cofibration, p extends to r: X x I + K. cj> and H can then be defined by cj>(x) H(x,t) = sup tEI (Min(t,l)J(x)) - prir(x,t)) = prxr(x,t). It is clear that Conversely, cp if cj> and and H have the desired properties. H are given, we can define a retraction r: X x I/* x I --> (X x 0 U A x I)/~ x by . (H(x,t) ,0), r(x,t) tljJ(x) =< cj>(x) = (H(x,t),t- cj> (x)/ljJ(x)), tljJ(x) > cj>(x). I - 10 - Top w · Our main interest will be in the full subcategory lie of well-pointed spaces, rather than the whole category Top • It is an easy consequence of the product theorem for cofibrations ( [6] , Theorem 6) Top * that the mapping cylinder in of a map between well-pointed spaces is well-pointed. Consequently, Top w is a cofibration in a map in Top w if and only if it Top * • is a cofibration in Dually, it follows from Lemmas 4 and 6 above that the mapping track of a map in in Topw Top is a fibration in w is well-pointed, and so a map Topw if and only if it is a Top * • fibration in A necessary and sufficient condition for a pointed space to be well-pointed is that there exist and a homotopy and (P,~) P(x,t) =* t > ~(x) P: X x I ([6], a well-pointing couple for Lemma 8. exi~s when If A c. X a ~: X I with X with + Lemma 4). P0 = ~X Let us 0all X. Top is a cofibration in (P,~) a well-pointing couple + X for w there X such that P(Axi)cA. Proofo for X and Let (PX,~X) and A, respectively. (PA'~A) Choose be well-pointing couples ~: X + I and H: X x I satisfying the conditions of Proposition 7 with respect to and define a: X - {!k} +I A c X is a cofibration, with P0 = ~x· Also, ~A by PA a(x) = 1- ~(x)/~X(x). extends to a homotopy P: + ~X' Since X x I can be extended to a continuous + X X - 11 - w: X + I by putting r(x)lj! AH(x,l) + ¢(x)' vHx) = cp(x) = l/Jx(x). (!x(x), (Recall that H(x,l) = ¢(x) < wx(x).) For, {*}. l/JAH(x,l) = ¢(x) = we should have =* A when rl co) We then have H(x,l) € and X E. A. However, o. if X '¢ Wx(x) = 0, '¢ :k and w(x) = o, But this would imply it is clear that for all and therefore X x' E. A, contradicting the assumption that *• The required couple (P,$) t/$(x)), P(x,t) t < l/J(x), = Px(P(x,l) ,t- l[i(x)), l/J(x) is now given by = t ~ iV(x), Min ( 1 , l/J ( x) + wxP ( x, 1) ) • We shall use this lemma to prove Proposition 9. A map i: A + X in Top w is a cofibration if and only if it is a free cofibration. Proof. Only "only if" needs proof. Suppose, then, that - 12 - AC.X i: that i Topw• is a cofibration in is an inclusion.) Let (For simplicity we assume (P,w) as described in Lemma 8, and then let be a well-pointing couple ~ and H be as in Pro- position 7. Define H': X xI+ X and P ( H (X, t ) , H'(x,t) ~ H' 9( X) and X+ I ~': Min [ t , ~ (X) / W(X)] ) , by X -# *, X : *, = = ~ (X ) - ~~ l/J ( X) + Sup WH ( X, t ) • t 4L I then satisfy the conditions of it follows that i r6J, Lemma 4, and is a free cofibration. The dual statement is also true: Proposition 10. A map p: E + B in Top w is a fibration if and only if it is a free fibration. Proof. It follows from [5] , Theorem 4 that if p is a free fibration, then it has the pointed homotopy lifting property with respect to all well-pointed spaces. It is also true that a map in Topw is a homotopy equivalence if and only if it is a free homotopy equivalence ([1], 2.18) - 13 - Theorem 11. The category Topw , with the classes of (pointed) fibrations, closed cofibrations, and homotppy equivalences, satisfies the axioms This follows from Theorem 3 and Propositions 9 and Proof. 10. Ml - M6. It is not hard to show that the constructions in the proof of Proposition 2, when performed on well-pointed spaces, yield well-pointed spaces. One could hardly expect MO to hold in Top w, but does have sums, finite products, pullbacks of fibrations, pushouts of cofibrations, smash products, suspensions, loop spaces, etc., and this goes a long way. A simple consequence of the product theorem for cofibrations and Proposition 9 is the following "smash product theorem". Proposition 12. in Top w If AC X Bc Y and are cofibrations and at least one of them is closed, then XI\ B UA/\. YC XI\ Y is also a cofibration in Topw• (X 1\ B x/\Y.) be given the subspace topology induced by Remark. Results analogous to 10, 11 in the lar~ category for which Top 0 A 1\ Y should here and and 12 above hold consisting of all pointed spaces there exist functions ~: X ~ I with but the proofs become a bit more complicated. ~ -1( 0) X = { *} , - 14 - References. l. T.Tom Dieck, K. H. Kamps, D. Puppe, Homotopietheorie, Lecture Notes in Mathematics 157, Berlin-Heidelberg-New York, 1970. 2. A. Dold, Die Homotopieerweiterungseigenschaft (= HEP) is eine lokale Eigenschaft, Invent. Math. ~ (1968), .185-189. 3. I. M. Hall, The generalized Whitney sum, Quart.J. Math. Oxford Ser. (2) 16 (1965),360-384. 4. D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Berlin-HeidelbergNew York, 1967. 5. A. Str¢m, Note on cofibrations, Math.Scand. 19 (1966), 11·14. 6. A. Str¢m, Note on cofibrations II, Math. Scand. 22 (l9v8), 130-142.