Reparametrizations of Continuous Paths

Journal of Homotopy and R elate d Structu r es , vol. 2(1), 2007, pp.1–24 REP ARAMETRIZA TIONS OF CONT INUOUS P A THS ULRICH F AHRENBE R G a nd MAR TI N RAUSSEN ( c ommunic ate d by R onnie Br own ) Abstr a ct A repara metr ization (of a co ntin uous path) is g iven by a surjective weakly incr easing self-map of the unit interv al. W e show that the monoid of reparametrizatio ns (with res pect to comp ositions) can b e unders too d via “stop-maps” that allow to inv e stigate comp ositions and factor izations, and w e compar e it to the dis tributiv e lattice of countable subsets o f the unit int erv al. The results obtained are used to analyse the space of traces in a top olo gical spa ce, i.e., the space of contin uous paths up to repa rametrizatio n equiv a lence. This spa ce is shown to b e homeomorphic to the s pace of regular paths (without stops) up to incr easing repara metrizations. Directed versions of the results are impo rtant in direc ted homotopy theory . 1. In tro duction and Outline 1.1. In tro duction In elementary differ ential ge ometry , the most basic ob jects studied (after p oints per haps) are p aths , i.e., differ entiable maps p : I → R n defined on the clo sed int erv al I = [0 , 1]. Suc h a path is called r e gular if p ′ ( t ) 6 = 0 for all t ∈ ]0 , 1[. A r ep ar ametrization of the unit interv al I is a surjective differentiable map ϕ : I → I with ϕ ′ ( t ) > 0 fo r all t ∈ ]0 , 1[, i.e. a (strictly increasing ) s elf-diffeomorphism of the unit interv al. Given a path p : I → R n and a reparametriza tion ϕ : I → I , the paths p and p ◦ ϕ repre s en t the same g eometric ob ject. In differential geometry one inv estiga tes equiv alenc e classes (identifying p with p ◦ ϕ for any reparametriza tion ϕ ) and their inv a r iants, like cur v ature and torsio n. Motiv ated b y applications in c oncurr ency the ory , a branch of theo retical Com- puter Sc ie nc e trying to mo del and to understand the co or dination b e t ween ma n y different pr o cessors working on a common task, we are interested in c ontinuous paths p : I → X in more gener al top ological spaces up to more gener al repa ramet- rizations ϕ : I → I . When the st ate sp ac e of a concurrent prog ram is viewed as The authors would like to thank the r eferee f or man y useful comments and in particular f or p ointing out an i naccuracy in the ori ginal pro of of Theorem 3.6. Receiv ed June 9, 2006, revised March 15, 2007; published on June 30, 2007. 2000 Mathematics Subj ect Classi fication: 55,68 Key words and phrases: path, r egular path, reparametrization, reparametrization equiv alence, trace, stop m ap, d-space, concu rr ency c  2007, U lrich F ahrenberg and Martin Raussen. P ermiss i on to cop y f or priv ate use gran ted. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 2 a top olog ical space (t ypically a cubical co mplex; cf. [ 6 ]), “dir ected” paths in that space resp ecting certain “monoto nic ity” prop erties cor resp ond to ex e cut ions . A nice framework to handle dir e cte d top ologica l spaces (with a n eye to homotopy prop er- ties) is the concept of a d-space propo sed and in vestigated by Marco Gra ndis in [ 9 ]. Essentially , a top olog ic al space co mes equipp ed with a subset of preferred d-p aths in the set of all paths in X , cf. Definition 4.1. Note in particular , tha t the r everse of a dir ected path in genera l is not directed; the slo gan is “bre a king symmetries” . W e do not tr y to ca pture the quantitative b ehaviour o f executions, co rresp onding to par ticular para metrizations of paths, but merely the qualitative be haviour, suc h as the or der of sha red resour ces us e d, or the result of a computation. Hence the ob jects of study a re pa ths up to certain repar ametrizations which 1. do not alter the image of a path, and 2. do not alter the or der of events . W e are thus interested in general paths in topolo gical spac es, up to su r je ctive re- parametriza tions ϕ : I → I whic h are incr e asing (and thus contin uous!—cf. Lemma 2.7), but not nece s sarily strictly increasing . Two paths ar e consider ed to have the same b ehaviour if they are r ep ar ametrization e quivalent , c f. Definition 1.2 . T o under s tand this equiv alence r elation, we hav e to inv estiga te the space of all repara metr izations which includes strange (e.g. nowhere differentiable) elements. Nevertheless, it enjoys r emark able proper ties: It is a monoid, in which composi- tions a nd factoriza tions can b e completely ana lysed through an investigation of stop intervals and of stop values . The quotient space after dividing out the self- homeomorphisms has nice a lgebraic la ttice prop erties. A pa th is called re gular if it do es no t “stop”; and we ar e able to show that the space of genera l pa ths mo dulo r ep ar ametr izations is homeomorphic to the space of r e gular pa ths mo dulo incr e asing aut o-home omorphisms of the interv al. Hence to inv e stigate prop erties of the for mer, it suffices to co nsider the latter. This is one of the star ting po in ts in the homotop y theoretica l a nd ca tegorical inv estigation of inv a r iants o f d-spaces in [ 15 ]. F urther poss ible area s of application of the results (and of higher - dimensionsal generalisa tions still to be investigated) include catego rical homotopy theory as in [ 8 ], c a tegorified ga uge theor y as in [ 1 ] and n -transp ort theory; cf. the blo g “The n -ca tegory caf´ e” at http:/ /golem .ph.utexas.edu/category . This a rticle do es not build on any sophistica ted machinery . Most o f the conce pts and pro o fs c a n b e understo o d with an undergr aduate mathematical background. There are certain para llels to the elemen tary theory of distr ibution functions in probability theory , cf. e.g . [ 13 ]. The flav our is nevertheless differen t, s inc e co n tinuit y (no jumps, i.e., s urjectivity) is essential for us. F or the s ake of completeness, we ha ve chosen to include also e lemen tary r esults and their pr o o fs (so me o f whic h may b e well-kno wn). Marco Grandis has studied piece wise linear repara metrizations in [ 1 0 ] for differ- ent purp oses, but a ls o in the framework of “dir ected alge br aic top ology” . 1.2. Basic de fi ni tions Let alw ays X denote a Ha usdorff topolo gical s pace and I = [0 , 1] the unit interv al. The set of a ll (nondegener a te) closed subinterv als of I will be denoted by P [ ] ( I ) = Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 3 { [ a, b ] | 0 6 a < b 6 1 } . Let p : I → X deno te a con tinuous ma p (a path), and remark tha t the pre - image p − 1 ( x ) of any element x ∈ X is a closed set. Definition 1.1. 1. An in terv al J ∈ P [ ] ( I ) is called a p - stop interval if the r e- striction p | J is constant and if J is a maximal interv al with that prop erty . 2. The set of all p -stop interv als will be denoted as ∆ p ⊆ P [ ] ( I ). Rema rk that the interv als in ∆ p are disjoint and that ∆ p carries a na tur al total o rder. W e let D p := S J ∈ ∆ p J ⊂ I denote the stop set of p . 3. A path p : I → X is called re gular if ∆ p = ∅ or if ∆ p = { I } (no stop or constant). 4. A con tinuous map ϕ : I → I is ca lled a r ep ar ametrization if ϕ (0) = 0 , ϕ (1) = 1 and if ϕ is incr e asing , i.e. if s 6 t ∈ I implies ϕ ( s ) 6 ϕ ( t ). Remark that neither a reg ular path nor a repara metr ization need b e injective. Definition 1.2. Two paths p, q : I → X are called r ep ar ametrization e quivalent if there exis t r eparametriza tio ns ϕ , ψ such that p ◦ ϕ = q ◦ ψ . W e will show later (Coro llary 3.3) that repa rametrizatio n equiv alence is indeed an equiv alence r e la tion. As in differential g eometry , we ar e interested in equiv alenc e classes of pa ths mo dulo reparametriza tion equiv alence. W e call these equiv a lence classes tra c es 1 in the space X . In particula r, we would like to know whether every tr ac e c an b e r epr esent e d by a r e gular p ath . The (p ositive) answer to this question in Prop osition 3 .7 is based on a closer lo ok a t the space of repar ametrizations of the unit interv al. 1.3. Outline of the article Section 2 contains a detailed study of repara metrizations (in their own right) and characterizes their b ehaviour essen tially b y an order-pre serving bijection b etw een the s e t of stop interv als a nd the set of stop v alues (Definition 1.1 and Pr o po sition 2.13). This pattern analysis allows to study comp ositions, and in particular, fac - torizations in the monoid o f repa rametrizations from an alg ebraic p oint of view. In particular , Pro po sition 2.1 8 sho ws tha t the spa ce o f all repar a metrizations “up to homeo morphisms” is a distributive lattice iso morphic to the lattice of countable subsets o f the unit interv al. Section 3 inv estiga tes the sp ac e of all paths in a Hausdorff space up to r eparamet- rization equiv alence. The main result (Theo rem 3 .6) states that tw o quotient spaces are in fact homeomor phic: the orbit space arising from the a ction of the group of all oriented homeomo r phisms of the unit interv al on the spa c e of r e gular paths (with given end points, cf. Definition 1.1) on the one side, and the spa c e of al l paths with given end points up to repar a metrization eq uiv alence (Definit ion 1.2); in particular, every tr ac e c an b e r epr esente d by a re gular p ath . It might b e a bit surprising tha t the pro of makes ess e n tial use of the res ults o n facto rizations of re pa rametrizatio ns from Sectio n 2. 1 with a geometric meaning; the not ion has nothing to do wi th algebraic traces. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 4 The fina l Section 4 deals with spaces of dir e cte d trac es (directed paths up to repara metr ization equiv alence) on a d-space (cf. Section 1.1 and Definition 4.1). Corollar y 4.5 confirms that the res ult o f Theo rem 3.6 has a n analo gue for directed paths in satur ate d (cf. Definition 4.3) d-spaces. This result is one of the starting po in ts for the (categor ical) inv estigations into inv a riants of directed spa ces in [ 15 ]. F urther mo re, it is sho wn how to re late reparametr ization equiv alence o f directed paths to thin dihomotopi es ; this re s ult is needed in the study [ 5 ] o f directed squares (“tw o -dimensional paths” ). 2. Reparametrizations 2.1. Stop and mo v e in terv als, stop v alues , stop maps The following definitio ns (extending Definition 1.1) and elementary results will mainly b e used for repara metrizations. F or the sa ke of generality , we will state and prov e them for general paths p : I → X in a Hausdor ff space X . Definition 2.1. 1. An element c ∈ X is called a p - st op value if there is a p -sto p int erv al J ∈ ∆ p with p ( J ) = { c } . W e let C p ⊆ X denote the se t of a ll p -stop v alues. 2. The map p induces the p - stop map F p : ∆ p → C p with F p ( J ) = c ⇔ p ( J ) = { c } . 3. An in terv al J ∈ P [ ] ( I ) is called a p - move int erval if it does no t contain any p -stop int erv al and if it is maxima l with that prop erty . 4. The set of all p -move in terv als will be denoted Γ p ⊆ P [ ] ( I ), a co lle ction of disjoint closed in terv als. W e let O p := S J ∈ Γ p int J ⊆ I denote the p - move set . Lemma 2. 2 . F or any p ath p : I → X , the set s of p -stop intervals ∆ p , of p -m ove intervals Γ P and of p -st op values C p ar e at most c ountable. Pr o of. The set O p and S J ∈ ∆ p int J of interior p oints in move, resp. stop interv als are op en subsets of I and thus unions of at most c oun tably many maximal open int erv als. Their clo sures co nstitute Γ p , resp. ∆ p . The stop v a lue set C p is at most countable as ima g e of ∆ p under the p -stop map F p . R emark 2.3 . This r esult is similar in spirit to the assertio n (relev ant fo r dis tr ibution functions in probability theory) that a nondecreasing function to an interv al has at most co untably many discontin uity p o in ts, cf. e.g. [ 13 , Sec. 11]. It is imp ortant to a nalyse the b oundary ∂ D p of the p -sto p set: It ca n b e decom- po sed as ∂ D p = ∂ 1 D p ∪ ∂ 2 D p as follows: • ∂ 1 D p = ∂ D p ∩ D p – the set of a ll b oundary points of in terv als in D p , a n at most countable set; • ∂ 2 D p = ∂ D p \ D p – the set of a ll (honest) accumulation p oints of these b ound- ary p o in ts. ∂ 2 D p can b e uncountable; compare Ex . 2.11. The mo ve set O p is the co mplemen t O p = I \ D p ⊂ I of the closure of D p . It do es o ccur that O p is empty; compare E x. 2.1 1. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 5 The following elementary tec hnical lemma concer ning stop sets will b e needed in the pr o of of Pr opo sition 3.7. Lemma 2.4. L et p : I → X denote a p ath and U ⊆ X an op en su bsp ac e. Then p − 1 ( U ) is a un ion of ( at most ) c ount ably many disjoint op en intervals, and for any maximal op en interval ] a, b [ in p − 1 ( U ) , [ a , c ] 6⊆ D p and [ c, b ] 6⊆ D p for every c ∈ ] a, b [ . Pr o of. As an o pen subset of I , p − 1 ( U ) is a union of (at most) countably man y disjoint op en interv als. Let ] a, b [ b e one of these, and let c ∈ ] a, b [. Then p ( c ) ∈ U, p ( a ) 6∈ U, p ( b ) 6∈ U . In particula r, p is neither constant on [ a, c ] nor on [ c, b ]. 2.2. Spaces of reparametrizations Within the set o f all self-maps o f the unit int erv al I fixing its b oundary po in ts, we study the following subs e ts : Definition 2.5. • Mo n + ( I ) := { ϕ : I → I | ϕ increa sing , ϕ (0) = 0 , ϕ (1) = 1 } ; • Rep + ( I ) := { ϕ ∈ Mon + ( I ) | ϕ contin uous } – the s et o f all increa s ing repara metrizations; • Homeo + ( I ) = { ρ ∈ Rep + ( I ) | ρ strictly increas ing } – the s et o f all increa s ing auto-homeo morphisms of the interv al. Note that Homeo + ( I ) ⊂ Rep + ( I ) ⊂ Mon + ( I ). The compact-op en to polo gy on the spa ce of al l con tinuous maps C ( I , I ) induces topolo g ies o n the la tter tw o spac e s Rep + ( I ) and Homeo + ( I ). Comp osition ◦ of maps tur ns Mon + ( I ) into a monoid, Rep + ( I ) into a top olog ical monoid and Homeo + ( I ) into a top olog ical group (con- sisting o f the units in Rep + ( I )). All three ma pping sets come equipp e d with a natural partial order: ϕ 6 ψ if and only if ϕ ( t ) 6 ψ ( t ) for all t ∈ I , and they for m complete lattices with resp ect to 6 . Least upp er bounds, resp. gr eatest low er b ounds are given by the max , resp. min of the functions inv olved: ( ϕ ∨ ψ )( t ) := max { ϕ ( t ) , ψ ( t ) } ( ϕ ∧ ψ )( t ) := min { ϕ ( t ) , ψ ( t ) } Lemma 2.6. 1. A l l thr e e sets Mon + ( I ) , Rep + ( I ) , Homeo + ( I ) ar e c onvex. In p ar- ticular, the latt er two sp ac es ar e c ont ra ctible. 2. Any two r ep ar ametrizations ϕ, ψ ∈ Rep + ( I ) ar e d-homotopic (cf. Definition 4.7 for the gener al definition), i.e. ther e ex ists a r ep ar ametrization ϕ, ψ 6 η ∈ Rep + ( I ) and incr e asing p aths G, H : ~ I → Rep + ( I ) with G (0) = ϕ, H (0 ) = ψ , G (1) = H (1) = η . Pr o of. 1. The sets are closed under conv ex combinations (1 − s ) ϕ + sψ . 2. F or η = ϕ ∨ ψ , define G ( s ) = (1 − s ) ϕ + sη and H ( s ) = (1 − s ) ψ + s η . A characterization of the e le men ts of Mon + ( I ), Rep + ( I ), a nd Homeo + ( I ) is achiev ed in the elementary Lemma 2.7. L et ϕ ∈ Mo n + ( I ) . Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 6 1. F or every interval J ⊆ I , the pr e-image ϕ − 1 ( J ) ⊆ I is an interval, as wel l. In p articular, ϕ − 1 ( a ) is an interval (p ossibly de gener ate) for every a ∈ I . 2. ϕ ∈ Rep + ( I ) if and only if ϕ is surjective . 3. ϕ ∈ Homeo + ( I ) if and only if ϕ is bijective . Pr o of. The o nly non-obvious s tatemen t is that surjectivity of ϕ ∈ Mon + ( I ) implies contin uit y; we show that the pre- image ϕ − 1 ( J ) of an o pen interv al J ⊂ I is o pen: Let d ∈ ϕ − 1 ( J ) and ϕ ( d ) = c ∈ J . Then ther e ex is t ε > 0 s uc h that [ c − ε, c + ε ] ⊆ J and d 1 , d 2 ∈ I such that ϕ ( d 1 ) = c − ε, ϕ ( d 2 ) = c + ε . Monotonicit y implies: ϕ ([ d 1 , d 2 ]) ⊆ [ c − ε , c + ε ] a nd d ′ 6∈ [ d 1 , d 2 ] ⇒ ϕ ( d ′ ) 6∈ ] c − ε, c + ε [. Surjectivit y implies: ϕ ([ d 1 , d 2 ]) = [ c − ε, c + ε ] ⊆ J ; hence d has an open neighbo urho o d in ϕ − 1 ( J ). The following information a bo ut ima ges of interv als under r eparametriza tions is needed in the pr o of of Prop ositio n 3.7: Lemma 2.8. L et a, b ∈ I and ϕ ∈ Rep + ( I ) . Then 1. ϕ ([ a, b ]) = [ ϕ ( a ) , ϕ ( b )] . 2. ] ϕ ( a ) , ϕ ( b )[ ⊆ ϕ (] a, b [) ⊆ [ ϕ ( a ) , ϕ ( b )] . 3. ϕ (] a , b [) 6 = ] ϕ ( a ) , ϕ ( b )[ if and only if ther e is c ∈ ] a, b [ such that [ a, c ] ⊆ D ϕ or [ c, b ] ⊆ D ϕ . Pr o of. Only the last assertio n re quires pro of. If [ a, c ] ⊆ D ϕ for some c ∈ ] a, b [, then ϕ ( a ) = ϕ ( c ) ∈ ϕ (] a, b [); similarly , if [ c, b ] ⊆ D ϕ , then ϕ ( b ) ∈ ϕ (] a, b [). F or the reverse direction, a ssume ϕ ( a ) ∈ ϕ (] a, b [), and let c ∈ ] a , b [ such that ϕ ( a ) = ϕ ( c ). Then ϕ ( a ) 6 ϕ ( t ) 6 ϕ ( c ) = ϕ ( a ) for any t ∈ [ a, c ], hence [ a, c ] ⊆ D ϕ . The other implication is s imilar. The fo llowing r esult deals with the rela tiv e size of the homeomo r phisms within the r e pa rametrizatio ns . It will b e needed in the pro of of the main result in Section 3 . Lemma 2.9 . In the top olo gy induc e d fr om the c omp act-op en top olo gy, b oth Homeo + ( I ) and its c omplement ar e dense in Rep + ( I ) . Pr o of. The compact-op en top o logy is induced by the supremum metric o n the space C ( I , I ) of a ll s e lf-maps of the interv al. Hence, for a given ϕ ∈ Rep + ( I ) and n ∈ N , we need to co ns truct ρ ∈ Homeo + ( I ) such that k ϕ − ρ k 6 1 n : Cho ose c k , 0 6 k 6 n , such tha t c 0 = 0 , c n = 1 a nd ϕ ( c k ) = k n ; clearly c k is strictly incr easing with k . Hence the piecewise linear map ρ giv en by ρ ( c k ) = k n is contained in Homeo + ( I ). F urther mo re, for x ∈ [ c k , c k +1 ] , k < n, w e have k n 6 ρ ( x ) , ϕ ( x ) 6 k +1 n , and thus k ϕ − ρ k 6 1 n . F or the same ϕ ∈ Rep + ( I ) and the same definition for c 0 and c 1 as ab ove, let ψ ∈ Rep + ( I ) \ Homeo + ( I ) b e given by ψ ( x ) = ϕ ( x ) , x > c 1 ; on the interv al [0 , c 1 ], we let ψ b e the piec e wise linear ma p with ψ (0) = ψ ( c 1 2 ) = 0 and ψ ( c 1 ) = ϕ ( c 1 ). See also Figure 1. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 7 1 3 2 3 1 c 1 c 2 c 3 Figure 1: Reparametriza tion ϕ (full line), homeo morphism ρ (broken line), and repara metr ization ψ (dotted line). c 1 c 2 c 3 ∆ 1 ∆ 2 ∆ 3 Figure 2 : Stop interv als and s top v a lues 2.3. Classification of reparametrizations In the following, we ar e mainly in terested in a n inv estiga tion of the algebraic monoid structure on Rep + ( I ) induced by c omp osition ◦ of maps. Note that there is another structure on the sets (spaces) Mon + ( I ), Rep + ( I ), a nd Homeo + ( I ), induced by c onc atenation of paths ( ϕ, ψ ) 7→ ϕ ∗ ψ ; ( ϕ ∗ ψ )( t ) = ( ϕ (2 t ) for t 6 1 2 ψ (2 t − 1) for t > 1 2 (2.1) This c o mpo sition do es no t induce a monoidal structure on these sets, as concate- nation is no t asso cia tiv e and do es not hav e units “o n the nose”. W e wish to describe a reparametriza tion ϕ ∈ Rep + ( I ) b y its ϕ -stop ma p F ϕ : ∆ ϕ → C ϕ illustrated in Figur e 2 and by the restriction of ϕ to its ϕ -mov e set O ϕ ⊆ I ; cf. Definition 2.1. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 8 c n +1 x + i x − k x − n +1 x + n +1 Figure 3 : Inserting the stop v alue c n +1 2.3.1. All c ount able sets in the interv al are stop v a lue sets Lemma 2.2 tells us that the ϕ -stop v alue set C ϕ ⊂ I of a reparametr ization ϕ is an a t most countable ordere d s ubs et of I . Rema rk also tha t automorphisms ρ ∈ Homeo + ( I ) are characterized b y the prop er ties ∆ ρ = C ρ = ∅ , resp. O ρ = I . Whic h (countable) subsets o f the unit in terv a l can b e rea lized as ϕ -stop sets of some ϕ ∈ Rep + ( I )? It is easy to construct (piecewise linear) repar a metrizations with a finite s et of stop v alues . Rather surprisingly , this c o nstruction can b e extended to arbitrar y (at most) c ountable sets of stop v alues: Lemma 2.10. F or every c ountable set C ⊂ I , ther e is a r ep ar ametrization ϕ ∈ Rep + ( I ) with C ϕ = C . Pr o of. Let C = { c 1 , c 2 , . . . } ⊂ I denote an injective enumeration o f the co un table set C . W e shall first construct a uniformly conv erg ent sequence o f piecewise linear maps ϕ n ∈ Rep + ( I ) , n > 0 with C ( ϕ n ) = { c 1 , . . . , c n } and th us ∆ ϕ n = { ϕ − 1 n ( c i ) | 1 6 i 6 n } . Let [ x − i , x + i ] = ϕ − 1 n ( c i ) , 1 6 i ; moreover, x − 0 = 0 , x + 0 = 1. W e start w ith ϕ 0 = id I . Inductively , assume ϕ n given as ab ov e. Among the x ± j , 1 6 j 6 n, c ho ose x + i , x − k such that c n +1 ∈ ϕ (] x + i , x − k [) and such that the restriction of ϕ n on that interv al is str ic tly inc r easing (and linear ). The ma p ϕ n +1 will differ from ϕ n only on (the in terior of ) that subinterv al [ c i , c k ]. The linear ma p on that interv al is replaced by a piecewise linear map, which comes in three pieces. The middle one tak es the cons tan t v alue c n +1 on a subinterv al [ x − n +1 , x + n +1 ]. On the left and right subinterv al, we connect linea rly to the v alues c i , c k on the boundar ies. See also Figur e 3. The interv a l [ x − n +1 , x + n +1 ] is chosen s o small that || ϕ n +1 − ϕ n || ∞ < 1 2 n ensuring uniform con vergence of the maps ϕ n to a contin uous map ϕ ∈ Rep + ( I ). F or this map ϕ , we have ∆ ϕ = { [ x − n , x + n ] | n > 0 } and C ϕ = C . Example 2.11 . If one choo s es a dense countable subset C ⊂ I , e.g ., C = Q ∩ I , then ϕ cannot b e injective on any non-trivial in terv al; he nc e O ϕ = ∅ and D ϕ = I . Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 9 The (uncountable) complement I \ D ϕ = ∂ 2 D ϕ do es not con tain any non-trivial int erv al. 2.3.2. Classificatio n What a re the essential data to descr ibe a repara metrization in terms of stop maps and move sets? Prop osition 2. 12. L et ϕ ∈ Rep + ( I ) denote a r ep ar ametrization. 1. The r ep ar ametrization ϕ induc es an order- preserving bijectio n F ϕ : ∆ ϕ → C ϕ . The r estriction ϕ | J : J → ϕ ( J ) to every move interval J ∈ Γ ϕ (Definition 2.1) is an (incr e asing) home omorphism onto its image. 2. The r estriction ϕ | D ϕ : D ϕ → C ϕ of ϕ to D ϕ is on to . 3. Two r ep ar ametrizations ϕ, ψ ∈ Rep + ( I ) with ∆ ϕ = ∆ ψ , C ϕ = C ψ , F ϕ = F ψ agr e e on D ϕ ; if, mor e over, ϕ | O ϕ = ψ | O ψ , then ϕ and ψ agr e e on al l of I . Pr o of. 1. The first statemen t is obvious from the definitions. F or the second, note that J ∩ D ϕ = ∂ J (or empty) for every such in terv al J . 2. E very element b ∈ C ϕ is the limit of a mono tone sequence of elements in C ϕ which is the image of a monoto ne a nd b ounded s e quence of e le ments in D ϕ ; the limit of such a seq ue nce exists and maps to b under ϕ . 3. By definition, ϕ and ψ agree on D ϕ ; by contin uity , they hav e to agree o n its closure D ϕ , as well. The last statement is obvious. A r eparametriza tion ϕ ∈ Rep + ( I ) is thus u niquely character iz ed b y its stop map F ϕ : ∆ ϕ → C ϕ and by a (fitting) c o llection of homeomor phisms ϕ | J : J → ϕ ( J ) , J ∈ Γ ϕ . Now we ask which conditions an “abstra ct” stop ma p ha s to satisfy in order to aris e from a g enu ine repa rametrization. W e star t with the following data : • ∆ ⊆ P [ ] ( I ) denotes an (at most) countable subset of disjoint close d in terv als – with a na tur al total order . • C ⊆ I deno tes a subset with the same cardinality as ∆. • F : ∆ → C deno tes an order- preserving bijection. Let ∆ − , ∆ + ⊆ I denote the set of lower, resp. upp er b ounda r ies of interv als in ∆. Define D := S J ∈ ∆ ∆ ⊂ I . Let O = I \ D . Since O is o pen, it is a disjoin t union O = S J ∈ Γ J of maximal op en interv als indexed by an (at most) co untable set Γ – po ssibly empty . F or every map G : ∆ → C w e define a map ϕ G : D → C b y ϕ G ( t ) = G ( J ) ⇔ t ∈ J . If F is order-pr eserving, then ϕ F is increasing . Moreov er: Prop osition 2. 13. 1. A r ep ar ametrization ϕ ∈ Rep + ( I ) satisfies t he fol lowi ng for eve ry p air of strictly monotonely c onver ging se quenc es x n ↑ x , y n ↓ y for Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 10 which x, x n ∈ (∆ ϕ ) + and y , y n ∈ (∆ ϕ ) − : x = y ⇒ lim ϕ ( x n ) = lim ϕ ( y n ) , (2.2) x < y ⇒ lim ϕ ( x n ) < lim ϕ ( y n ) , (2.3) x = 1 ⇒ lim ϕ ( x n ) = 1 , (2.4) y = 0 ⇒ lim ϕ ( y n ) = 0 , (2.5) 2. F or every or der pr eserving bije ction F : ∆ → C with ϕ F satisfying (1 ) – (4) ab ove fo r every p air of strictly monotonely c onver ging se quenc es x n ↑ x , y n ↓ y for which x, x n ∈ (∆ ϕ ) + and y , y n ∈ (∆ ϕ ) − , ther e exists a r ep ar ametriza- tion ψ ∈ Rep + ( I ) with ∆ ψ = ∆ , C ψ = C and F ψ = F . The set of al l su ch r ep ar ametrizations is in one-to-one c orr esp ondenc e with Q Γ Homeo + ( I ) . Pr o of. 1. By contin uit y , lim ϕ ( x n ) = ϕ ( x ) and lim ϕ ( y n ) = ϕ ( y ); this settles all but (2 .3). Supp ose ϕ ( x ) = ϕ ( y ) in (2). Then [ x, y ] is contained in a stop int erv al, hence x 6∈ ∆ + and y 6∈ ∆ − . 2. Fir st, we ex tend ϕ F to D : there is a unique c ontinuous (and incr easing!) extension o f ϕ F from D to D : lim x n ↑ x ϕ F ( x n ) ex is ts and is indep endent of the sequence x n by mo notonicity and agrees with lim y n ↓ x ϕ F ( y n ) by condi- tion (2.2) o f Prop os itio n 2.13. Mor eov er, we let ϕ F (0) = 0 and ϕ F (1) = 1, in accorda nce with (3 ) a nd (4) ab ov e. Let J = ] a J − , a J + [ ∈ Γ deno te a maximal open interv al. Its boundar y points a J − , a J + are contained in ∂ D unless p ossibly if a J − = 0 and/or a J + = 1, in whic h case we ar e cov er e d by (2.4) and/or (2.5) a bove. In conclusion, ϕ F is defined on ∂ J . Moreover, ϕ F ( a J − ) < ϕ F ( a J + ), since F is or der preser ving and injective and b ecause o f condition (2.3) ab ov e. Hence, ev ery co llection of strictly increa sing homeomor phisms be tw een [ a J − , a J + ] and [ ϕ F ( a J − ) , ϕ F ( a J + )] – prese rving endpoints – extends ϕ F to a contin uous increasing map ψ : I → I with ∆ ψ = ∆ , C ψ = C and F ψ = F . The set of all collections of such homeomorphisms is easily s een to b e in one- to -one corres p ondence with Q Γ Homeo + ( I ). 2.4. Comp osi tions and F actorizations W e shall now inv estigate the behaviour of Rep + ( I ) under comp osition and fac- torization in view of the description and cla ssification from Prop osition 2.1 3 a bove. W e need to in tro duce the follo wing notation: F or a (co n tinuous) map ψ : I → I , let ψ ∗ , ( ψ − 1 ) ∗ : P [ ] ( I ) → P [ ] ( I ) denote the maps induced on subinterv als: ψ ∗ ( J ) = ψ ( J ) ∈ P [ ] ( I ) , ( ψ − 1 ) ∗ ( J ) = ψ − 1 ( J ). 2.4.1. Comp osition o f repar ametrizations The results below follow ea sily from the definitions of s top-interv als, sto p-v alues and stop- maps: Lemma 2.14. L et ϕ, ψ ∈ Rep + ( I ) denote r ep ar ametrizations with asso ciate d stop maps F ϕ : ∆ ϕ → C ϕ , F ψ : ∆ ψ → C ψ . Then Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 11 1. ∆ ϕ ◦ ψ = { J ∈ ∆ ψ | F ψ ( J ) 6∈ D ϕ } ∪ ( ψ − 1 ) ∗ (∆ ϕ ) , 2. C ϕ ◦ ψ = ϕ ( C ψ ) ∪ C ϕ , 3. F ϕ ◦ ψ : ∆ ϕ ◦ ψ → C ϕ ◦ ψ is given by F ϕ ◦ ψ ( J ) = ( ϕ ( F ψ ( J )) , J ∈ ∆ ψ F ϕ ( ψ ∗ ( J )) , J ∈ ( ψ − 1 ) ∗ (∆ ϕ ) . Corollary 2. 15. L et ϕ, ψ ∈ Rep + ( I ) as in L emma 2.14. If ψ ∈ Homeo + ( I ) , r esp. ϕ ∈ Homeo + ( I ) , then 1. ∆ ϕ ◦ ψ = ( ψ − 1 ) ∗ (∆ ϕ ) , r esp. ∆ ϕ ◦ ψ = ∆ ψ , 2. C ϕ ◦ ψ = C ϕ , r esp. C ϕ ◦ ψ = ϕ ( C ψ ) , 3. F ϕ ◦ ψ = F ϕ ◦ ψ ∗ : ψ − 1 ∗ (∆ ϕ ) → C ϕ , r esp. F ϕ ◦ ψ = ϕ ◦ F ψ : ∆ ψ → ϕ ( C ψ ) . 2.4.2. F ac to rizations of repar ametrizations Also facto rizations can b e studied effectively using sto p- data of the repar a metriza- tions inv olved. It turns out that the following result on factor izations on the right will be a n essential to ol in Section 3: Prop osition 2. 16. L et η , ϕ ∈ Rep + ( I ) denote r ep ar ametrizations. 1. Ther e exists a lift ψ ∈ Re p + ( I ) in t he diagr am I ϕ   I η / / ψ @ @     I (2.6) if and only if C ϕ ⊆ C η . 2. If C ϕ ⊆ C η and C is any (at most) c ountable set with ϕ − 1 ( C η \ C ϕ ) ⊆ C ⊆ ϕ − 1 ( C η \ C ϕ ) ∪ D ϕ , then ther e exists su ch a lift ψ ∈ Rep + ( I ) with C ψ = C . In p articular, if C ϕ = C η , ther e exist a lift ψ ∈ Homeo + ( I ) . 3. Assu me C ϕ ⊆ C η and let ∆ 1 := F − 1 η ( C ϕ ) ⊆ ∆ η . Then the sp ac e of al l lifts { ψ | η = ϕ ◦ ψ } ⊆ Rep + ( I ) is in one-to-one-c orr esp ondenc e with Q L ∈ ∆ 1 Rep + ( L ) = Q ∆ 1 Rep + ( I ) . Pr o of. The “o nly if ” part of 1. follows immedia tely fro m Lemma 2.14.2. F or the “if ” pa rt, we analyse first the set-theoretic require men ts to a lift ψ on r elev ant subint erv als. T o this end, decompose ∆ η =: ∆ 1 ⊔ ∆ 2 with ∆ 1 := F − 1 η ( C ϕ ) and ∆ 2 = F − 1 η ( C η \ C ϕ ), and D η = D 1 ⊔ D 2 with D 1 := S J ∈ ∆ 1 and D 2 = S J ∈ ∆ 2 . W e construct a lift ψ : I = D 1 ∪ D 2 ∪ O η → I by co nsidering ea c h o f these three subse ts of I : Remark that ∆ 2 necessarily has to b e a s ubs et of ∆ ψ and that for J ∈ ∆ 2 , F ψ ( J ) has to b e the unique element of ϕ − 1 ( F η ( J )). On any move interv al K ∈ Γ η (cf. Definition 2.1.4), the restriction η | K : K → η ( K ) is an incr easing homeomorphism; in par ticular, η ( K ) ∩ C η consists at most of the tw o boundary po in ts. Hence, the restric tion ϕ | ϕ − 1 η ( K ) : ϕ − 1 η ( K ) → η ( K ) Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 12 is als o an increas ing homeomor phis m, since η ( K ) ∩ C ϕ ⊆ η ( K ) ∩ C η again co ns ists at most of the t wo bo undary po in ts. The restrictio n of ψ to K has to b e defined as ψ | K = ( ϕ | ϕ − 1 ηK ) − 1 ◦ η | K ; it is onto ϕ − 1 η K . On any in terv al L ∈ ∆ 1 , the restriction of ψ to L ca n b e defined as any increa sing contin uous map ψ | L : L → F − 1 ϕ ( F η ( L )) ∈ ∆ ϕ resp ecting the b oundar y p oints. The map ψ : ~ I → ~ I th us defined a ltogether is by definition a lift, it is increa s ing and sur jectiv e. Lemma 2.7.2 settles 1. The only freedom in the cons truction of a lift ψ is the choice of increa sing con- tin uous maps on the interv als L ∈ ∆ 1 (with given end p oints). As in Lemma 2 .10, we can construct the set of stop v a lues (on D 1 ) to b e any countable subset of D ϕ . T o these o ne has of course to add the pre-images of the stop v alues in C η \ C ϕ . This settles 2. a nd 3. W e will also make use o f the following result on factoriza tions on the left . Prop osition 2. 17. L et η , ϕ ∈ Rep + ( I ) denote r ep ar ametrizations. 1. Ther e exists a factorization with ψ ∈ Rep + ( I ) in t he diagr am I ϕ   η / / I I ψ @ @     (2.7) if and only if ther e exists a map i ϕη : ∆ ϕ → ∆ η such that J ⊆ i ϕη ( J ) for every J ∈ ∆ ϕ . (∆ ϕ is a r efinement of ∆ η ) . 2. If it exists, the factor ψ ∈ Rep + ( I ) is uniquely determine d and satisfies • C ψ = C η \ { F η ( J ) | J ∈ ∆ η ∩ ∆ ϕ } , • ∆ ψ = { ϕ ( K ) | K ∈ ∆ η \ ∆ ϕ } , • if K ∈ ∆ ψ , then F ψ ( K ) is the unique element of η ( ϕ − 1 ( K )) , K ∈ ∆ ψ . Pr o of. A lift ψ a s in (2.7) has to sa tisfy ψ ( x ) := η ( ϕ − 1 ( x )). It is well-defined (and then unique and increasing ) if and o nly if the condition of Pro po s ition 2.17 is satisfied. Since η is on to, ψ is onto as well, and th us c ont inuous by Lemma 2.7.2. The description of the inv ariants of ψ follows b y insp ection. 2.5. The alge bra of repara metrizations up to homeomo rphi sms Consider the gr oup action Rep + ( I ) × Ho meo + ( I ) → Rep + ( I ) given by c o mpo si- tion o n the right. An element in the quo tient space Rep + ( I ) / Homeo + ( I ) preserves the set of stop v alues, whereas the exact distr ibution of stop interv als ov er the interv al is factor ed out. Using the fa ctorization to ols from Section 2.4 ab ov e, this intu ition will be ma de more formal in Pro po s itions 2.18 a nd 2.22 b elow. Consider the preorder on Rep + ( I ) (differen t from the one co nsidered in Se c - tion 2 .2) given b y ϕ 6 ψ ⇔ ∃ η ∈ Rep + ( I ) : ψ = ϕ ◦ η ( ⇔ C ϕ ⊆ C ψ by Prop osition 2 .1 6.1). This preorder factors to yield a partial order on the quotient Rep + ( I ) / Homeo + ( I ) since ψ = ϕ ◦ η = ( ϕ ◦ ρ ) ◦ ( ρ − 1 ◦ η ) for ρ ∈ Homeo + ( I ). Moreov er, let us consider the set P c ( I ) of c ount able subs ets of I with the par tial order g iv en by inclusion. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 13 Prop osition 2.18 . The map C : Rep + ( I ) / Homeo + ( I ) → P c ( I ) given by C ( ϕ ) = C ϕ is an or der-pr eserving bije ction. Pr o of. By Coro llary 2.15.2, the map C is well-defined; b y Lemma 2.14, it is order - preserving , and by Lemma 2 .10, it is surjective. Given t wo r eparametriza tions with the s ame set of stop v a lues, Pr op osition 2 .16 shows that o ne can co ns truct a lift ψ from one into the other that is a homeomor phism ( C ψ = ∅ ); as a c o nsequence, C is also injective. Prop osition 2.19. F or every ϕ 1 , ϕ 2 ∈ Rep + ( I ) , ther e ex ist ψ 1 , ψ 2 ∈ Rep + ( I ) c ompleting the diagr am I ψ 1 / / _ _ _ ψ 2      I ϕ 1   I ϕ 2 / / I with C ϕ 1 ◦ ψ 1 = C ϕ 2 ◦ ψ 2 = C ϕ 1 ∪ C ϕ 2 . Pr o of. Using Lemma 2.10, construct ψ 1 ∈ Rep + ( I ) with C ψ 1 = ϕ − 1 1 ( C ϕ 2 \ C ϕ 1 ) and hence C ϕ 1 ◦ ψ 1 = C ϕ 1 ∪ C ϕ 2 (cf. Lemma 2.1 4.2). Using Pr opo sition 2.16, construct a lift ψ 2 in the dia gram I ϕ 2   I ϕ 1 ◦ ψ 1 / / ψ 2 @ @     I with C ψ 2 = ϕ − 1 2 ( C ϕ 1 \ C ϕ 2 ), i.e., without in tro ducing sup e r fluous extra stop v alues. R emark 2.20 . In gener al, it is not p oss ible to complete the dual diagr a m I ϕ 1 / / ϕ 2   I ψ 1      I ψ 2 / / _ _ _ I since D ψ 1 ◦ ϕ 1 = D ψ 2 ◦ ϕ 2 ⊇ D ϕ 1 ∪ D ϕ 2 . The la tter set might b e the entire interv a l I which is imp ossible for a repar ametrization. Is there a natur a l way to construct fr om tw o re parametrizatio ns a third one (a common fac to r) with a set o f stop v alues that is just the interse ction of the sets of stop v alue s of the g iven ones? In o rder to hav e the stop interv als a ppea r in prop er order, it is neces sary to mo dify one of the repa rametrizatio ns by a homeo mo rphism first (which is no t a problem if one works in the quo tien t Rep + ( I ) / Homeo + ( I ) !) Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 14 Prop osition 2.21. F or every ϕ 1 , ϕ 2 ∈ Rep + ( I ) , ther e exist ρ ∈ Homeo + ( I ) , ψ 1 , ψ 2 , ϕ ∈ Rep + ( I ) c ompleting the diagr am I ϕ 1 / / ρ      ϕ   . . . . . . . I I ϕ 2   I I ψ 2 o o _ _ _ ψ 1 O O        and such that C ϕ = C ϕ 1 ∩ C ϕ 2 . Pr o of. Define ∆ ϕ := F − 1 ϕ 1 ( C ϕ 1 ∩ C ϕ 2 ) ⊆ ∆ ϕ 1 , C ϕ := C ϕ 1 ∩ C ϕ 2 and define F ϕ : ∆ ϕ → C ϕ as the re striction of F ϕ 1 . By Pr op osition 2 .13, ther e exists ϕ ∈ Rep + ( I ) with F ϕ as its stop ma p. By Pr op osition 2 .17, there exis ts a lift ψ 1 ∈ Rep + ( I ) in the r ight triangle of the diagr a m ab ov e. In general, the reparametrization ϕ constr uc ted ab ov e does not factor ov er ϕ 2 immediately , cf. P rop osition 2.1 7. W e need a ” correction” homeo morphism ρ ∈ Homeo + ( I ) whose restr iction to D ϕ fits in to D ϕ ρ / / _ _ _ F ϕ   D ϕ 2 F ϕ 2   C ϕ ⊆ / / C ϕ 2 . On interv als J ∈ ∆ ϕ , ρ can b e chosen as the increa sing linea r map sending J onto F − 1 ϕ 2 ( F ϕ ( J )) and then extended from D ϕ to I as a homeo mo rphism as in the pro of of Prop ositio n 2.1 3. The c o ndition of Prop os itio n 2.1 7 is now satisfied to gua rantee a lift o f ϕ ov er ϕ 2 ◦ ρ in the left triangle of the diagram a bove. Using the bijection from Pro pos ition 2.18, one may introduce binar y op era tions on the quotient Rep + ( I ) / Homeo + ( I ) in a purely a lgebraic manner, i.e., one may pull back the op era tions given by set union a nd intersection o n P c ( I ). The results of this section allow us to give these o per ations an int rins ic meaning in terms of repara metr izations. Using the notation from Prop osition 2 .19, an oper ation ∨ (“least common multiple”) is defined by [ ϕ 1 ] ∨ [ ϕ 2 ] := [ ϕ 1 ◦ ψ 1 ] = [ ϕ 2 ◦ ψ 2 ]. Likewise, [ ϕ 1 ] ∧ [ ϕ 2 ] can be re pr esented by the r eparametriza tio n ϕ from Pro p os itio n 2.2 1. Altogether we obtain: Prop osition 2. 22. The op er ations ∨ , ∧ : Rep + ( I ) / Homeo + ( I ) × Rep + ( I ) / Homeo + ( I ) → Rep + ( I ) / Homeo + ( I ) turn Rep + ( I ) / Homeo + ( I ) into a distributive lattic e with the class r epr esente d by Homeo + ( I ) as a glob al minimum. The map C : (Rep + ( I ) / Homeo + ( I ) , ∨ , ∧ ) → ( P c ( I ) , ∪ , ∩ ) fr om Pr op osition 2. 18 is then an isomorphi sm of distributive lattic es. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 15 3. Spaces of traces 3.1. The compact-op e n top ology on path spaces W e no w tur n to spaces of equiv alence classes of paths in a Hausdorff space X . T o start with, w e give a handy c har acterization of the compac t-op en top olog y o f the space of all pa ths P ( X ) = X I in X . By definition, is has as a basis the sets P ( X )( C , V ) := { p ∈ P ( X ) | p ( C i ) ⊂ V i } , indexed b y c ollections C = { C 1 , . . . , C n } of compact s ubsets C i ⊆ I , resp. V = { V 1 , . . . , V n } o f o p en subsets V i ⊆ X , n ∈ N . A par tition A = { 0 = a 0 < a 1 < · · · < a n − 1 < a n = 1 } of the unit interv al I gives rise to the collection A = { [ a j − 1 , a j ] } of close d in terv a ls. W e write P ( X )( A, U ) = P ( X )( A , U ) ⊆ P ( X ). Lemma 3 . 1. The sets P ( X )( A, U ) ⊂ P ( X ) form a b asis for the c omp act-op en top olo gy of P ( X ) . Pr o of. Compare [ 4 , Ch. XII] for the first pa r t of this pro of. Every path q in a basis se t P ( X )( C , V ) satisfies C i ⊂ q − 1 ( V i ); hence C i is cov ere d by the c o nnected comp onents of q − 1 ( V i ), a set of o pen disjoint interv als. Finitely many of those, say the in terv als J ij , cov er C i ; for each of those let I ij = [min( J ij ∩ C i ) , max( J ij ∩ C i )]. Then I ij ⊂ J ij ⊆ q − 1 ( V i ) whenc e q ( I ij ) ⊆ V i ; moreov er, C i ⊆ S j I ij . Let A ′ = S i,j ∂ I ij ⊂ I deno te the finite subset of interv al b oundary p oints. Then, for t wo successive p oints a, b ∈ A ′ we hav e: q ([ a, b ]) ⊆ T [ a,b ] ⊆ I ij V i ; this intersection is to be in terpreted as the set X if the index set is empty . W e slig h tly alter the partition A ′ and arrive at a new par tition A if o ne or several b o undary p oints are bo th the upp er and the lower boundary of some of the interv als I ij : If a is such an upper and a low er b oundar y of some of the in terv als I ij and hence q ( a ) ∈ T a ∈ I ij V i , we replace a with a pair a − , a + satisfying a ′ < a − < a < a + < a ′′ for all a ′ < a < a ′′ in A ′ and such that q ([ a − , a + ]) ⊆ T a ∈ I ij V i . Let 0 = a 0 < a 1 < · · · < a n − 1 < a n = 1 deno te the elements of A in the induced order, and let U denote the collection of op e n sets U j := T q ([ a j − 1 ,a j ]) ⊆ V i V i . The o pen set P ( X )( C , V ) fro m ab ov e then contains the op en neig h b ourho o d P ( X )( A, U ) of q . 3.2. Regular traces v ersus trac es In this section we compare se veral spa ces of paths in a Hausdo rff space X up t o r ep ar ametrization . Extending Definition 1.1, we get Definition 3.2. 1. A path p : I → X in a top olog ical space X is s a id to be r e gular if ∆ p = ∅ or if ∆ p = { I } . 2. The set of re g ular paths in X is denoted R ( X ) and re garded as a subs pace of P ( X ) = X I . 3. The spa ces of (regular) paths p starting in x ∈ X and ending in y ∈ X ( p (0) = x , p (1) = y ) are denoted by R ( X )( x, y ) ⊂ P ( X )( x, y ); they are equipp e d with the induced to po lo gies. Comp osition on the right yields a gro up a ction of the top olog ical group Homeo + ( I ) on R ( X ) and a mono id a c tion of the to p olo gical monoid Rep + ( I ) on P ( X ). Thes e actions resp ect the decomp ositio ns in subspaces R ( X )( x, y ), resp. P ( X )( x, y ). Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 16 In Definition 1.2, we called pa ths p, q ∈ Rep + ( I ) repara metrization equiv alent if there exis t r eparametriza tio ns ϕ, ψ ∈ Rep + ( I ) such that p ◦ ϕ = q ◦ ψ . Corollary 3.3. 1. R ep ar ametrization e quivalenc e of p aths is an e quivalenc e r e- lation. 2. Two r ep ar ametrization e quivalent p aths ar e thinly homotopic. F or a definition of thin homotopy see [ 11 ]; esse ntially , a homotopy H : I × I → X fixing the endp oint s is thin if it facto rs thr o ugh a tr e e (the geometr ic realisatio n of a n a cyclic one- dimensional simplicia l set), i.e. if H : I × I → J → X for a tr ee J . Remark that reparametriz a tion eq uiv alent pa ths hav e the same image: p ( I ) = q ( I ) ⊆ X . This is not necessarily true for thinly homo topic paths; e.g., the cancella tion homotopy [ 16 , p. 48] b etw een the co ncatenation of a path with its inv e rse and the constant path is thin. Pr o of. 1. Reparametrization equiv ale nce is clearly a reflexive and symmetric re- lation. F or transitivity , let p, q , r ∈ P ( X ) denote thre e paths and a ssume that p ◦ ϕ = q ◦ ψ and q ◦ ϕ ′ = r ◦ ψ ′ for repar ametrizations ϕ, ϕ ′ , ψ , ψ ′ ∈ Rep + ( I ). By Prop osition 2.19, there ar e η , η ′ ∈ Rep + ( I ) suc h that ψ ◦ η = ϕ ′ ◦ η ′ ; hence p ◦ ϕ ◦ η = r ◦ ψ ′ ◦ η ′ . 2. It is enough to sho w that p and p ◦ ϕ are thinly homotopic for every ϕ ∈ Rep + ( I ); consider the ho motopy H : I × I → I p → X, H ( s, t ) = p ((1 − s ) t + sϕ ( t )), that even factors ov er I . F acto ring out the resp ective e q uiv alence relations given by the actions ab ov e, we arrive at quotien t s paces T R ( X ) = R ( X ) / Homeo + ( I ) , resp. T ( X ) = P ( X ) / Rep + ( I ) with s ubs paces T R ( X )( x, y ) = R ( X )( x, y ) / Homeo + ( I ) , resp. T ( X )( x, y ) = P ( X )( x, y ) / Rep + ( I ) for x, y ∈ X . They are considered as spaces of (regula r) tr ac es of paths in X and should b e compa r ed to the notio ns of curves or regula r curves in elementary differ- ent ial geometr y . These space s ca n be org anised in a top ological categor y T ( X ) (and likewise T R ( X )) with the elements of X as o b jects, with the top olo gical spaces T ( X )( x, y ) as morphism fr o m x to y and with a compo sition T ( X )( x, y ) × T ( X )( y , z ) → T ( X )( x, z ) induced by co ncatenation. Remar k that one do es n ot o btain a category structure on P ( X ) since concatenation is not a s so ciative “o n the nos e ” . The catego ries T ( X ) and their dir ected rela tives ar e used as impo rtant too ls in [ 15 ]. Lemma 3.4. L et x, y b e elements of a Hausdorff sp ac e X and let p ∈ R ( X )( x, y ) , ϕ ∈ Rep + ( I ) . If p ◦ ϕ = p , then ϕ = id I or p is c onstant. In p articular, for x 6 = y , the action of Homeo + ( I ) on R ( X )( x, y ) is free . Pr o of. If ϕ 6 = id I , there exists an in terv al J = [ a, b ] ⊆ I with ϕ ( a ) = a, ϕ ( b ) = b and, without los s of generality , ϕ ( t ) < t for all a < t < b . F or all these t we conclude that p ( t ) = p ( ϕ ( t )) = p ( ϕ n ( t )) for all n > 0, a nd hence that p ( t ) = p (lim n →∞ ( ϕ n ( t ))) = p ( a ). In particular, there is a non-trivial interv al on which p is co nstant ; this is not allow ed for a regular path unless p is constant on the en tire unit int er v al I (and th us x = y ). Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 17 Corollary 3.5. L et x 6 = y b e elements of a top olo gic al sp ac e X . The quotient map R ( X )( x, y ) → T R ( X )( x, y ) is a we ak homotopy e quivalenc e. Pr o of. The free g roup a ction yields a fibratio n with contractible fib er Homeo + ( I ). It is not clear to the autho r s whether one ca n sort out conditions under which the q uotien t map is a genuine homotopy equiv alence. 3.3. Spaces of (regular) traces F or x, y ∈ X , the inclusion map R ( X )( x, y ) ֒ → P ( X )( x, y ) induces a natura l map i : T R ( X )( x, y ) → T ( X )( x, y ) b etw ee n the corr espo nding quotient trace spaces. The main aim o f this section is a pro of o f Theorem 3.6. F or every two elements x, y ∈ X of a Hausdorff sp ac e X , t he map i : T R ( X )( x, y ) → T ( X )( x, y ) is a ho meomorphism . In particula r, every tra ce can b e r epresented by a regular trace (cf. Pro pos ition 3.7 below). It turns out that man y of the results on reparametriz ations from the preceding section w ill b e used in the pro o f. In a first step, we show that the map i is sur jective: Prop osition 3.7. F or every p ath p ∈ P ( X ) , t her e exists a r e gular p ath q ∈ R ( X ) and a r ep ar ametrization ϕ ∈ Rep + ( I ) s uch that p = q ◦ ϕ . Pr o of. F or ev ery interv al J ⊆ I let m ( J ) denote its midp oint. Let m : ∆ p → I denote the map J 7→ m ( J ) with image C := m (∆ p ) ⊆ I . In order to arrive a t a repara metr ization with stop map m , we check the co nditions fr o m Prop osition 2.1 3 for the order -preserv ing bijection m : ∆ p → C : Let x n = max ( J n ) ↑ x ∈ I denote a strictly monotone sequence. Then the midpoints con verge a s w ell: m ( J n ) ↑ x . Likewise for a decr easing sequenc e o f low er b oundar ies a nd co r resp onding mid- po in ts. F rom Prop osition 2.13 we conclude that there exis ts a reparametriza tion ϕ ∈ Rep + ( I ) with ∆ ϕ = ∆ p . Hence, there is a set-theoretic factoriza tion I p / / ϕ   X I q ? ?     through a r e gular map q : I → X . T o chec k that q is contin uous, choose an op en s e t U ⊂ X and note that q − 1 ( U ) = ϕp − 1 ( U ). Com bining Lemma 2.4 and Lemma 2.8.3, w e conclude that p − 1 ( U ) is a union of maximal disjoint o pen interv als that are a ll mapp ed ont o op e n in terv als under ϕ . Hence q − 1 ( U ) is op en as well. Prop osition 3.8. Two r e gular p aths p, q ∈ R ( X )( x, y ) that ar e r ep ar ametrization e quivalent ar e in fact strictly r ep ar ametrization e quivalent, i.e., ther e exist s a home- omorphism η ∈ Homeo + ( I ) s uch that q = p ◦ η . The pro o f of Pro po s ition 3.8 (and also that of Theorem 3.6) makes use o f Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 18 Lemma 3. 9 . L et p ∈ R ( X )( x, y ) , p ′ ∈ P ( X )( x, y ) , ϕ, ϕ ′ ∈ Rep + ( I ) with p ◦ ϕ = p ′ ◦ ϕ ′ . Then ther e exists η ∈ Rep + ( I ) with p ◦ η = p ′ . Unless p is c onstant, η is unique. Pr o of. W e apply P rop osition 2 .17 to prov e the existenc e of such a re parametrizatio n η : F or every in terv a l J ′ ∈ ∆ ϕ ′ , there is a unique J ∈ ∆ p ′ ◦ ϕ ′ = ∆ p ◦ ϕ = ∆ ϕ such that J ′ ⊆ J . Hence, there exists a unique η ∈ Rep + ( I ) with ϕ = η ◦ ϕ ′ , whence p ′ ◦ ϕ ′ = p ◦ ϕ = ( p ◦ η ) ◦ ϕ ′ . Since ϕ ′ is onto, we conclude that p ′ = p ◦ η . T o prove uniqueness o f the factor η , suppose that η 1 , η 2 ∈ Rep + ( I ) with p ◦ η 1 = p ◦ η 2 . If η 1 6 = η 2 , o ne can choo se an interv al J = [ a, b ] suc h that η 1 ( a ) = η 2 ( a ) , η 1 ( b ) = η 2 ( b ) and η 1 ( t ) < η 2 ( t ) for a < t < b (or vice v ersa ). Given a < t 0 = t ′ 0 < b , c ho ose an increa sing sequence t i and a decreas ing sequence t ′ i such tha t η 1 ( t i +1 ) = η 2 ( t i ), resp. η 2 ( t ′ i +1 ) = η 1 ( t i ). Both s equences converge to , say , a 6 T ′ < T 6 b . Since η 1 ( T ) = η 2 ( T ) a nd η 1 ( T ′ ) = η 2 ( T ′ ), we must hav e T ′ = a, T = b . On the other hand, p ( η 2 ( t i )) = p ( η 1 ( t i +1 )) = p ( η 2 ( t i +1 )) and hence p ( t 0 ) = p ( b ), and, by a s imilar argument: p ( t 0 ) = p ( a ). Since t 0 ∈ ] a, b [ was chosen arbitra rily , p has to b e co ns tan t on the int er v al J = [ a , b ]; this is imp ossible for a r egular path p unless it is co nstant . Pr o of. (of Pr op o sition 3.8) Let p, q ∈ R ( X )( x, y ) be r eparametriza tio n equiv alent, i.e., there exist reparametrizations ϕ, ψ ∈ Rep + ( I ) s uch tha t p ◦ ϕ = q ◦ ψ . If one of p, q is co nstant, the other is as w ell. Assume that neither of them is constant. By Lemma 3.9, there is a repa rametrization ρ ∈ Rep + ( I ) with q = p ◦ ρ . Since q is r egular and non-consta n t, ρ has to b e injectiv e and thus a homeomor phism; in particular, p and q r epresent the same element in T R ( X )( x, y ). Pr o of. (of Theo rem 3 .6) F rom Prop ositions 3.7 and 3.8 w e conclude that the map i from Theor em 3.6 is a co n tinuous bijection. T o s ee that it is also o pen, the following diagram will b e useful: R ( X )( x, y ) ⊆ / / Q R   P ( X )( x, y ) Q   T R ( X )( x, y ) i / / T ( X )( x, y ) (3.1) By Lemma 3.1, a basis for the to p olo gy o f R ( X )( x, y ) is given by the s ets R ( X )( A, U ; x, y ) = { p ∈ R ( X )( x, y ) | p ([ a j − 1 , a j ]) ⊆ U j } indexed b y all partitio ns A = { 0 = a 0 < · · · < a n = 1 } of I and collections of n op en s ets U = { U 1 , . . . , U n } , n ∈ N . The sets Q R ( R ( X )( A, U ; x, y )) for m a basis for the quotien t topolog y on T R ( X )( x, y ) since Q − 1 R ( Q R ( R ( X )( A, U ; x, y ))) = [ ρ ∈ Homeo + ( I ) ρ · R ( X )( A, U ; x, y )) = [ ρ ∈ Homeo + ( I ) R ( X )( ρ − 1 ( A ) , U ; x, y )) , and the latter s et is op en in R ( X )( x, y ). Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 19 W e need to show that the sets i ( Q R ( R ( X )( A, U ; x, y ))) = Q ( R ( X )( A, U ; x, y )) are open in T ( X )( x, y ), i.e., that the sets Q − 1 ( Q ( R ( X )( A, U ; x, y ))) are o pen in P ( X )( x, y ). T o this end, w e note first that Q − 1 ( Q ( R ( X )( A, U ; x, y ))) = Re p + ( I ) · R ( X )( A, U ; x, y ): The inclusion ⊇ is o b vio us . If p ∈ R ( X )( A, U ; x, y ) and p ′ ∈ P ( X )( x, y ), ϕ, ϕ ′ ∈ Rep + ( I ) are such tha t p ◦ ϕ = p ′ ◦ ϕ ′ , then Lemma 3.9 yields η ∈ Rep + ( I ) such that p ′ = p ◦ η , hence p ′ ∈ Rep + ( I ) · R ( X )( A, U ; x, y ). In the next tw o steps we show that every element q ∈ Rep + ( I ) · R ( X )( A, U ; x, y ) has a n op en neig h b ourho o d in that set. F or elements in R ( X )( A, U ; x, y ) it suffices to chec k that P ( X )( A, U ; x, y ) ⊆ Rep + ( I ) · R ( X )( A, U ; x, y ): Acco rding to Pr op o- sition 3.7, a path p ∈ P ( X )( A, U ; x, y ) can b e factored in the form p = q ◦ ϕ with q ∈ R ( X )( x, y ) , ϕ ∈ Rep + ( I ); the factor s then hav e to satisfy ϕ ([ a i − 1 , a i ]) ⊆ q − 1 ( U i ). By Lemma 2.9, there is an appr oximation ρ ∈ Ho meo + ( I ) to ϕ s atisfying ρ ([ a i − 1 , a i ]) ⊆ q − 1 ( U i ) for all i . W e can th us factor p = ( q ◦ ρ ) ◦ ( ρ − 1 ◦ ϕ ) with q ◦ ρ ∈ R ( X )( A, U ; x, y ), a nd hence p ∈ Rep + ( I ) · R ( X )( A, U ; x, y ). Now let q ∈ R ( X )( A, U ; x, y ) and ϕ ∈ Rep + ( I ). Let B = { b 0 = 0 < · · · < b n = 1 } denote any partition o f I with ϕ ( B ) = A . Then q ◦ ϕ ∈ P ( X )( B , U ; x, y ), which is op en in P ( X )( x, y ). Hence it will s uffice to show that P ( X )( B , U ; x, y ) ⊆ Rep + ( I ) · R ( X )( A, U ; x, y ). T o this end, cho ose ρ ∈ Homeo + ( I ) such that ρ ( b i ) = a i . Let p ∈ P ( X )( B , U ; x, y ) a nd write p = p 1 ◦ ρ . Then p 1 ∈ P ( X )( A, U ; x, y ) ⊆ Rep + ( I ) · R ( X )( A, U ; x, y ) by the previous step, a nd hence also p ∈ Rep + ( I ) · R ( X )( A, U ; x, y ). R emark 3.10 . All four spaces in (3.1) a re (a t least) weakly ho motopy equiv a le n t: If x, y are not in the sa me path c o mpo nen t, they are all empty . Otherwise, P ( X )( x, y ) is homotopy equiv alent to the lo op spac e Ω( X )( x ) based at x ; likewise, R ( X )( x, y ) is homotopy eq uiv alent to the s pace o f regular lo ops Ω R ( X )( x ) based at x . Both lo op spaces a re fibres in fibrations P R ( X ) ↓ X , resp. P ( X ) ↓ X ov er X with contractible total s pa ces P R ( X ; x ), r esp. P ( X ; x ) of (regula r) paths starting at x . The Five Lemma shows that the inclusion ma p Ω R ( X )( x ) ֒ → Ω( X )( x ) is a w eak homotopy equiv alenc e . Note that Lemma 3.9 allows us to g ive the following “backw ards ” characteriza tio n of repara metrization equiv alence : Prop osition 3.11 . Paths p, q ∈ P ( X ) ar e r ep ar ametrization e qu ivalent if and only if ther e exists r ∈ P ( X ) and ϕ, ψ ∈ Rep + ( I ) s uch that p = r ◦ ϕ and q = r ◦ ψ . Pr o of. F or the “only if ” part, supp ose p ◦ ϕ 1 = q ◦ ψ 1 , ϕ 1 , ψ 1 ∈ Rep + ( I ). W e use Prop osition 3.7 to write p = r 1 ◦ η 1 and q = r 2 ◦ η 2 , with r 1 , r 2 ∈ R ( X ) and η 1 , η 2 ∈ Rep + ( I ). Then r 1 ◦ η 1 ◦ ϕ 1 = r 2 ◦ η 2 ◦ ψ 1 ; b y Lemma 3.9, there exists η ∈ Rep + ( I ) such that r 2 = r 1 ◦ η , whence q = r 1 ◦ η ◦ η 2 . The r everse implication is clear by P rop osition 2 .19; if ϕ 2 , ψ 2 ∈ Rep + ( I ) a re such that ϕ ◦ ϕ 2 = ψ ◦ ψ 2 , then p ◦ ϕ 2 = r ◦ ϕ ◦ ϕ 2 = r ◦ ψ ◦ ψ 2 = q ◦ ψ 2 . Let us finally note the following consequence of P rop osition 3.7: Definition 3.12. A path p : I → X is called lo op-fr e e if p ( s ) = p ( t ) for an y s < t ∈ I implies that the restric tion p | [ s,t ] is the constant path. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 20 Note that a lo op-free r e gu lar path is either co nstant or injective. Corollary 3.13. A lo op-fr e e p ath p : I → X in a Hausdorff sp ac e X has an image p ( I ) ⊆ X that is either a p oint or home omorphic to I . Pr o of. By Prop osition 3.7, there is a facto rization p = q ◦ ϕ w ith q a lo op-free regular and th us e ither constant or injective path. In the second case, q is a cont inuous bijection from the compact spa ce I to its image q ( I ) = p ( I ) ⊆ X . The claim follo ws since X is Ha usdorff. 4. Directed traces Originally motiv a ted by mo dels aris ing in concurrency theo r y in theor etical c om- puter science, an in vestigation of topolo gical spaces w ith “ preferred directions” has bee n launched under the title “Dir e cted Algebr aic T o po logy”. V a r ious frameworks (lo cal p o-spa ces [ 6 ], d-spaces [ 9 ], flows [ 7 ] and others) have been suggested, all mo difying in v arious ways co ncepts fr om elementary alg e braic top olo gy , in partic- ular r eplacing relev ant groups o r gr oup oids by catego ries. The d-spaces intro duced by M. Gr andis [ 9 ] hav e turned out to giv e rise to a particular ly success ful wa y to combine homoto p y theo retical and categorical metho ds for the study o f spaces with preferred directions. The que s tion whether one can neglect re pa rametrizatio ns in a homotopy theo- retical study of spaces of dir ected paths w as one of the or iginal motiv ations for this article. This is – under a certain natural condition – affirmed in Co rollary 4 .5, which is used as a sta rting p oint for the homotopy theor etical study of “directed spa ces” in [ 1 5 ]. Definition 4.1 ([ 9 ]) . A d-sp ac e is a top ological s pa ce X together with a set ~ P ( X ) ⊆ P ( X ) of contin uous pa ths I → X such that 1. ~ P ( X ) contains all constant paths; 2. p ◦ ϕ ∈ ~ P ( X ) fo r any p ∈ ~ P ( X ) a nd any co n tinuous increa sing (not nece s sarily surjective)s map ϕ : I → I ; 3. for all p , q ∈ ~ P ( X ) such that p (1) = q (0), their concatenation p ∗ q ∈ ~ P ( X ), cf. (2 .1). Elements of ~ P ( X ) are called d- p aths . ~ P ( X ) ⊆ P ( X ) is given the subspace top ology (of the co mpact-op en top olog y). Definition 4.2. A d-map b etw een d-spaces X , Y is a co n tinuous mapping f : X → Y satisfying p ∈ ~ P ( X ) ⇒ f ◦ p ∈ ~ P ( Y ). Isomor phisms in the categor y of d-spaces are called d-home omorphisms . The d-interv al ~ I is g iven the standa r d d-structur e ~ P ( I ) = Rep + ( I ). In ge neral d-spaces, it may o ccur that a non-d-path becomes directed after repa- rametrizatio n. T o exclude this p ossibility , we add Definition 4. 3 . A d-space X is called satura te d if it has the following additional prop erty: Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 21 4. If p ∈ P ( X ) , ϕ ∈ Rep + ( I ) and p ◦ ϕ ∈ ~ P ( X ), then p ∈ ~ P ( X ). In words: If a path b ecomes a d-pa th after r eparametriza tion, then it has to b e a d-path itself already . Remark that for a s aturated d-space, tw o trace equiv alent paths are either both d-paths, or neither of them is. The d-space ~ I is sa tur ated. It is easy to turn a given d-s pace X into a sa turated one: one just adds all paths p ∈ P ( X ) for which there is a reparametrization ϕ ∈ Rep + ( I ) with p ◦ ϕ ∈ ~ P ( X ) to the d-paths in a new structure S ~ P ( X ), whic h is e a sily seen to satisfy the prop erties of a satura ted d-space. So there is no harm in ass uming that a d-space is satura ted right aw ay . Among the d-paths in X , we pa y par ticular a tten tion to the r e gu lar d-pa ths, cf. Definition 1.1.3; the set of all those will be denoted ~ R ( X ) = R ( X ) ∩ ~ P ( X ) and equipp e d with the subspace top ology . Again, Homeo + ( I ) a cts (ess e ntially freely) on ~ R ( X ), a nd Rep + ( I ) acts on ~ P ( X ). W e can now sp eak o f spa c e s of (reg ula r) tra c es in a sa turated d-spa ce X : Definition 4.4. • ~ T R ( X )( x, y ) := ~ R ( X )( x, y ) / Homeo + ( I ) ⊆ T R ( X )( x, y ) • ~ T ( X )( x, y ) := ~ P ( X )( x, y ) / Rep + ( I ) ⊆ T ( X )( x, y ) These form the morphisms of ca tegories ~ T R ( X ), re s p. ~ T ( X ) that a re inv estigated from a homotopy theory po in t of v iew in [ 15 ]. The following cons e quence of Theorem 3.6 tells us that it makes no difference in top ology which of the tw o (quotient) trace spaces is chosen: Corollary 4. 5. L et X denote a satu r ate d d-sp ac e and let x, y ∈ X . The map ~ i : ~ T R ( X )( x, y ) → ~ T ( X )( x, y ) induc e d by inclusion ~ R ( X )( x, y ) ֒ → ~ P ( X )( x, y ) is a home omorphism. R emark 4.6 . I t is no longer clea r whether the inclusion map ~ R ( X )( x, y ) ֒ → ~ P ( X )( x, y ) is a w eak ho motopy equiv alence. The (weak) homotopy types of b oth spaces dep end on the choice of x and y and it is therefo r e not p ossible to argue using lo op spa ces as in 3 .10. F r om the diag ram ~ R ( X )( x, y ) ⊆ / / Q R ≃   ~ P ( X )( x, y ) Q   ~ T R ( X )( x, y ) i ∼ = / / ~ T ( X )( x, y ) , obtained from (3 .1) by restricting to d-paths, w e can o nly deduce that the inclusion maps ~ R ( X )( x, y ) ֒ → ~ P ( X )( x, y ) induce injections and that the quotien t maps Q : ~ P ( X )( x, y ) → ~ T ( X )( x, y ) induce surjectio ns on all homotopy groups. Definition 4.7. 1. [ 9 ] A d-homotopy from a d-path p ∈ ~ P ( X ) to a d-path q ∈ ~ P ( X ) is a d-map H : ~ I × ~ I → X for which H (0 , · ) = p , H (1 , · ) = q , and H ( · , 0), H ( · , 1) a r e consta n t. 2. A d-homotopy is said to b e thin if it factors through the d-in terv al ~ I , i.e. if there are d-ma ps Φ : ~ I × ~ I → ~ I , r : ~ I → X suc h that H = r ◦ Φ. Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 22 3. Two d-paths p, q ∈ ~ P ( X ) are said to b e d-homotopic , re s pectively thinly d- homotopic , if there exists a sequence H 1 , . . . , H 2 n +1 of d-homotopies, re s pec- tively thin d-homotopies, such that H 1 (0 , · ) = p , H 2 n +1 (1 , · ) = q , H 2 i − 1 (1 , · ) = H 2 i (1 , · ), and H 2 i (0 , · ) = H 2 i +1 (0 , · ). R emark 4.8 . • The directed structure o n a pr o duct of d-spac es X and Y is given by P ( X × Y ) ⊇ ~ P ( X × Y ) = ~ P ( X ) × ~ P ( Y ) ⊆ P ( X ) × P ( Y ) under the natural iden tification P ( X × Y ) ∼ = P ( X ) × P ( Y ). In particular, ~ P ( ~ I × ~ I ) consists of all paths p : I → I × I that are (weakly) increasing in bo th co o rdinates. • The r elations 4 , 4 T on d-paths given by existence of (thin) d-homotopies are preorder s o n ~ P ( X ). The relations ≃ , ≃ T on d-paths g iven by b eing (thinly) d-homotopic a re equiv alence r elations on ~ P ( X ); they are the symmetric, tra n- sitive closures of 4 resp ectively 4 T . • In the differe ntiable setting, thin homotopies of differ en tiable lo ops and paths were defined (using homo to pies o f r a nk at most 1) and a resulting smo oth fundamen tal gr oup oid used for holonomy considerations in [ 2, 14 ]. Prop osition 4. 9. Two d-p aths p , q in a satur ate d d-sp ac e ar e r ep ar ametrization e quivalent if and only if they ar e t hinly d-homotopic. This result is needed in the study [ 5 ] o f directed squares. Note that the notion of d-homotopy factors over ~ T ( X ) (and ~ T R ( X )). Pr o of. W e use Pro po sition 3.11. F or the for w ar d implication, write p = r ◦ ϕ , q = r ◦ ψ , and let η = max( ϕ, ψ ). Define Φ , Ψ : ~ I × ~ I → ~ I by Φ( s, t ) = (1 − s ) ϕ ( t ) + sη ( t ) Ψ( s, t ) = (1 − s ) ψ ( t ) + sη ( t ) then r ◦ Φ, r ◦ Ψ a re thin d-homoto pies connecting p and q . T o show the back implication, it is enough to consider the c a se where p and q are connected by one thin d-homotopy H with H (0 , · ) = p and H (1 , · ) = q . W r ite H = r ◦ Φ : ~ I × ~ I → ~ I → X ; b y Co rollary 4.5, we can ass ume r to b e reg ular. Also, by reparametr izing if necessary , we ca n ass ume that Φ(0 , 0) = 0 and Φ(1 , 1 ) = 1 . Then r (Φ( s, 0)) = H ( s, 0) = H (0 , 0) = r (0) r (Φ( s, 1)) = H ( s, 1) = H (1 , 1) = r (1) hence b y regularity of r and following the logic o f the final step in the pro of of Lemma 3.4, Φ( s, 0) = 0 and Φ( s, 1) = 1. Now define ϕ, ψ : ~ I → ~ I b y ϕ ( t ) = Φ(0 , t ), ψ ( t ) = Φ(1 , t ), then ϕ, ψ ∈ Rep + ( I ) and p = r ◦ ϕ , q = r ◦ ψ . Finally , w e mo dify the r esults from Se c tion 3 a bout lo o p-free paths (Definition 3.12) to the d-spa ce environment: Journal of Homotopy and R elate d Struct ur es , vol. 2(1), 2007 23 Definition 4.10. A d-space X is sa id to be lo c al ly lo op-fr e e provided tha t every po in t has a neighbourho o d in which all non-cons tan t d-paths a re lo op-free. Note that d-spa ces ar is ing fro m a space with a lo cally par tial o rder [ 6 ] ar e lo ca lly lo op-free. The following result applies such to such s paces, in particular . F or p o- spaces, a result similar to the following had previously b een obtained in [ 12 , Thm. 5 ]. Corollary 4.11. If p ∈ ~ P ( X ) is a lo op-fr e e d-p ath in a lo c al ly lo op-fr e e satur ate d Hausdorff d-sp ac e X , then its image p ( ~ I ) is either a p oint or d-home omorphic to ~ I . Pr o of. The statement is trivial for a constant d-pa th. Otherwise, Coro llary 4.5 provides us with a r e gular loo p-free d-pa th q : ~ I → X with p = q ◦ ϕ , ϕ ∈ Rep + ( I ) which by Co rollary 3.13 yields the homeomor phis m q : I → p ( ~ I ). All w e need to show is that its inv erse q − 1 : p ( ~ I ) → ~ I is a d-map. Let r : ~ I → p ( ~ I ) be a d- path; we need to show that q − 1 ◦ r ∈ ~ P ( I ) = Rep + ( I ). Let t 1 < t 2 ∈ I and supp o se that q − 1 ( r ( t 1 )) > q − 1 ( r ( t 2 )). Restricting to a s maller interv al, if neces s ary , will ensur e tha t r ([ t 1 , t 2 ]) ⊆ X is contained in a lo op- fr ee neighbourho o d U ⊆ X . The concatena tion r | [ t 1 ,t 2 ] ∗ q | [ q − 1 ( r ( t 2 )) ,q − 1 ( r ( t 1 ))] is a d- pa th and a lo op in U and hence constant. Then q | [ q − 1 ( r ( t 2 )) ,q − 1 ( r ( t 1 ))] is con- stant, in contradiction to b eing a regula r and non-consta n t pa th. Hence q − 1 ( r ( t 1 )) 6 q − 1 ( r ( t 2 )) whence q − 1 is a d- ma p. R emark 4.12 . It seems plausible that many of the metho ds and of the re sults from this article allow genera lizations to maps p : I n → X , r esp. p : ~ I n → X , fr o m (directed) cub e s to (d)-spa c es. The relev ant r eparametriza tions to inv estigate are the d-ma ps ϕ : ~ I n → ~ I n (monotone in every co o rdinate) that pr eserve b oundarie s in the following sense: ϕ ( x 1 , . . . , x n ) = ( y 1 , . . . , y n ) a nd x i = 0 , res p. 1 ⇒ y i = 0 , res p. 1 . In the differentiable se tting, suc h r eparametriza tions for so-called n -lo o ps and re- sulting smo oth ho mo topy g roups have been defined and studied in [ 3, 14 ]. References [1] J. Ba e z a nd U. Schreib er. Higher Ga uge Theory . arXiv:math.DG/0 511710 , 2005. [2] A . Caetano and R. Pic ken. An a xiomatic definition of holonomy . In ternat. J. Math. , 5 (6):835–8 4 8, 1 994. [3] A . Caetano and R. Pic ken. On a family of top olo gical inv ariants similar to homotopy groups. R end. Istit. Mat. Un iv. 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Hauco urt. T op olo gie alg ´ ebrique dirig ´ ee et c oncurr enc e . PhD thesis, Uni- versit ´ e Paris 7, UFR Informatique, 200 5. [13] M. Lo ` eve. Pr ob ability The ory . Univers. Series in Higher Math. v an Norstrand, Princeton, NJ, 3 edition, 19 63. [14] M. Mac k a ay and R. Pick en. Holonomy and pa rallel transp ort for abelia n gerb es. A dv. Math. , 1 70:287 –339, 2002. [15] M. Raussen. In v ariants of dir ected s paces. Appl. Cate g. Stru ctur es , 2007 . to app ear. [16] E.H. Spanier. Alg ebr aic Top olo gy . Mc Graw-Hill, 1966. Ulrich F ahrenber g uli@cs .aau.d k www.cs .aau.d k/ ~ uli Department of Co mputer Science Aalb o rg Universit y Denmark F redr ik Ba jersvej 7B DK-9220 Aalb or g Øst Martin Ra ussen rausse n@math .aau.dk www.ma th.aau .dk/ ~ rausse n Department of Mathema tical Sciences Aalb o rg Universit y Denmark F redr ik Ba jersvej 7G DK-9220 Aalb or g Øst