Saturating directed spaces

SA TURA TING DIRECTED SP A CES ANDR ´ E HIRSCHOW ITZ, M ICHEL HIRSCHOWITZ, AND TOM HIRSCHOWI TZ 1. Intr oduction Directed algebraic top olog y [4] (see also , e.g., [7, 3, 4, 2, 1, 5, 8]) ha s recently emerged as a v ariant of algebra ic top olog y . In the appro a ch prop osed b y Grandis , a dir e cte d top ological space (or d-sp ac e for shor t), is a top ologica l space equipp ed with a set of dir e cte d paths s atisfying three conditions. These conditions are the three conditions necessary for constructing the so-c a lled fundamen tal ca tegory: constant paths ar e directed fo r having iden tities, sta bilit y under concatena tion is r equired for having comp ositio n, a nd re pa rameterisa tion is requir ed for having a sso ciativ- it y . These are somehow minimal conditions, which leave ro om for a lo t of exotic examples (see E xamples 2.1). In the prese n t work w e pro p o s e an additional condition of satura tion for distin- guished sets of paths and show how it allows to rule out exotic e x amples without any serious collateral damag e. Our condition inv olves “dir ected” functions (to the unit in terv a l I ), namely those which ar e non-decr easing along each dir ected path. And it asserts that a path along which a ny suc h (lo ca l) directed function is non-decreasing should be directed itself. Our saturation condition is lo cal in a natural sense, a nd is satisfied b y the di- rected interv al (a nd the directed circle). F urthermore w e show in whic h sense it is the stro ng est condition fulfilling these tw o basic req uir ements. Our satura tion condition s elects a full sub categor y SDT op o f the catego ry DT op of d-spaces, and w e show that this new catego r y has all standar d desirable proper - ties, namely: • S DT op is a full, reflective sub categor y of DT op , wic h means that there is a nice saturatio n functor from DT op to SDT op ; • it is closed under arbitr ary limits; • a lthough it is not clo sed under colimits (as a s ubca tegory), it ha s ar bi- trary colimits, e ach of whic h is obtained by satura tion of the cor resp onding colimit in DT op ; • S DT op is a dIP1 - categor y in the sense of [4] which essen tially means that it has nice cylinder a nd co cylinder co ns tructions; • the forgetful functor fro m SDT op to T op has both a right a nd a left adjoint. Altogether these prop er ties satisfy the gener al principles which, according to Grandis [4, Section 1.9], should b e satisfied by a go o d top olo gic al setting for directed algebraic top olo gy . In Sec tio n 2 w e describ e our sheaf o f “ directed” functions, and introduce our sat- uration co ndition. In Section 3, w e ex hibit adjunctions relating our new catego ry SDT op to T op and DT op . In Section 4, we pr ove the completeness and co co m- pleteness prop erties of SDT op . In Section 5, we pro ve that SDT op admits cylinder and co cylinder constructio ns with the desired prop e rties. Finally in Section 6 , we discuss other saturation conditions, and show in which sense ours is max imal. This work ow es muc h to th e second aut hor’s w edding on October 30, 2010. 1 2 ANDR ´ E HIRSCHO WITZ, MICHEL HIRSCHOWITZ, AND TOM HIRSCHOWITZ 2. S a tura ted d-sp aces W e denote by I the sta ndard closed unit in terv al, a nd by DT op the catego ry where • o b jects are all dir e cte d spaces, i.e., pairs ( X , d X ) of a top olo gical spa ce X and a set dX of co n tinuous maps I → X , sub ject to the following three conditions – co nstant paths are in dX , – dX is stable under co ncatenation, – dX is sta ble under preco mpo sition with contin uo us, non-decr e a sing maps I → I ; • mo rphisms from ( X , dX ) to ( Y , d Y ) are a ll contin uous maps f : X → Y satisfying f ◦ dX ⊆ d Y . The set dX is called the set of dir e cte d paths, o r d-p aths in ( X , dX ). In the sequel, for a d -space X , we will also write X for the underlying top ologica l space, and we will write dX for its set of dire c ted paths. W e denote b y I the d -space obtained by equipping the standa rd closed unit int erv al with the set of no n-decreasing (contin uous) paths. Examples 2.1. As pr omise d, her e ar e a few exotic examples. F or e ach of them, the underlying sp ac e is either the usu al plane X := R 2 or its quotient, the st andar d torus X := R 2 / Z 2 , which we c onsider e quipp e d with the usual pr o duct or der (or lo c al or der). Henc e we jus t sp e cify the distinguishe d subset of p aths, either dX or d X . (1) dX c onsists of al l horizontal p aths with r ational or dinate (i.e., c ontinuous maps p : I → R 2 with p ( t ) = ( q ( t ) , a ) , for some r ational a and c ontinuous q : I → R ). (2) d X c onsists of al l horizontal p aths with r ational or dinate (i.e., c ontinu ous maps p : I → R 2 / Z 2 with p ( t ) = ( q ( t ) , a ) , for s ome r ational a and c ontinu- ous q : I → R / Z ). (3) dX c onsists of al l horizontal nonde cr e asing p aths with r ational or dinate (i.e., c ontinu ous maps p : I → R 2 with p ( t ) = ( q ( t ) , a ) , for some ra tional a and c ontinu ous nonde cr e asing q : I → R ). (4) d X c onsists of al l horizontal lo c al ly nonde cr e asing p aths with r ational or di- nate (i.e., c ontinuous maps p : I → R 2 / Z 2 with p ( t ) = ( q ( t ) , a ) , for some r ational a and c ontinuous lo c al ly nonde cre asing q : I → R / Z ). (5) dX c onsists of pie c ewise horizo ntal or vertic al p aths, i.e., (finite) c onc ate- nations of vertic al and horizontal p aths. (6) d X c onsists of pie c ewise horizontal or vertic al p aths. (7) dX c onsists of c ontinu ous p : I → R 2 whose r estriction to some dense op en U ⊆ R 2 is lo c al ly pie c ewise horizontal or vertic al. (8) dX c onsists of pie c ewise r e ctiline ar p aths with r ational slop e. (Mor e gen- er al ly, for any subset P of R c ontaining at le ast two distinct elements , pie c ewise r e ctiline ar p aths in R 2 with slop e in P form a d-sp ac e.) (9) dX c onsists of pie c ewise cir cular p aths. (10) dX c onsists of pie c ewise horizontal or vertic al nonde cr e asing p aths. (11) dX c onsists of pie c ewise horizontal or vertic al lo c al ly n onde cr e asing p aths. W e now describ e our sa turation pro cess, which will rule out such examples. F or any d -spa ce X , we have the sheaf ˆ I X on X which ass igns to a ny op en U ⊆ X the set of contin uous functions U → I . Note that each such U inher its a s tr ucture of d -space. W e refer to this str ucture by sa ying that U is an op en subspace of X . SA TURA TING DIRECTED SP AC ES 3 Definition 2.2. F or any d-sp ac e X , we denote by ˆ I X the su bshe af of ˆ I X c onsisting, on any op en subsp ac e U ⊆ X , of al l morphisms of d -sp ac es c : U → I . W e say that such a section c : U → I of this sheaf is a dir e cte d function (on U ). The statement that this is indeed a s ubsheaf needs a pro of: Pr o of. Let us consider an op en subspace U ⊆ X , a contin uous map c : U → I , and an op en cov ering ( U j ) j ∈ J of U such that any r estriction c j of c to a U j is a directed function. In o rder to prove that c is dire c ted, w e consider an arbitrar y directed pa th p : I → U in dX and show that c ◦ p is non-decre asing. P ulling back the co vering along p gives a cov ering ( V j ) j ∈ J of I , and the restrictions p j : V j → U j of p are lo cally non-decr easing. W e conclude by r ecalling that a function which is lo cally non-dec r easing on I is globally non-decrea sing.  Examples 2.3. • On the dir e cte d interval I , the she af of dir e cte d functions is the she af of lo c al ly non-de cr e asing fu n ctions. • The dir e cte d cir cle is a lo c al ly or der e d sp ac e, and its she af of dir e cte d func- tions is the she af of lo c al ly non-de cr e asing functions. • On Examples 2.1, for items 1 and 2, t he she af of dir e cte d functions is the she af of al l c ontinu ous functions which ar e lo c al ly horizontal ly c onst ant. • On Examples 2.1, for items 3 and 4, t he she af of dir e cte d functions is the she af of fun ctions which ar e lo c al ly nonde cr e asing in the first variable. • On Examples 2.1, for items 5 to 9, the she af of dir e cte d fun ctions is the she af of lo c al ly c onstant functions. • On Examples 2.1, for items 10 and 11, the she af of dir e cte d functions is t he she af of lo c al ly nonde cr e asing functions. Morphisms of d -s paces resp ect dir ected functions in the following sense: Prop ositio n 2.4. L et f : X → Y b e a morphism of d -sp ac es. If X ′ ⊆ X and Y ′ ⊆ Y ar e op en subsp ac es with f ( X ′ ) ⊆ Y ′ , then for any dir e cte d funct ion c : Y ′ → I on Y ′ , c ◦ f is a dir e cte d function on X ′ . Pr o of. The po int is that f induces a mo rphism fro m X ′ to Y ′ . Since p is a mo rphism from Y ′ to I , the comp osite p ◦ f is a morphism from X ′ to I .  Remark 2. 5 . The pr evious st atement has a she af-the or etic formulation as fol lows: the c ont inuous f : X → Y yields a c omp anion she af morphism f ∗ : ˆ I Y → f ∗ ˆ I X and if f is a morphism, then f ∗ sends the she af of dir e cte d funct ions on Y into the (dir e ct image of the) she af of dir e cte d functions on X . Next we in tro duce our notion of w eakly directed paths: Definition 2.6. We say that a p ath c : I → X in a d -sp ac e X is weakly dir ected if, given any dir e cte d function f : U → I on an op en subsp ac e U ⊆ X , f ◦ c : c − 1 ( U ) → I is again dir e cte d, that is t o say lo c al ly non-de cr e asing. We denote by ˆ dX t he s et of we akly dir e cte d p aths in X . Remark 2.7. Note that the inverse image c − 1 ( U ) ne e d not b e c onne cte d, so that the pul l-b ack f ◦ c may b e lo c al ly non-de cr e asing without b eing glob al ly non-de cr e asing. Example 2.8. Of c ourse dir e cte d p aths ar e also we akly dir e cte d. Her e we sketch an example of a we akly dir e cte d p ath which is not dir e cte d. Consider t he plane R 2 , e quipp e d with the s et of pie c ewise horizontal or vertic al p aths. Its dir e cte d functions ar e lo c al ly c onstant fu n ctions. As a c onse quenc e, al l its p aths ar e we akly dir e cte d. W e are now ready for the in tro duction of our saturatio n condition. 4 ANDR ´ E HIRSCHO WITZ, MICHEL HIRSCHOWITZ, AND TOM HIRSCHOWITZ Definition 2.9. We say t hat a d -sp ac e X is s a turated if e ach we akly dir e cte d p ath in X is dir e cte d, in other wor ds if ˆ dX = dX . Examples 2.10. • On the dir e cte d interval I , the she af of dir e cte d functions is the she af of lo c al ly non-de cr e asing fu n ctions and I is satura te d. • Sinc e its she af of dir e cte d functions is the she af of lo c al ly n on-de cr e asing functions, the dir e cte d cir cle is satur ate d. F or t his example, the c onsid- er ation of the she af inste ad of only glob al dir e cte d fun ct ions is obviously crucial. • The Examples 2.1 ar e nonsatu r ate d. We wil l se e b elow what is their satu- ration . 3. A djunctions W e now hav e the full sub ca tegory SDT op consisting o f saturated d -spaces, which is equippe d with the forge tful functor U : SDT op → T op . This functor has a right adjoint which sends a space X to the d -space o bta ined b y equipping X with the full set of paths in X . W e will see below tha t U also has a left adjoin t, obtained as a comp osite of the left adjoint to U : DT op → T op a nd the left adjoint L to SDT op → DT op which we build no w. Definition 3. 1. Given a d -sp ac e X := ( X , dX ) , we build its satu r ation ˆ X := ( X, ˆ dX ) (r e c al l that ˆ dX is the set of we akly dir e cte d p aths in X ). Examples 3.2. • On Examples 2.1 , for items 1 and 2, we akly dir e cte d p aths ar e horizontal p aths. • On Examples 2.1, for items 3 and 4, we akly dir e cte d p aths ar e (lo c al ly) nonde cr e asing horizo ntal p aths. • On Examples 2.1, for items 5 to 9, al l p aths ar e we akly dir e cte d. • On Examples 2.1, for items 10 and 11, we akly dir e ct e d p aths ar e al l (lo c al ly) nonde cr e asing p aths. In the pre vious definition, the verification that ˆ X is indeed a saturated d -space is straightforw ard. It is also eas y to chec k that this construction is functorial, that is to say that if the contin uous map c : X → Y is a morphism of d -spaces, then it is a morphism from ˆ X to ˆ Y as well. This y ields our le ft adjoint L : DT op → SDT op for the inclusion J : SDT op → DT op . Indeed L ◦ J is the identit y , a nd we tak e the ident ity as co unit of o ur adjuntion. While for the unit η e v aluated at X , we take the ident ity map: i d X : X → ˆ X . The eq ua tions for these data to yie ld an adjunction (see [6, Chapter IV, Thm 2 (v)]) a re easily verified. Since J is a n inclusion, this adjunction is a so-called r efle ction (a ful l one since S DT op is by definition full in T op ). Remark 3.3. We c ould build a mor e symmetric pictur e as fol lows. Th er e is a c ate gory ST op c onsisting of sp ac es e quipp e d with a “st r u ctur al” subshe af of their she af ˆ I X of functions to I . Morphisms ar e those c ontinuous maps f : X → Y which, by c omp osition, send the structur al subshe af on Y into the structur al subshe af of X . (In she af-the or etic al t erms, maps f such that D Y ֒ → ˆ I Y → f ∗ ( ˆ I X ) factors thr ough f ∗ ( D X ) ֒ → f ∗ ( ˆ I X ) , wher e D X and D Y ar e the st ructur al she aves of X and Y , r esp e ctively.) Ou r c onstruction of t he s he af of dir e cte d fun ct ions c an b e u p gr ade d into a functor D : DT op → ST op . Du al ly, our c onstruction of the satur ation c an b e up gr ade d into a functor S : ST op → DT op which is right adjoint to D . This adjunction factors thr ough SDT op , which is thus not only a r efle ctive sub c ate gory SA TURA TING DIRECTED SP AC ES 5 of DT op bu t also a c or efle ctive sub c ate gory of ST op . We cho ose n ot to develop this material her e and s ele ct only the n ext two statement s . Prop ositio n 3.4 . L et X b e a top olo gic al sp ac e, and D a subshe af of ˆ I X . Then the set d D X of p aths in X along which lo c al se ctions of D ar e lo c al ly non-de cr e asing turns X int o a satur ate d d - sp ac e. Pr o of. Indeed, it is ea sily chec ked that ( X , d D X ) is a d -space, a nd that D is co n- tained in the s hea f o f directed functions o n this d -space. Hence weakly directed paths in ( X , d D X ) are a utomatically in d D X .  Remark 3.5. The pr evious st atement al lows to che ck e asily t hat a d -sp ac e is sat- ur ate d after having gu esse d (inst e ad of pr ove d) what ar e its dir e cte d fu nctions. Prop ositio n 3.6. L et f : X → Y b e a c ontinuous map b etwe en satur ate d d -s p ac es. If f t r ansforms, by c omp osition, (lo c al) dir e cte d fun ctions on Y into (lo c al) dir e cte d functions on X , then f is a morphism of d -sp ac es. Pr o of. Indeed, consider a directed path c : I → X . w e prov e that f ◦ c is weakly directed (hence directed). for this we take a (lo ca l) directed function p : Y ′ → I . W e know that p ◦ f is a (loca l) dir ected function o n X , hence p ◦ f ◦ c is directed.  Remark 3.7. In the pr evious s t atement, the assumption that X is satur ate d is useless, but we pr efer to se e this as a char acterisation of morphisms in SDT op . 4. Compl eteness and co completeness In this section, we prove that SDT op is complete and co complete. W e further- more show that limits may b e computed as in DT op . First, we ha ve easily: Prop ositio n 4.1. SDT op is c o c omplete. Pr o of. DT op is c o complete [4 ], so given any diag ram D : J → SDT op , we may compute its colimit d in DT op . The left adjoint L then pr eserves co limits, o f course, but it also r estores the or iginal diag ram by idemp otency , so that L ( d ) is a colimit of D in SDT op .  Example 4. 2. Her e we sketch an example showing that a c olimit of satu r ate d d - sp ac es ne e d not b e satur ate d. This involves four differ ent dir e cte d planes. The first one P 0 has only c onstant dir e cte d p aths. The next two ones P ′ and P ′′ have only horizontal (r esp. vertic al) dir e cte d p aths. The fourt h one is the c opr o duct P 1 of P ′ and P ′′ along P 0 . Its di r e cte d p aths ar e pie c ewise horizontal or vertic al. As we have se en ab ove, al l its p aths ar e we akly dir e cte d, henc e it is n ot satur ate d. Example 4.3. The pr o duct I × I in DT op , which has as underlying sp ac e the pr o duct I × I and as dir e cte d p aths al l c ontinuous maps p : I → I × I with non-de cr e asing pr oje ctions, is satu r ate d. F urthermor e, for any op en U ⊆ I × I , a map c : U → I is dir e cte d iff it is lo c a lly non-decreasing , i.e., for any x ∈ U , ther e is a n eighb ourho o d V of x on which c is non- de cr e asing. This is of c ourse e quivalent to b eing lo c al ly separately non-de cr e asing, i.e., lo c al ly non-de cr e asing in e ach variable. Next we also hav e: Prop ositio n 4.4. SDT op is c omplete as a sub c ate gory of DT op . Pr o of. First, rec a ll that DT op is co mplete [4]. Then, c o nsider a ny diagra m D : J → SDT op , and its limiting cone u j : d → D j in DT op . Let d ′ = sat ( d ). By universal prop erty of η , this yields a cone u ′ j : d ′ → D j in SDT op . By univ ersal prop erty o f d , we also have a compatible morphism from d ′ to d , which has to be the identit y .  6 ANDR ´ E HIRSCHO WITZ, MICHEL HIRSCHOWITZ, AND TOM HIRSCHOWITZ 5. Tow ards directed homotopy As explained by Grandis [4], the bas ic requirement for building directed homo - topy is the existence o f conv enient cy linder and co cylinder construc tio ns. In the present section, we chec k that our categor y SDT op is stable under the cylinder and co cylinder cons tructions in DT op . In Grandis’s terminolo g y , this reads as fo llows: Theorem 5.1. SDT op is a c artesian dIP1- c ate gory. Pr o of. Since SD T op is a full ca r tesian sub ca teg ory of DT op which is a c a rtesian dIP1- category (see [4, Se c tio n 1.5.1]), w e just ha ve to chec k that it is stable under the cylinder, the co cylinder a nd the reversor constructions . It is clear for the cylinder, since it is the pro duct with the dir ected interv al, which is an ob ject of SD T op . F or the c o cylinder constructio n, w e must check that for X in S D T op , its path- ob ject X I is again in SDT op . Thus we ha ve to prov e that weakly dir e cted paths in X I are directed. On the wa y , we hav e “sepa rately directed” paths. Firs t, r ecall from Grandis [4 , Section 1.5.1] that for an y d-space X and t ∈ I , e v aluatio n at t yields a directed morphism ev t : X I → X . Say that a path p : I → X I is sep ar ately dir e cte d iff for all t ∈ I , ev t ◦ p is directed in X . W e firs t pr ov e that a ny weakly dir ected path p : I → X I is separ ately dir ected. Thu s we ha ve a p oint t ∈ I and we m ust prov e that p t := ev t ◦ p is dir ected in X . Since X is saturated, it is enough to show that it is weakly directed. F or this, w e consider a directed function f : U → I o n an op en set U ⊆ X , and we must prove that, wher e defined, f ◦ p t is lo cally non-decre a sing. In pictures, w e must prov e that the top row of U ′′ U ′ U I I X I X i j f p ev t p t is lo cally non- decreasing. Since p is weakly directed in X I , it is enough to show that f ◦ j is directed, w hich holds by Pr op osition 2.4. Now we prove that any separ ately dir ected path is directed. Consider any sep- arately dir ected p : I → X I . It is dir e cted in Grandis’s sense iff its uncurrying p ′ : I × I → X is. F or this, by Ex ample 4.3 ( I × I is saturated) and Prop osition 3.6, it is enoug h to show that for any direc ted map f on X , the compo site f ◦ p ′ , where defined, is directed on I × I . By E xample 4.3 again, this is equiv alent to both f ◦ p ′ ◦ h id , t i and f ◦ p ′ ◦ h t, id i b eing lo cally non-de c reasing, for a ny t ∈ I . F or the firs t map, observe that p ′ ◦ h id , t i = ev t ◦ p is dir ected in X by hypothesis. Hence, b e cause X is sa turated, its comp osition with f , whe r e defined, is lo cally non-decreas ing. F or the second map, we hav e p ′ ◦ h t, id i = p ( t ), which is directed in X by construction of X I , hence, again its co mpo sition with f , where defined, is lo cally non-dec r easing. Finally we chec k that SDT op is s ta ble under reversion. F or this we take a saturated d -space X a nd prov e that RX is satur ated. W e fir st chec k that directed functions on R X are exactly obtained by reversion from directed functions on X . Then w e tak e a w eakly dir ected path c in RX and chec k easily that its rev ersio n is weakly directed on X hence dir ected, which means that c is directed in RX .  SA TURA TING DIRECTED SP AC ES 7 6. A universal proper ty of sa tura tion In this final section, we discuss other pos sible s aturation pr o cesses, showing in which s ense o ur c hoice is the b est one. As a first naiv e attempt, we could ha ve defined weakly directed paths b y testing only aga inst g lo bal directed functions. In this cas e, the directed interv al would hav e remained saturated; but the saturation of the directed cir cle would ha ve pro- duced the reversible circle, whic h is highly undesired. This explains wh y w e have considered lo cal directed functions. As a seco nd, m uch more rea s onable attempt, we ca n define almost dir e cte d paths to b e limits (in the co mpact-op en to po logy) of directed paths. It is e a sily check ed that almost directed paths a re weakly dir ected. But we observe that almost directed paths are no t in g eneral stable b y concatenatio n. T o see this, just equip the real line L with the set dL of paths whic h are constant or a void 0: almost directed paths are those whic h stay in the nonnegative, or in the no np o s itive half-line. Of cours e we could nevertheless de fine the small saturation of a d -spa ce X to be obtained by eq uipping X with the smallest set a dX of paths in X co nt aining almost directed paths and stable b y repara meterisation and concatenation. This is in genera l strictly smaller than the set of weakly directed paths. T o show this we sketc h an ad ho c example. Example 6.1 . Our example is a subsp ac e H (for harp) of R 3 . It c onsists of a skew curve C , to gether with some of its chor ds L a,b (her e, by the chor d, we me an the close d se gment) . F or t he curve C , we take the r ational cubic curve: C := { ( t, t 2 , t 3 ) | t ∈ R } . The inter est ing pr op erty of this curve is t hat its chor ds me et C only at two p oints, and two of these chor ds c annot me et outside C . Inde e d, otherwise, the plane c ontaining two such chor ds would me et our cubic curve in four p oints. We p ose C t := ( t, t 2 , t 3 ) and write L a,b for the chor d thr ough C a and C b . We take for H the union of C with the chor ds L a,b for a < b , a r ational and b irr ational (t he p oint her e is t hat t hese two subsets ar e dense and disjoint). We take for dH the set of p at hs which ar e either c onstant or dir e cte d p aths in one of the chor ds L a,b e quipp e d with t he usu al or der with C a < C b . These ar e cle arly stable by r ep ar ameterisation and ther e is cle arly no p ossibility for c onc atenation exc ept within a chor d. Thus this yields a d -sp ac e. Conc erning t his d -sp ac e, we have two claims. We first claim that this s et of p aths is close d. In de e d, a (simple) limit of p aths e ach c ontaine d in a line is c ontaine d in a line to o and if the limiting p ath is not c onstant, the line for the limit has to b e the limit of the lines. Se c ond ly we claim that this d -sp ac e is not satur ate d. Inde e d, lo c al dir e cte d functions ar e non-de cr e asing along C (e quipp e d with the obvious or der, wher e C a < C b me ans a < b ). T o se e this, c onsider a lo c al fun ction f : U → I wher e U is an op en neighb ourho o d of C a . W e may cho ose a n eighb orho o d V of C a on C such that U c ont ains any chor d joining two p oints in V . S inc e f is c ontinu ous (in p articular along C ) and non-de cr e asing along t hese chor ds (whose endp oints ar e dense in V ), it has to b e non-de cr e asing along V . Thus dir e cte d p aths on C (e quipp e d with the ab ove or der) ar e we akly dir e cte d in H but n ot dir e cte d. Now we wish to s how in whic h sense our satur a tion pro cess is maximal a mong reasona ble saturation pro cesses , in the following sense. Definition 6.2. A d-satura tion pro cess is any functor S : DT op → DT op which c ommut es with the for getful functor DT op → T op , e quipp e d with a natu r al tr ans- formation fr om the identity η S : id → S , such that 8 ANDR ´ E HIRSCHO WITZ, MICHEL HIRSCHOWITZ, AND TOM HIRSCHOWITZ • η S is mapp e d to t he identity by the for getful fun ctor to T op ; • its c omp onent η S I : I → S I at I is the identity; • S satisfies the fol lowing “lo c ality” c ondition: for any d -s p ac e X and sub- sp ac e Y ⊆ X , dir e ct e d p aths in S X with image c ontaine d in Y ar e also dir e cte d in S Y . Remark 6.3. L et us c ommen t on t he pr evious c ondition. First note that dir e cte d p aths in S Y ar e automatic al ly dir e cte d in S X thanks to functoriality. Next let u s exp ain why our c ondition c onc erns lo c ality: if X is c over e d by op en su bsp ac es Y i , then S X is determine d by t he S Y i ’s. In de e d , by c onc atenation (and c omp actness of I ), a p ath in S X is dir e cte d if and only if e ach of its re strictions c ontaine d in a Y i is dir e cte d in this S Y i . (The lo c ality c ondition is her e use d in the “only if ” dir e ction.) Example 6. 4. Our functor L : X 7→ ( X, ˆ dX ) is obviously a d-satur ation pr o c ess, with η L the unit of L ⊣ J . Now we hav e an order on d-s aturation pr o cesses, which says S ≤ T whe ne ver, for each X ∈ D T op , the set-theor etic identit y of X is directed from S X to T X . Observe in pa rticular tha t the induced p oset contains the (oppo site o f the) p ose t of fully reflective s ub ca tegories of DT op . Theorem 6.5. Ou r fun ct or L is maximal among d-satu r ation pr o c esses. Pr o of. Let us consider a d-s aturation pro cess S . What we have to prove is that, given a d -space X , a n y directed path c in S X is weakly directed in X . F or this, we tak e a directed function f : U → I on X and pr ov e that for any closed directed subpath c ′ of c with ima ge contained in U , f ◦ c ′ is non-dec r easing. By functoriality of S , f is also a morphism from S U to I = S I , and b y locality , c ′ is also dir ected in S U , so that f ◦ c ′ is an endomorphism of I , hence non-dec r easing.  References [1] Lisb eth F a jstr up, Martin Rauen & Er ic Goubault (2006): Algebr aic top olo gy and c oncurr ency . Theoretical Computer Scienc e 357(1-3), pp. 241 – 278 , doi:10.1016/j.tcs.2006.03.022. Clifford Lectures and the Mathematical F oundations of Programming Semantics. [2] P . Gauche r (2 003): A mo del c ate gory for the ho motopy the ory of c oncurr ency . ArX iv Mathe- matics e-prints . [3] Eri c Goubault & Thomas P . Jensen (1992) : Homolo gy of Higher Dimensional Autom ata . 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Curr ent addr ess : Universit ´ e de Nice - Soph ia A n tip olis Curr ent addr ess : CEA - LIS T Curr ent addr ess : CNRS, U ni v ersit´ e de Sa voie