Abstract sectional category

ABSTRACT SECTIONAL CA TEGOR Y F. D ´ IAZ D ´ IAZ, J.M. GARC ´ IA CALCINES, P .R. GARC ´ IA D ´ IAZ, A. MURILLO MAS AND J. REMEDIOS G ´ OMEZ Abstract. W e study , in an abstract ax iomatic setting, the notion o f sectiona l category of a morphism. F rom this, we unify and generalize known results ab out this inv a riant in d ifferent settings as well as we deduce new applica tions. Intr oduction The sectional category secat ( p ) o f a fibration p : E ։ B , originally in tro duced b y A. Sch w arz [20], is defined a s the least in teger n suc h that B a dmits a co v er constituted b y n + 1 op en subsets, on eac h of which p has a lo cal section. It is a lo w er b ound of the Lusternik-Sc hnirelmann category of the ba se space and it is also a generalization of this in v arian t since secat ( p ) = cat ( B ) when E is con tra ctible. Apart fro m the original a pplications o f the sectional category in the classification of bundles or the em b edding problem [20], this nume rical inv ariant has prov ed to b e useful in differen t settings. F or instance, Smale [21] show ed that the sectional cate- gory of a certain fibration pro vides a low er b ound for the complexit y of algo rithms computing the ro ots o f a complex p olynomial. W e can also men tion the work of M. F arb er [8, 9] who introduced the top ological complexity of a giv en space X as the sectional category of the pat h fibration X I → X × X , α 7→ ( α (0) , α (1)) . In rob otics, when X is thoug ht to b e the configuratio n space asso ciated to the motion o f a giv en mec ha nical system, this in v ar ia n t measures, roug hly sp eaking, the minimu m amoun t of instructions of a n y algo rithm con trolling the giv en system. In general, the sectional catego r y of a fibration is hard to compute. The notion of Lusternik-Sc hnirelm ann category (L.-S. category , for short) has the same disad- v an tage. In order to f a ce this problem for L.-S. catego r y t here hav e b een sev eral attempts to describe it in a more functorial and therefore manageable form; among the most success ful ones w e can men t io n the Whitehead and G anea characteriza- tions. Man y other approx imations of L.-S. category ha v e b een in tro duced. One of them relies in an imp ortan t algebraic techn ique for obtaining lo w er bounds. It 2000 Mathematics Su bje ct Classific atio n. 5 5U35, 55M30. Key wor ds and phr ases. Sectional categ ory , mo del category , J -categor y , Lus ternik-Schnirelmann category . Partially suppor ted by FEDER, the Ministerio de Educaci´ on y Ciencia grants MTM2009 -120 8 1, MTM2010- 18089 and b y the J unt a de Andaluc ´ ıa grant F QM-213. 1 2 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS consists of t a king mo dels of spaces in an alg ebraic category where a notion of L.- S.- category t yp e in v a rian t is g iv en. Suc h algebraic category m ust p osses an a bstract notion of ho motop y , usually established in an axiomatic homotopy setting, suc h a s a Quillen mo del category . Then the algebraic L.-S. category o f the mo del of X is a low er b ound o f t he original L.-S. category of X . D ur ing the progress of this tec h- nique, sev eral algebraic notions of L.-S. category hav e b een app earing. In 199 3 , in order to giv e a common p oint for all of them, Do eraene [6] introduced the not ion of L.-S. category in a Quillen mo del category . Actually , in his w ork Do eraene dev elops t w o differen t notions of L.-S. category , which are the a nalogous to the Ganea and Whitehead c haracterizations in the top ological case and pro v es tha t , under the cru- cial cub e axiom, these notio ns agr ee, as exp ected. As far as the sectional categor y is concerned, not m uc h has b een done in this direction. In t he w ork o f A. Sc h w arz [20] it w as established a G anea-t yp e c haracterization of sectional category . Namely , if p : E ։ B is a fibration w e can consider j n : ∗ n B E → B , whic h is the n -th fold join of p. If the base space B is paracompact, then A. Sch w arz prov ed that secat ( p ) ≤ n if and o nly if j n admits a (homotopy ) section. Clapp and Pupp e [5, Cor. 4 .9] also obtained a Whitehead-t yp e c haracterization of sectional catego r y; more precisely , for a giv en map p : E → B with asso ciated cofibratio n ˆ p : E → ˆ B , secat ( p ) ≤ n if and only if the diagonal map ∆ n +1 : ˆ B → ˆ B n +1 factors, up t o homotopy , through the n -t h f a t w edge T n ( ˆ p ) = { ( b 0 , b 1 , ..., b n ) ∈ ˆ B n +1 : x i ∈ E , for some i } . With this c haracterization F ass` o [10 ] studied the sectional category of the corresp onding a l- gebraic mo del of p in rational homotopy . These functor ia l c haracterizations in the top ological case op en a doo r t hrough an axiomatization o f sectional category . In this direction an initial adv ance has b een made b y T. Kahl in [15]. In his work he giv es the notion of abstract sectional category thr o ugh a certain v ariation of inductiv e L.-S. category in the sense of Hess-Lemaire [12]. Our aim in this pap er is to dev elop, in the same spirit as Do eraene did in [6] with the L.-S. catego r y , the notion of sectional category in an abstract homotop y setting and to deduce some applications. In the first section w e recall some background to set the axiomatic framew ork in whic h we shall w ork as w ell as the main to ols that will b e used. In § 2 w e in tro duce, under t w o differen t approac hes, the concept of sectional category of a giv en mo r phism. Then, in § 3 w e pr esen t the main prop erties of this inv a r ia n t a nd finally , in the fourth section, w e give some applications. 1. Preliminaries: J -ca tegor y and main notions. In this pap er we shall w ork in a J - category [6], whic h includes the cases of a p oin ted cofibration and fibration category in the sense of Baues [2] or a p ointe d prop er mo del category [1 8, 19] as long as they satisfy the “ cub e lemma”. The aim of this section is to pro vide some of the most imp o rtan t notions and pro p erties give n in suc h a homotopy setting. F or pro ofs a nd more details the reader is referred to Do eraene’s pap er [6] or his thesis [7 ]. ABSTRACT SECTIONAL CA TEGOR Y 3 Explicitly , a J - category C is a category with a zero ob ject 0 a nd endow ed with three classes o f mor phisms called fibrations ( ։ ), cofibra t ions ( ֌ ) and weak equi- v alences ( ∼ → ), satisfying the following set of axioms (J1)-(J5) b elo w. Recall that a morphism whic h is b o t h a fibra tion (resp. cofibration) and a w eak equiv alence is called trivial fibr ation (resp. trivial c ofibr ation ). An ob ject B is called c ofibr ant mo del if ev ery trivial fibrat ion p : E ∼ ։ B admits a section. (J1) Isomorphisms are trivial cofibrations and also trivial fibrations. Fibrations and cofibrations a r e closed b y comp osition. If an y tw o of f , g , g f are we ak equiv alences, then so is the third. (J2) The pullbac k of a fibration p : E ։ B and a n y morphism f : B ′ → B E ′ p     f / / E p     B ′ f / / B alw a y s exists and p is a fibration. Moreo v er, if f (resp ec. p ) is a w eak equi- v alence, then so is f (respec. p ). The dual assertion is also required. (J3) F or any map f : X → Y there exist an F -factorization (i.e., f = pτ where τ is a w eak equiv alence and p is a fibration) a nd a C -factorization (i.e., f = σ i, where i is a cofibration a nd σ is a w eak equiv alence). (J4) F or an y ob ject X in C , there exists a t r ivial fibration p X : X ∼ ։ X, in whic h X is a cofibrant mo del. The morphism p X : X ∼ ։ X is called c ofi br ant r eplac ement for X . A comm utativ e square D g ′   f ′ / / C g   A f / / B is said to b e a homotopy pul lb ack if for some (equiv alently any ) F - f actorization o f g ( equiv alently f or b oth), the induced map from D to the pullbac k E ′ = A × B E is a we ak equiv alence D g ′   f ′ / / # # C g   τ ∼ | | ② ② ② ② ② ② E ′ p | | | | ① ① ① ① ① ① f / / E p " " " " ❊ ❊ ❊ ❊ ❊ ❊ A f / / B The not ion of homotopy pushout is dually defined. 4 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS (J5) The cub e axio m . Give n any comm utativ e cub e where the b ott om fa ce is a homotop y pushout and the v ertical faces a r e homotop y pullbac ks , then the top face is a homot op y pushout. Remark 1. As p ointe d out by Do er aene, (J1)- ( J4 ) axioms al low us to r eplac e ’some’ by ’any’ in the definition of homotopy pul l b ack, or to use an F -factoriza tion of f inste ad of g . W e are particularly in terested in kno wledge of o b jects and morphisms up to we ak e quivalenc e . Tw o ob jects A and A ′ in C are said to b e w e akly e quival e n t if there exists a finite c hain of w eak equiv alences joining A and A ′ A ∼ • ∼ • · · · · · · • ∼ A ′ where the sym bo l • • m eans an a rro w with either left or rig h t orientation. One can analogously define the notion of we akly e quivalent morph i s ms by considering a finite c hain of w eak equiv alences in the category P air( C ) of mo r phisms in C ([2] Def. I I.1.3) A f   ∼ •   ∼ •   ∼ •   ∼ •   ∼ A ′ f ′   B ∼ • ∼ • ∼ • ∼ • ∼ B ′ Definition 2. Giv en t w o morphisms f : A → B and g : C → B , consider an y F - factorization of g = pτ and the pullbac k of f and p. Let f and p the base extensions of f a nd p r esp ective ly . Then, tak e any C -fa ctorization of f = σ i and the pushout of p and i. This pushout o b ject is denoted b y A ∗ B C and is called the join of A a nd C o v er B . The dotted induced map from A ∗ B C to B is called the joi n morphism of f and g . E ′ f / / % % i % % ❏ ❏ ❏ ❏ ❏ ❏ ❏ p     E p     C τ ∼ o o g   ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Z σ ∼ : : t t t t t t t   A ∗ B C $ $ A : : : : t t t t t t f / / B The ob ject A ∗ B C and the join map a re w ell defined and they are symmetrical up to weak equiv alence [6, 7 ]. An import an t result that allows us to see that if a prop erty holds fo r some F - factorization, then it also holds for an y F -factorization is t he follo wing lemma. Recall from [2] that in a fibrat io n category a r elative c o cylinder of a fibration p : ABSTRACT SECTIONAL CA TEGOR Y 5 E ։ B is just an F - factorization of the morphism ( id E , id E ) : E → E × B E , where E × B E denotes the pullbac k of p with itself E ( id E ,id E ) / / ∼   ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ E × B E Z p ( d 0 ,d 1 ) ; ; ; ; ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ Then, giv en f , g : X → E such that pf = pg , it is said that f is homotopic to g r elative to p ( f ≃ g rel. p ) if there exists a morphism F : X → Z p suc h t hat d 0 F = f and d 1 F = g . When p = 0 : E ։ 0 is the zero morphism w e obtain the notion o f non relativ e homotopy (a nd write f ≃ g ). In this case, the co cylinder of 0 : E ։ 0 will b e denoted b y E I . Lemma 3. [2, I I.1.11] Consider a comm utativ e diagram of un brok en arrows: D ∼ τ   g / / E p     A f / / l > > B (a) If A is a cofibran t mo del, then there is a morphism l : A → E suc h that pl = f . (b) If A a nd D are cofibrant mo dels, then there is a morphism l : A → E for whic h pl = f and l τ ≃ g rel. p. Moreo v er, if g is a w eak equiv alence, then so is l. W e also recall the notion o f w eak lifting. Definition 4. Let f : A → B and g : C → B b e morphisms in C . W e sa y tha t f admits a we ak lifting along g if for some F - factorization g = pτ of g and for some cofibran t r eplacemen t p A : A ∼ → A of A there exists a comm utativ e diagram C g   τ ∼ ~ ~ ⑦ ⑦ ⑦ ⑦ E p     ❄ ❄ ❄ ❄ A s ? ? f p A / / B In the particular case f = id B w e sa y that g : C → B admits a we ak se ction . This notion do es not dep end on the c hoice of the F - factorization nor on t he cofibran t replacemen t. In order to c hec k this fact one has to use Lemma 3 ab ov e and the following result. The details are left to the reader. 6 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS Lemma 5. [2, I I.1.6] Let p : X ∼ ։ Y b e a trivial fibration a nd f : A → Y a n y morphism, with A a cofibrant mo del. Then there exists a lift of f with resp ect to p, i.e. a morphism ˜ f : A → X suc h that p ˜ f = f X p ∼     A f / / ˜ f > > Y Another importa n t notion that will b e used in this paper is the one of weak pullbac k. Definition 6. Let f : A → B , f ′ : A ′ → B ′ and b : B → B ′ b e morphisms in C . It is said tha t A - A ′ - B ′ - B is a we ak pul lb ack if for some F -factorization f ′ = pτ and some cofibra n t replacemen t p A : A ∼ ։ A of A there exists a homotopy pullbac k A f p A h.p.b.   x / / X p ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ A ′ f ′   τ ∼ o o B b / / B ′ Remark 7. A ny homotopy pul lb ack i s a we ak pul lb ack. A gain, L emma 3, axio m (J4) and L emma 5 al low us to r eplac e the wor d ’some’ by ’any’ in the ab ove definition. We also have to take into ac c ount that the c omp osition of h omotopy p ul lb acks is a homotopy pul lb ack (in f a ct ther e is a Prism L emma for ho motopy pul lb acks [6, Prop. 1.1] ) and that the we ak e quivalen c es in the c ate g o ry Pair ( C ) of morphisms in C ar e homotopy pul lb a cks. 2. Sectional ca tegor y. Gane a and Whitehe ad app ro aches. As in Do eraene’s w ork, from no w on w e will assume that C is a J -c ate gory in which al l obje cts ar e c ofi b r ant mo d els . Therefore w e will tak e as cofibran t replacemen ts the corresp onding iden tities. It is imp ortan t to remark tha t in a general J -category we will also obtain the same results. Ho wev er, the exp osition and/or the arguments in this general case w ould b e affected by unessen tial tec hnical complications. So just for the sak e of simplicit y and comfort we a dmit this assumption without lost of generalit y . Essen tia lly , the k ey p oint for the pass from our assumption to the general case is established by considering cofibrant replacemen ts: • A ny obje ct X in C has a c ofibr ant r eplac em ent, that is, a trivial fibr a tion p X : X ∼ ։ X , in which X is a c ofibr ant mo del. ((J4) axiom ) • Any morp h ism f : X → Y in C has a c ofibr ant r eplac ement, that is, given c ofibr ant r eplac ements p X , p Y of X and Y , ther e exists an ind uc e d morphism ABSTRACT SECTIONAL CA TEGOR Y 7 f : X → Y making c ommutative the fo l lowing squar e X ∼ p X     f / / Y ∼ p Y     X f / / Y Observ e that the second item holds thanks to Lemma 5. Using these simple f a cts when necessary and w orking a little bit harder t he reader should b e a ble to prov e our results when not a ll o b jects are cofibrant mo dels. W e a re now prepared for the definition of sectional catego r y of a mo r phism in C under tw o differen t approac hes. In the follo wing definition, only axioms (J1)-(J4) are needed. Definition 8. Let p : E → B b e a ny morphism in C (not necessarily a fibration). W e consider fo r eac h n a morphism h n : ∗ n B E → B inductiv ely as follo ws: (1) h 0 = p : E → B (so ∗ 0 B E = E ) (2) Assume t hat h n − 1 : ∗ n − 1 B E → B is already constructed. Then h n is the join morphism of p and h n − 1 : • / / & & & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲     E ′     E ∼ o o p   ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ • ∼ ; ; ✈ ✈ ✈ ✈ ✈ ✈   ∗ n B E h n ! ! ∗ n − 1 B E : : : : ✉ ✉ ✉ ✉ ✉ h n − 1 / / B Then, the Gane a se ction a l c ate g ory of p , Gsecat( p ), is the least in teger n ≤ ∞ suc h that h n admits a weak section ∗ n B E h n   ∼ | | ② ② ② ② • # # # # ❋ ❋ ❋ ❋ ❋ B > > id B ∼ / / / / B Remark 9. Observe that Gse c at ( p ) = 0 if and only if p has a we ak se ction. Mor e- over, in the top olo gic al setting this invariant c oi n cides w ith s e c at ( p ) , the classic al se ctional c ate gory of a given fi b r ation p : E ։ B , w i th B p ar ac omp act. In fact, the n -th iter a te d join of p over B , h n : ∗ n B E → B has a homotopy se ction if and o nly if B c an b e c over e d by n + 1 o p en subsets, e ach of them having a lo c al homotopy se ction [14, 20] . 8 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS No w we show that this is an in v arian t up to weak equiv alence. Prop osition 10. If p : E → B and p ′ : E ′ → B ′ are w eakly equiv alent morphisms, then Gsecat( p ) = G secat( p ′ ) . F or the pro of w e shall use the following result. Lemma 11. [6, Lemma 3.5] Consider the f ollo wing comm uta tiv e diagram in C A x   f / / B b   C g o o y   • ' ' ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ • w w w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ A ′ ∼ O O f ′ / / B ′ C ′ g ′ o o ∼ O O That is, bf admits a w eak lifting along f ′ and bg admits a w eak lifting a long g ′ . Let j : A ∗ B C → B and j ′ : A ′ ∗ B ′ C ′ → B ′ denote the corresp onding join maps. Then bj admits a w eak lifting along j ′ A ∗ B C j   / / • # # # # ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ A ′ ∗ B ′ C ′ j ′   ∼ o o B b / / B ′ F urthermore, if b, x a nd y are weak equiv alences, then A ∗ B C is w eakly equiv alent to A ′ ∗ B ′ C ′ via the ab o v e diagram. Pr o o f of Pr op osition 10. W e can supp ose without losing generalit y that there is a comm utativ e diagram of the follo wing form E p   u ∼ / / E ′ p ′   B ∼ v / / B ′ Let us see by induction on n t ha t h n : ∗ n B E → B and h ′ n : ∗ n B ′ E ′ → B ′ are w eakly equiv alen t morphisms. Indeed, f o r n = 0 it is certainly true. No w supp ose that h n − 1 and h ′ n − 1 are w eakly equiv alent. Again w e can assume, without losing generalit y , that there is a commutativ e square ∗ n − 1 B E h n − 1   w ∼ / / ∗ n − 1 B ′ E ′ h ′ n − 1   B ∼ v / / B ′ ABSTRACT SECTIONAL CA TEGOR Y 9 No w tak e h ′ n − 1 = q λ and p ′ = r µ F -factorizations. Then w e hav e a comm utativ e diagram ∗ n − 1 B E λw ∼   h n − 1 / / B v ∼   E p o o µu ∼   • q ' ' ' ' P P P P P P P P P P P P P P P P • r x x x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ∗ n − 1 B ′ E ′ ∼ λ O O h ′ n − 1 / / B ′ E ′ p ′ o o ∼ µ O O whic h, applying Lemma 11, giv es rise to this one ∗ n B E h n   ∼ / / • ! ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ∗ n B ′ E ′ h ′ n   ∼ o o B v ∼ / / B ′ No w w e ha v e t hat h n admits a w eak section if and only if h ′ n admits a w eak section. In order to che c k this assertion, one has just to tak e into accoun t Lemma 3 and the fact t hat the pullbac k of • ։ B ′ and v : B ∼ → B ′ giv es rise to an F -factorizatio n of h n in a na t ural w a y .  No w we give a Whitehead-ty p e definition o f sectional category . Definition 12. Let p : E → B b e an y morphism in C where B is e -fibran t, that is, the zero morphism B → 0 is a fibration. W e define j n : T n ( p ) → B n +1 inductiv ely as fo llows: (1) j 0 = p : E → B ( so T 0 ( p ) = E ) (2) If j n − 1 : T n − 1 ( p ) → B n is constructed, then j n is the following join construc- tion: • / / ( ( ( ( ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘     •     B n × E ∼ o o id B n × p   ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ • ∼ 8 8 r r r r r r r r r   T n ( p ) j n $ $ T n − 1 ( p ) × B 7 7 7 7 ♦ ♦ ♦ ♦ ♦ ♦ j n − 1 × id B / / B n +1 Then the Whitehe ad se ctional c ate gory of p , Wsecat ( p ) , is the least in teger n ≤ ∞ suc h that the diagonal morphism ∆ n +1 : B → B n +1 admits a w eak section along 10 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS j n : T n ( p ) → B n +1 : T n ( p ) j n   ∼ { { ✇ ✇ ✇ ✇ ✇ • $ $ $ $ ❍ ❍ ❍ ❍ B ? ? ∆ n +1 / / B n +1 Observ e tha t , in or der to define Wsecat ( p ) , w e hav e to consider B a n e-fibrant ob ject to ensure that all pro ducts B n , T n ( p ) × B and B n × E exist ( n ≥ 0). Now w e extend Wsecat( p ) to the general case, in whic h B need not b e e-fibrant. F or it consider an F -fa ctorization B ∼ τ / / F / / / / 0 of the zero morphism. T hen w e define Wsecat( p ) := Wsecat( τ p ) Lemma 13. If p : E → B is an y morphism, then Wsecat( p ) do es not dep end on the c hoice of the F -fa ctorization for B → 0 . Pr o o f . Consider B ∼ τ / / F / / / / 0 and B ∼ τ ′ / / F ′ / / / / 0 tw o suc h F -factorizations. Then, by L emma 3(b) a pplied to the f o llo wing commutativ e diagram B τ ∼ / / τ ′ ∼   F     F ′ ∼ h > > / / / / 0 there exists a w eak equiv alence h : F ′ ∼ → F suc h that hτ ′ ≃ τ . T ak e a homotop y H : B → F I v erifying that d 0 H = hτ ′ and d 1 H = τ a nd consider the comm utativ e diagram, where the co domain of eac h v ertical arrow is an e-fibran t ob ject E τ ′ p   E hτ ′ p   E H p   E τ p   F ′ ∼ h / / F F I ∼ d 0 o o ∼ d 1 / / F This diagram sho ws that τ p and τ ′ p are w eakly equiv alen t morphisms. Observ e that, since F × F is e-fibrant and b y definition there is a fibra t io n ( d 0 , d 1 ) : F I ։ F × F , we hav e that the co cylinder ob ject F I is also e-fibrant. Finally , considering a similar a rgumen t to that g iven in the pro of of Prop osition 10 we obtain the iden tit y Wsecat( τ p ) = Wsecat( τ ′ p ) .  Prop osition 14. If p : E → B and p ′ : E ′ → B ′ are w eakly equiv alent morphisms, then Wsecat( p ) = Wsecat( p ′ ) . ABSTRACT SECTIONAL CA TEGOR Y 11 Pr o o f . W e can suppose, without losing generalit y , that there is a comm utat iv e square E p   u ∼ / / E ′ p ′   B ∼ v / / B ′ No w, if B ′ ∼ τ ′ / / F ′ / / / / 0 is a n F - factorization of the zero morphism, then an F -factorization B ∼ τ / / F w / / / / F ′ of τ ′ v giv es rise to B ∼ τ / / F / / / / 0 , another F -factorization, and a commutativ e square E τ p   u ∼ / / E ′ τ ′ p ′   F ∼ w / / / / F ′ Again, the result follow s considering a similar argument to that giv en in the pro o f of Prop osition 10.  W e no w see that Gsecat and Wsecat coincide in a J -category . Theorem 15. If p : E → B is an y morphism, then Gsecat( p ) = Wsecat( p ) . F or it w e recall some useful prop erties ab out w eak pullbac ks. Again we refer the reader to [6 ]. Lemma 16 ( Prism Lemma for weak pullbac ks ) . [6 , Prop. 2 .5] Consider the follo wing diagra m A   B   C   X / / Y / / Z If B - C - Z - Y is a w eak pullbac k, then A - B - Y - X is a weak pullbac k if and only if A - C - Z - X is a we ak pullbac k. Lemma 17. [6, Lemma 3.5] Consider a w eak pullback D g h.p.b.   / / •     ❄ ❄ ❄ ❄ ❄ ❄ ❄ C g ′   ∼ o o A f / / B and let h : X → A b e an y morphism. Then h admits a w eak lifting along g if and only if f h admits a w eak lifting along g ′ . 12 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS And no w t he Join Theorem. This result strongly relies on the cube axiom (J5 axiom) a nd therefore it do es not admit a dual v ersion. Lemma 18 ( Join Theorem ) . [6, Th. 2.7] Consider the w eak pullbac ks A f h.p.b.   / / X p ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ A ′   ∼ o o C g h.p.b.   / / Y q ❆ ❆ ❆ ❆ ❆ ❆ ❆ C ′   ∼ o o B b / / B ′ B b / / B ′ Then there is a we ak pullbac k A ∗ B C h.p.b.   / / • # # # # ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ A ′ ∗ B ′ C ′   ∼ o o B b / / B ′ Pr o o f of The or e m 15. F irst supp ose that B is e-fibrant. W e will see by induction on n ≥ 0 that for any map p : E → B , there is w eak pullback: ∗ n B E h n h.p.b.   / / • ! ! ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ T n ( p ) j n   ∼ o o B ∆ n +1 / / B n +1 F or n = 0 it is trivially t rue. Supp ose the statemen t true for n − 1 and consider the diagram ∗ n − 1 B E h n − 1   1  T n − 1 ( p ) × B pr / / j n − 1 × id B   T n − 1 ( p ) j n − 1   B ∆ n +1 / / B n × B pr / / / / B n where the right square is a pullbac k in whic h pr : B n × B ։ B n is a fibration (observ e t hat B is e-fibran t and use (J2) axiom). Therefore this pullbac k is also a homotop y pullbac k and a weak pullbac k. No w, applying the Prism Lemma together with the induction hypothesis w e deduce that diagra m 1  is also a w eak pullbac k. The same argumen t applied to the diag r a m E p   ( p,p,.. .,p,id E ) / / 2  B n × E id B n × p   pr / / / / E p   B ∆ n +1 / / B n × B pr / / / / B ABSTRACT SECTIONAL CA TEGOR Y 13 implies tha t 2  is a weak pullbac k. W e obtain the exp ected result by applying the Join Theorem to the w eak pullbac ks 1  and 2  . The theorem easily follows no w from this fa ct together with Lemma 17. When B is not e-fibrant, consider B ∼ τ / / F / / / / 0 an F - f actorization. Then w e ha v e that Gsecat( p ) = Gsecat( τ p ) b y Prop osition 10. But we hav e a lready pro v ed that Gsecat( τ p ) = Wsecat( τ p ) (=Wsecat( p )).  Remark 19. Whe n our c ate gory C do e s not satisfy the cub e axiom (J5), the most we c an say is that Wse c at ( p ) ≤ Gse c at ( p ) . I n de e d, a similar ar gument that the on e use d in The or em 15 using L emma 11 inste ad of L emm a 18, pr oves that for e ach n ≥ 0 , ∆ n +1 h n admits a we ak lifting alo ng j n , i . e ., ther e is a c ommutative diagr am ∗ n B E h n   / / • ! ! ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ T n ( p ) j n   ∼ o o B ∆ n +1 / / B n +1 The gene r al c ase, in w h ich B is not ne c essarily e-fibr a nt, fol lows e asily. Now, if id B admits a we ak lifting along h n , then i t is e a sy to che ck that ∆ n +1 = ∆ n +1 id B admits a we ak lifting along ∆ n +1 h n . Using L emma 21 b elow we obtain that ∆ n +1 admits a we ak lifting alo ng j n . F rom no w on w e will denote b y secat ( p ) b oth equiv alen t inv a rian ts and call it the se ctional c ate gory of p. 3. Main p roper ties of t he sectional ca tegor y W e b egin b y o bserving that t he Lusternik Schnire lmann catego r y of an ob ject B in C is t he sectional categor y of the zero morphism 0 → B . Indeed (see [6]) the n -th Ganea map p n : G n B → B is precisely the n - th join ov er B , h n : ∗ n B E → B , of 0 → B a nd therefore, cat( B ) = secat(0 → B ) . On the other hand, giv en b : B → B ′ an y morphism, w e define cat( b ) as the least in teger n ≤ ∞ suc h that b admits a we ak lifting alo ng p ′ n : G n B ′ → B ′ . Compare the next result with [15]. Theorem 20. Let p : E → B , p ′ : E ′ → B ′ and b : B → B ′ b e morphisms in C defining a we ak pullbac k. Then, secat( p ) ≤ min { cat( b ) , secat( p ′ ) } . F or its pro of w e shall need t he fo llowing lemma. Lemma 21. [6, Lemma 3.4] Let f : A → B , g : C → B and h : D → B b e morphisms. If f admits a w eak lifting along g and g admits a w eak lifting a lo ng h , then f admits a weak lifting along h. 14 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS Pr o o f of 20. By induction, using rep eatedly the Join Theorem (Lemma 18) on the giv en w eak pullbac k E p   / / •     ❄ ❄ ❄ ❄ ❄ ❄ ❄ E ′ p ′   ∼ o o B b / / B ′ w e obtain, fo r ev ery n ≥ 0 , a weak pullback o f the form ∗ n B E h n   / / • ! ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ∗ n B ′ E ′ h ′ n   ∼ o o B b / / B ′ Hence, if secat( p ′ ) ≤ n , h ′ n admits a w eak section: ∗ n B ′ E ′ h ′ n   ∼ { { ✇ ✇ ✇ ✇ ✇ • $ $ $ $ ❍ ❍ ❍ ❍ ❍ B ′ s > > id B ′ / / B ′ In particular, b : B → B ′ admits a w eak lifting alo ng h ′ n through the morphism sb : B → • . By Lemma 17, h n admits a w eak section and secat( p ) ≤ n. No w supp ose that cat( b ) ≤ n, t ha t is, b admits a weak lifting along p ′ n : G n B ′ → B ′ . Consider the fo llowing diagram obtained b y simply c ho osing any F -fa ctorization of p ′ : 0   / / •     ❄ ❄ ❄ ❄ ❄ ❄ ❄ E ′ p ′   ∼ o o B ′ id / / B ′ As this is not in general a w eak pullbac k, apply this time Lemma 11 inductiv ely to obtain that p ′ n : G n B ′ → B ′ admits a weak lifting along h ′ n : ∗ n B ′ E ′ → B ′ . Finally , b y Lema 21 w e conclude that b admits a we ak lifting along h ′ n : ∗ n B ′ E ′ → B ′ , whic h b y Lemma 17, is equiv alent t o the fact that h n : ∗ n B E → B admits a w eak section.  Ev en if our data is not a w eak pullbac k, w e can pro v e a similar result. Compare with [15]. Theorem 22. Let p : E → B and p ′ : E ′ → B b e morphisms in C . If p admits a w eak lifting along p ′ , then secat( p ′ ) ≤ secat( p ) . In particular, secat ( p ) ≤ cat ( B ) . ABSTRACT SECTIONAL CA TEGOR Y 15 Moreo v er, if p : E → B a dmits a w eak lifting along the zero morphism 0 → B (in particular, when E is we akly con tractible, i.e., E and 0 are w eak ly equiv a lent) then secat ( p ) = cat ( B ). Pr o o f . F or the first assertion, apply Lemma 11 inductive ly to the diagram E p   / / •     ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ E ′ p ′   ∼ o o B id B / / B to conclude that, for ev ery n ≥ 0 , h n admits a w eak lifting along h ′ n . If secat ( p ) ≤ n , id B admits a w eak lif ting along h n and, by Lemma 21, id B admits a w eak lif ting along h ′ n . Hence, secat ( p ′ ) ≤ n . On the other hand, recall tha t cat( B ) = secat(0 → B ) and observ e that the zero morphism admits a w eak lift ing along any morphism. Th us, secat ( p ) ≤ cat ( B ) . Finally note that, if E is a weak ly trivial ob ject, by Lemma 3, p admits a we ak lifting along 0 : 0 → B .  3.1. Modelization functors. W e no w study the b ehav iour of secat through a mo deliz a tion functor . Recall from [6 ] that a co v arian t functor µ : C → D b et w een categories satisfying ( J1)-(J4) axioms is called a mo delization functor if it preserv es w eak equiv alences, homotopy pullbac ks and homotopy pushouts. W e say that µ is p ointe d if µ (0) = 0 . If µ : C → D is contra v arian t , it is said to b e a mo delization functor if the corresp o nding cov a r ian t functor µ : C op → D is a mo delization functor. Here w e pro v e: Theorem 23. If µ : C → D is a mo delization functor b et wee n J -categories, then for an y morphism p : E → B of C secat( µ ( p )) ≤ secat( p ) F or it w e shall need the following Lemma 24. [6, Prop. 6.7] Let µ : C → D b e a mo delization functor and let j : A ∗ B C → B denote the jo in ma p of f : A → B and g : C → B . Then, there is a commutativ e diagram µ ( A ∗ B C ) µ ( j ) ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ • ∼ O O ∼   / / µ ( B ) µ ( A ) ∗ µ ( B ) µ ( C ) j ′ 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ where j ′ denotes the join mo r phism of µ ( f ) and µ ( g ) . 16 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS Pr o o f of The or e m 23. In view of Lemma 21 it is sufficien t to prov e that, for each n , h µ ( p ) n admits a w eak lif t ing a long µ ( h p n ) µ ( ∗ n B E ) ∼ z z t t t t t µ ( h p n )   • $ $ $ $ ■ ■ ■ ■ ■ ■ ∗ n µ ( B ) µ ( E ) 9 9 r r r r r r h µ ( p ) n / / µ ( B ) where h p n and h µ ( p ) n are the n -th join morphisms p and µ ( p ) respective ly . F or n = 0 is trivially true. By assuming t he assertion true for n − 1 , a nd c ho osing an y F - factorization of µ ( p ) we o btain a commu tativ e diagram of the form ∗ n − 1 µ ( B ) µ ( E )   h µ ( p ) n − 1 / / µ ( B ) id   µ ( E ) µ ( p ) o o ∼   • ) ) ) ) ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ • v v v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ µ ( ∗ n − 1 B E ) ∼ O O µ ( h p n − 1 ) / / µ ( B ) µ ( E ) µ ( p ) o o ∼ O O By Lemma 11 h µ ( p ) n admits a w eak section along the j o in morphism of µ ( h p n − 1 ) and µ ( p ) : ∗ n µ ( B ) µ ( E ) h µ ( p ) n   / / • ' ' ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ µ ( ∗ n − 1 B E ) ∗ µ ( B ) µ ( E )   ∼ o o µ ( B ) id / / µ ( B ) (3) On the other hand, applying Prop osition 24 ab ov e to the morphisms h p n − 1 : ∗ n − 1 B E → B and p : E → B , w e obtain a comm utativ e diagram • ∼ / / ∼   µ ( ∗ n B E ) µ ( h p n )   µ ( ∗ n − 1 B E ) ∗ µ ( B ) µ ( E ) / / µ ( B ) T aking any F -factorizatio n of µ ( h p n ) and applying Lemma 3 w e deduce that the join morphism µ ( ∗ n − 1 B E ) ∗ µ ( B ) µ ( E ) → µ ( B ) admits a w eak lifting along µ ( h p n ). Fina lly , b y Lemma 21 applied to (3), w e conclude the inductiv e step.  Remark 25. Observe that, for the pr o of of The or e m 23 we have use d the Gane a- typ e version of se ctional c ate gory. If (J5) axi o m is not s a tisfie d, then using s i m ilar ABSTRACT SECTIONAL CA TEGOR Y 17 ar guments we c an also obtain the sam e r esult for the Whi tehe ad-typ e version of se ctional c ate gory. T h e same als o applies for the r emaining r es ults of this se ction. Corollary 26. Consider µ : C → D and ν : D → C mo delization functors b et w een J -categories and let p : E → B b e a morphism in C suc h that ν ( µ ( p ) ) is w eakly equiv alen t to p . Then secat( µ ( p )) = secat( p ) As an example w e apply the t heorem ab ov e to the abstr act top olo gic al c omplexity of a giv en o b ject. F or an y e-fibrant ob ject B w e define its top olog ical complexit y , TC( B ) as the sectional category of t he diagonal morphism ∆ B : B → B × B . If B is not e-fibrant consider a n y F - factorization B ∼ / / F / / / / 0 and set TC( B ) := TC( F ) . Then TC( B ) do es not dep end on the e-fibrant ob ject F ; indeed, if we tak e another F -factorization B ∼ / / F ′ / / / / 0 , then t here exists a w eak equiv alence h : F ′ ∼ → F (see the pro of of Lemma 1 3). The natura lity of the diagonal morphism applied to h together with the fa ct that h × h : F ′ × F ′ ∼ → F × F is a weak equiv alence (b y the dual of the Gluing Lemma [2, I I.1.2]) pro v e that ∆ F : F → F × F and ∆ F ′ : F ′ → F ′ × F ′ are weakly equiv alent morphisms. Therefore TC( F ) = secat(∆ F ) = secat(∆ F ′ ) = TC( F ′ ) . TC( B ) neithe r dep ends on the w eak t ype of B ; giv en f : B ∼ → B ′ a w eak equiv alence, if w e consider an F -factor ization B ′ ∼ τ ′ / / F ′ / / / / 0 , then an y F -facto r izat io n of the comp osite τ ′ f : B → F ′ B τ ′ f ∼ / / ∼ τ   ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ F ′ F g > > > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ giv es rise to a trivial fibration g : F ∼ ։ F ′ , whic h shows that TC( B ) = TC( F ) = TC( F ′ ) = TC( B ′ ) . Theorem 27. F or an y p oin ted mo delization functor µ : C → D a nd an y ob ject B , TC( µ ( B )) ≤ TC( B ) Pr o o f . T aking in to a ccount that µ preserv es w eak equiv alences and TC do es not dep end on the w eak t yp e, we can supp ose without losing generalit y that B is an 18 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS e-fibran t ob ject. Since µ ( B ) need not b e e-fibrant w e consider an y F - factorization µ ( B ) / / ∼ τ ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ 0 F @ @ @ @ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ so that TC( µ ( B )) = TC( F ) . Now take the followin g comm utativ e cub e: µ ( B × B ) µ ( pr 2 ) / / ω   µ ( pr 1 ) } } ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ µ ( B ) τ ∼     ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ µ ( B ) / / τ ∼   0 id ∼   F × F pr 2 / / / / pr 1 | | | | ② ② ② ② ② ② ② ② ② ② ② F     ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ F / / / / 0 where pr 1 and pr 2 denote the pro jection morphisms. As µ is a p oin ted mo delization functor, the top face is a homotop y pullbac k. On the other hand, the b ottom fa ce is a strict pullbac k (and a homotop y pullback) and ω = ( τ µ ( pr 1 ) , τ µ ( pr 2 )) is the induced morphism from the univ ersal prop ert y of the pullback . Since the to p and b ottom faces are homotopy pullbacks and the un brok en v ertical morphisms are weak equiv alences, b y [6, Cor. 1.12] (or the dual of the Gluing Lemma [2, I I.1.2]) w e hav e that ω is also a w eak equiv alence. F rom the following comm utativ e diagram µ ( B ) τ ∼ / / µ (∆ B )   F ∆ F   µ ( B × B ) ∼ ω / / F × F w e deduce tha t µ (∆ B ) and ∆ F are w eakly equiv alen t morphisms. Then, by Prop o- sition 10 we hav e that TC( µ ( B )) = secat(∆ F ) = secat( µ ( ∆ B )) while, by Theorem 23, secat( µ (∆ B )) ≤ secat(∆ B ) = TC( B ).  Corollary 28. Consider µ : C → D and ν : D → C p ointed mo delization functor s and let B b e an ob ject in C suc h that ν ( µ ( B )) is w eakly equiv alen t to B . Then TC( µ ( B )) = TC( B ) ABSTRACT SECTIONAL CA TEGOR Y 19 4. Some applica tions. W e start by an immediate a pplication in r ational homot o y theory . A classical fact [3, § 8 ] a ssures the existence of an adjunction CDGA ε SSet ∗ h · i o o A P L / / b et w e en the cat ego ries of aug mented comm utativ e differen tia l g raded a lg ebras o v er a field K of characteristic zero, and p oin ted simplicial sets. The category SSet ∗ is kno wn t o b e a J -category endow ed with Ka n fibrations, injectiv e maps and maps realizing to homotop y equiv alences [18, Chap.I I I § .3], [6, Prop.A.8]. The category CDGA ε is also a (pro p er) closed mo del category [3, § 4] (and thus J1-J4 are satis- fied) in whic h fibrations are surjectiv e morphisms, weak equiv alences are morphisms inducing homolog y isomorphisms (the so called “quasi-isomorphisms”) and cofibra- tions are “relative Sulliv an alg ebras” [11, § 14], i.e., inclusions A → A ⊗ Λ V in which Λ V denotes the f r ee comm utativ e alg ebra generated by the g r aded vec tor space V and the differen tial o n A ⊗ Λ V satisfies a certain “minimalit y” condition. Ho w- ev er, this is NOT a J -category and the Ec kmann-Hilton dual of a partial v ers ion of the cub e axiom is satisfied when restricting to 1- connected a lgebras [6, A.18]. The functors h · i and A P L do not in general respect w eak equiv alences altho ug h h · i sends cofibrations to fibratio ns and h · i can b e sligh tly mo dified to send fibrations to cofibratio ns [3, § 8]. Therefore, as they stand, they are not mo delization functors. Ho w eve r, it is also kno wn [3, § 8,9] that, r estricting those functors to the categories CDGA 1 cf Q Kan-Complexes 1 Q o o / / of cofibran t 1-connected comm utativ e differen tial graded algebras of finite type ov er Q (kno wn as Sulliv an a lgebras [11, § 12]) and 1-connected ra tional Kan complexes of finite t ype, then they do preserv e w eak equiv alences and via [6, Prop.6.5] they are mo delization functors. On the other hand, in [10, Ch.8], F a ss` o in tro duce, for a map of finite t ype 1- connected CW-complexes, or equiv alently for a simplicial map of finite t ype 1- connected Kan complexes E p → B the r ational se c tion al c ate gory of p , secat 0 ( p ) whic h can b e seen as the sectional category in the opp osite category of CDGA 1 cf Q of A P L ( p Q ) , b eing p Q the map in Kan-Complexes 1 Q obtained b y rationalization [3, § 11]. Th us , b y Corollary 26, secat 0 ( p ) = secat ( p Q ) Our second application conce rns localizatio n functors. Let P b e a (p ossibly empt y) set of primes and ( − ) P : CW N − → CW N 20 F.J. D ´ IAZ, J.M. CALCINES, P .R. D ´ IAZ, A. MUR ILLO MAS AND J. REMEDIOS denotes the P - lo calization functor (see [13, § 2] or [1, Chap.I I I] where it is sho wn that lo calization can c hosen to b e a functor as it stands, not just in the homotopy category) in the p oin ted category of spaces of the ho motop y ty p e of nilp o ten t CW- complexes. Then, this functor sends homotopy pushouts to homotop y pushouts and homotop y pullbacks (if the c hosen homotop y pullback sta ys in this category) t o homotop y pullback s [13, § 7 ]. (Note that, considering closed cofibrations, Hurewicz fibrations and ho mo t op y equiv alences, the categor y o f w ell p o in ted top olog ical spaces T op ∗ has the structure of a J -category; see [22, Thm.11] fo r axioms (J1 ) -(J4) plus [16, Thm.25] fo r (J5)). Thus, ev en though strictly sp eaking this is not a mo delization functor as it is defined on a certain sub category of T op ∗ , the argumen ts in Theorem 23 could be follo w ed m utatis m utandi as long as all constructions there remain within our category . But this is in fact the case as the homotop y pullbac k (o r pushout) of t w o ma ps in CW N can b e c hosen t o live a lso in this category [13, § 7]. Hence, secat ( f P ) ≤ secat f . Ho w eve r, the situation is drastically differen t in the g eneral case as all sort of p os- sible P -lo calizations (extending t he one on nilp otent complexes) do not, in general, preserv e homotopy pullbac ks and homotop y pushouts. Here, w e consider t he Casacub erta-P esc hke lo calization functor on T op ∗ [4] and start b y se tting some notat io n. Giv en a group G w e denote b y P[G] the ring lo caliza- tion of the group r ing Z P G obtained b y in v erting a ll of the eleme n ts 1 + g + · · · + g n − 1 , where g ∈ G , and ( n, p ) = 1 for an y p ∈ P (se e [4, § 2]). F ollowin g [17] we sa y that a P -torsion group G is an acting gr oup for a space X if there is a n epimorphism f : π 1 X ։ G suc h that, f or eac h m ≥ 2, the action π 1 X → Aut ( π m X ) factors through G . Prop osition 29. Let f : X → Y a map for whic h: (i) π 1 ( ∗ n Y f ) : π 1 ( ∗ n Y X ) ∼ = − → π 1 Y is an isomorphism of P -lo cal groups for any n ≥ 0. (ii) π 1 ( ∗ n Y X ) and π 1 Y hav e a common acting group G for an y n ≥ 0 . (iii) If we denote π 1 Y by π , the morphism Z P π → P [ π ] induce isomorphisms on homology with lo cal co efficien ts H ∗ ( − ; Z P π ) → H ∗ ( − ; P [ π ]). Then, secat ( f P ) ≤ secat ( f ) . Pr o o f . Again, note that the argumen t in Theorem 23 could b e applied if , for an y n ≥ 1, there is a homotopy comm utativ e diagram of the form: ( ∗ n − 1 Y X ) P ∗ Y P X P ( h n − 1 ) P ∗ f P ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ≃ / / ( ∗ n Y X ) P ( h n ) P { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ Y P ABSTRACT SECTIONAL CA TEGOR Y 21 T o this end, an inductiv e pro cess, as in [6, Prop.6 .7 ] will w ork as long as the f o llo wing t w o conditio ns hold: (1) The lo calization of the homotop y pullbac k Q n / /   X f   ∗ n − 1 Y X h n − 1 / / Y ( Q n ) P / /   X P f P   ( ∗ n − 1 Y X ) P h n − 1 P / / Y P is again a homotop y pullback. 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T ra nsl. 55 (1966), 4 9-140 [21] S Smale. On the top ology of algor ithms. I, J. Complexity 3 (1987) 81-89. [22] A. Strøm. The homotopy c ate gory is a homotopy c ate gory . Arch. Ma th., 23 (1972), 435-44 1. F.J. D ´ ıaz; J.M. Garc ´ ıa Calcines; P.R. G arc ´ ıa D ´ ıaz; J. Remedios G ´ omez Dep ar t amento de Ma tem ´ atica Fundament al Universidad d e La Laguna 38271 La Laguna, Sp ain. E-mail addr ess : fra diaz@ ull.e s E-mail addr ess : jmg arcal @ull. es E-mail addr ess : prg diaz@ ull.e s E-mail addr ess : jre med@u ll.es A. Muril lo Mas Dep ar t amento de Al gebra, Geometr ´ ıa y Topol o g ´ ıa Universidad d e M ´ alaga Ap.59, 2908 0 M ´ alaga, Sp ain. E-mail addr ess : ani ceto@ agt.c ie.uma.es