Weak morphisms of higher dimensional automata
aa r X i v : . [ c s . F L ] M a r WEAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA
THOMAS KAHL
Centro de Matem´atica, Universidade do Minho, Campus de Gualtar,4710-057 Braga, Portugal
Abstract.
We introduce weak morphisms of higher dimensional automataand use them to define preorder relations for HDAs, among which homeomor-phic abstraction and trace equivalent abstraction. It is shown that homeomor-phic abstraction is essentially always stronger than trace equivalent abstrac-tion. We also define the trace language of an HDA and show that, for a largeclass of HDAs, it is invariant under trace equivalent abstraction.
Introduction
One of the most expressive models of concurrency is the one of higher dimensionalautomata [8]. A higher dimensional automaton (HDA) over a monoid M is aprecubical set with initial and final states and with 1-cubes labelled by elements of M such that opposite edges of 2-cubes have the same label. Intuitively, an HDAcan be seen as an automaton with an independence relation represented by cubes.If two actions a and b are enabled in a state q and are independent in the sense thatthey may be executed in any order or even simultaneously without any observabledifference, then the HDA contains a 2-cube linking the two execution sequences ab and ba beginning in q . Similarly, the independence of n actions is represented by n -cubes. It has been shown in [8] that many classical models of concurrency canbe translated into the one of HDAs. HDA semantics for process algebras are givenin [6] and [10].In this paper, we introduce three preorder relations for HDAs. Whenever, asin figure 1, an HDA B is a subdivision of an HDA A , then A is related to B ineach of these preorders. The definitions of the preorder relations are based on theconcept of weak morphism, which is developed in section 2. Roughly speaking, aweak morphism between two HDAs is a continuous map between their geometricrealisations that sends subdivided cubes to subdivided cubes and that preserveslabels of paths. A morphism of HDAs, or, more precisely, its geometric realisation,is a weak morphism but not vice versa. For example, in figure 1, there exists aweak morphism from A to B , but there does not exist any morphism between the E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases.
Higher dimensional automata, precubical set, geometric realisation,trace language, preorder relation, abstraction.This research has been supported by FEDER funds through “Programa Operacional Factoresde Competitividade - COMPETE” and by FCT -
Funda¸c˜ao para a Ciˆencia e a Tecnologia throughprojects Est-C/MAT/UI0013/2011 and PTDC/MAT/0938317/2008. ababc c (a) A a ba bc cc (b) B Figure 1.
Two HDAs A and B over the free monoid on { a, b, c } such that B is a subdivision of A , A → B , A ∼ t → B and A ≈ → B two HDAs. If there exists a weak morphism from an HDA A to an HDA B , wewrite A → B . The relation → is the first preorder relation for HDAs we considerin this paper. It has the basic property that A → B implies that the language (thebehaviour) of A is contained in the language of B . One may consider → as a kindof simulation preorder: If there exists a weak morphism from A to B , then for every1-cube in A with label α from a vertex v to a vertex w there exists a path in B with label α from the image of v to the image of w .The higher dimensional structure of an HDA induces an independence relationon the monoid of labels and allows us to define the trace language of an HDAin section 3. The fundamental concept in this context is dihomotopy (short fordirected homotopy) of paths [5, 9]. Besides the trace language of an HDA, wealso consider the trace category of an HDA and trace equivalences between HDAs.The trace category of an HDA is a variant of the fundamental bipartite graph ofa d-space [1]. Its objects are certain important states of the HDA, including theinitial and the final ones, and the morphisms are the dihomotopy classes of pathsbetween these states. A trace equivalence is essentially defined as a weak morphismthat induces an isomorphism of trace categories. If there exists a trace equivalencefrom an HDA A to an HDA B , we say that A is a trace equivalent abstraction of B and write A ∼ t → B . Trace equivalent abstraction is our second preorder relation forHDAs. We show that for a large class of HDAs, A ∼ t → B implies that there exists abijection between the trace languages of A and B .The third preorder relation is called homeomorphic abstraction and is the sub-ject of section 4. We say that an HDA A is a homeomorphic abstraction of anHDA B and write A ≈ → B if there exists a weak morphism from A to B that isa homeomorphism and a bijection on initial and on final states. Homeomorphicabstraction may be seen as a labelled version of T-homotopy equivalence in thesense of [7]. We show that under a mild condition, A ≈ → B implies A ∼ t → B .1. Precubical sets and HDAs
This section contains some basic and well-known material on precubical sets andhigher dimensional automata.1.1.
Precubical sets. A precubical set is a graded set P = ( P n ) n ≥ with boundaryoperators d ki : P n → P n − ( n > , k = 0 , , i = 1 , . . . , n ) satisfying the relations d ki ◦ d lj = d lj − ◦ d ki ( k, l = 0 , , i < j ) [3, 4, 5, 7, 9]. The least n ≥ P i = ∅ for all i > n is called the dimension of P . If no such n exists, then thedimension of P is ∞ . If x ∈ P n , we say that x is of degree n and write deg( x ) = n .The elements of degree n are called the n -cubes of P . The elements of degree 0 EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 3 are also called the vertices or the nodes of P . A morphism of precubical sets is amorphism of graded sets that is compatible with the boundary operators.The category of precubical sets can be seen as the presheaf category of functors ✷ op → Set where ✷ is the small subcategory of the category of topological spaceswhose objects are the standard n -cubes [0 , n ( n ≥
0) and whose non-identitymorphisms are composites of the maps δ ki : [0 , n → [0 , n +1 ( k ∈ { , } , n ≥ i ∈ { , . . . , n + 1 } ) given by δ ki ( u , . . . , u n ) = ( u , . . . , u i − , k, u i . . . , u n ). Here, weuse the convention that given a topological space X , X denotes the one-pointspace { () } .1.2. Precubical subsets. A precubical subset of a precubical set P is a gradedsubset of P that is stable under the boundary operators. It is clear that a precubicalsubset is itself a precubical set. Note that unions and intersections of precubicalsubsets are precubical subsets and that the image of a morphism f : P → Q ofprecubical sets is a precubical subset of Q .1.3. Intervals.
Let k and l be integers such that k ≤ l . The precubical interval J k, l K is the at most 1-dimensional precubical set defined by J k, l K = { k, . . . , l } , J k, l K = { [ k, k + 1] , . . . , [ l − , l ] } , d [ j − , j ] = j − d [ j − , j ] = j .1.4. Tensor product.
Given two graded sets P and Q , the tensor product P ⊗ Q is the graded set defined by ( P ⊗ Q ) n = ` p + q = n P p × Q q . If P and Q are precubicalsets, then P ⊗ Q is a precubical set with respect to the boundary operators givenby d ki ( x, y ) = (cid:26) ( d ki x, y ) , ≤ i ≤ deg( x ) , ( x, d ki − deg( x ) y ) , deg( x ) + 1 ≤ i ≤ deg( x ) + deg( y )(cf. [3]). The tensor product turns the categories of graded and precubical sets intomonoidal categories.The n -fold tensor product of a graded or precubical set P is denoted by P ⊗ n .Here, we use the convention P ⊗ = J , K = { } . The precubical n -cube is theprecubical set J , K ⊗ n . The only element of degree n in J , K ⊗ n will be denotedby ι n . We thus have ι = 0 and ι n = ([0 , , . . . , [0 , | {z } n times ) for n > The morphism corresponding to an element.
Let x be an element ofdegree n of a precubical set P . Then there exists a unique morphism of pre-cubical sets x ♯ : J , K ⊗ n → P such that x ♯ ( ι n ) = x . Indeed, by the Yonedalemma, there exist unique morphisms of precubical sets f : ✷ ( − , [0 , n ) → P and g : ✷ ( − , [0 , n ) → J , K ⊗ n such that f ( id [0 , n ) = x and g ( id [0 , n ) = ι n . The map g is an isomorphism, and x ♯ = f ◦ g − .1.6. Paths. A path of length k in a precubical set P is a morphism of precubicalsets ω : J , k K → P . The set of paths in P is denoted by P I . If ω ∈ P I is a path oflength k , we write length( ω ) = k . The concatenation of two paths ω : J , k K → P and ν : J , l K → P with ω ( k ) = ν (0) is the path ω · ν : J , k + l K → P defined by ω · ν ( j ) = (cid:26) ω ( j ) , ≤ j ≤ k,ν ( j − k ) , k ≤ j ≤ k + l WEAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA and ω · ν ([ j − , j ]) = (cid:26) ω ([ j − , j ]) 0 < j ≤ k,ν ([ j − k − , j − k ]) k < j ≤ k + l. Clearly, concatenation is associative. Note that for any path ω ∈ P I of length k ≥ x , . . . , x k ) of elements of P such that d x j +1 = d x j for all 1 ≤ j < k and ω = x ♯ · · · x k♯ .1.7. Geometric realisation.
The geometric realisation of a precubical set P isthe quotient space | P | = ( ` n ≥ P n × [0 , n ) / ∼ where the sets P n are given thediscrete topology and the equivalence relation is given by( d ki x, u ) ∼ ( x, δ ki ( u )) , x ∈ P n +1 , u ∈ [0 , n , i ∈ { , . . . , n + 1 } , k ∈ { , } (see [3], [4], [5], [7], [9]). The geometric realisation of a morphism of precubical sets f : P → Q is the continuous map | f | : | P | → | Q | given by | f | ([ x, u ]) = [ f ( x ) , u ]. Weremark that the geometric realisation is a functor from the category of precubicalsets to the category of topological spaces. Examples. (i) The geometric realisation of the precubical n -cube can be identifiedwith the standard n -cube by means of the homeomorphism [0 , n → | J , K ⊗ n | , u [ ι n , u ].(ii) The geometric realisation of the precubical interval J k, l K can be identifiedwith the closed interval [ k, l ] by means of the homeomorphism | J k, l K | → [ k, l ] givenby [ j, ()] j and [[ j − , j ] , t ] j − t . Using this correspondence, the geometricrealisation of a precubical path J , k K → P can be seen as a path [0 , k ] → | P | , andunder this identification we have that | ω · ν | = | ω | · | ν | .We note that for every element a ∈ | P | there exist a unique integer n ≥
0, aunique element x ∈ P n and a unique element u ∈ ]0 , n such that a = [ x, u ].The geometric realisation of a precubical set P is a CW-complex [7]. The n -skeleton of | P | is the geometric realisation of the precubical subset P ≤ n of P definedby ( P ≤ n ) m = P m ( m ≤ n ). The closed n -cells of | P | are the spaces | x ♯ ( J , K ⊗ n ) | where x ∈ P n . The characteristic map of the cell | x ♯ ( J , K ⊗ n ) | is the map [0 , n ≈ →| J , K ⊗ n | | x ♯ | → | P | , u [ x, u ]. The geometric realisation of a precubical subset Q of P is a subcomplex of | P | .The geometric realisation is a comonoidal functor with respect to the naturalcontinuous map ψ P,Q : | P ⊗ Q | → | P | × | Q | given by ψ P,Q ([( x, y ) , ( u , . . . , u deg( x )+deg( y ) )])= ([ x, ( u , . . . , u deg( x ) )] , [ y, ( u deg( x )+1 , . . . u deg( x )+deg( y ) ]) . If P and Q are finite, then ψ P,Q is a homeomorphism and permits us to identify | P ⊗ Q | with | P |×| Q | . We may thus identify the geometric realisation of a precubicalset of the form J k , l K ⊗· · ·⊗ J k n , l n K ( k i < l i ) with the product [ k , l ] ×· · ·× [ k n , l n ]by means of the correspondence[([ i , i + 1] , . . . , [ i n , i n + 1]) , ( u , . . . , u n )] ( i + u , . . . , i n + u n ) . Higher dimensional automata.
Let M be a monoid. A higher dimensionalautomaton over M (abbreviated M -HDA or simply HDA) is a tuple A = ( P, I, F, λ )where P is a precubical set, I ⊆ P is a set of initial states , F ⊆ P is a set of final states , and λ : P → M is a map, called the labelling function , such that λ ( d i x ) = λ ( d i x ) for all x ∈ P and i ∈ { , } . A morphism from an M -HDA EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 5 A = ( P, I, F, λ ) to an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) is a morphism of precubical sets f : P → P ′ such that f ( I ) ⊆ I ′ , f ( F ) ⊆ F ′ and λ ′ ( f ( x )) = λ ( x ) for all x ∈ P .This definition of higher dimensional automata is essentially the same as the onein [8]. Besides the fact that we consider a monoid and not just a set of labels, theonly difference is that in [8] an HDA is supposed to have exactly one initial state.Note that 1-dimensional M -HDAs and morphisms of 1-dimensional M -HDAs arethe same as automata over M and automata morphisms as defined in [13].1.9. The language accepted by an HDA.
Let A = ( P, I, F, λ ) be an M -HDA.The extended labelling function of A is the map λ : P I → M defined as follows: If ω = x ♯ · · · x k♯ for a sequence ( x , . . . , x k ) of elements of P such that d x j +1 = d x j (1 ≤ j < k ), then we set λ ( ω ) = λ ( x ) · · · λ ( x k ); if ω is a path of length 0, then weset λ ( ω ) = 1. The language accepted by A is the set L ( A ) = { λ ( ω ) : ω ∈ P I , ω (0) ∈ I, ω (length( ω )) ∈ F } . Note that L ( A ) is the the behaviour in the sense of [13] of the 1-skeleton of A , i.e.the M -HDA ( P ≤ , I, F, λ ). A fundamental property of the language accepted byan HDA is that one has L ( A ) ⊆ L ( B ) if there exists a morphism of M -HDAs from A to B [13, prpty. II.3.1].Recall, for instance from [2] or [13], that a subset L of a monoid M is called • rational if it belongs to the smallest subset of the power set P ( M ) thatcontains all finite subsets of M and is closed under the operations union,product and star; • recognisable if there exist a right action of M on a finite set S , an element s ∈ S and a subset T ⊆ S such that L = { m ∈ M : s · m ∈ T } .A subset of a monoid M is rational if and only if it is the languague accepted byan M -HDA A = ( P, I, F, λ ) such that P and P are finite [13, thm. II.1.1]. ByKleene’s theorem, a subset of a finitely generated free monoid is rational if andonly if it is recognisable. Consequently, if M is a finitely generated free monoid and A = ( P, I, F, λ ) is an M -HDA such that P and P are finite, then L ( A ) is both arational and a recognisable subset of M .2. Weak morphisms
In this section, we introduce weak morphisms of precubical sets and HDAs anduse them to define the preorder relation → . We establish the basic properties ofweak morphisms and show, in particular, that a weak morphism induces a mapof path sets. We also define subdivisions, which provide an important exampleof weak morphisms. We begin by studying dihomeomorphisms of hyperrectangles,which are central to the concept of weak morphism.2.1. Dihomeomorphisms of hyperrectangles.
Let φ : [ a , b ] × · · · × [ a n , b n ] → [ c , d ] × · · · × [ c n , d n ] ( a i < b i , c i < d i )be a homeomorphism such that φ and φ − preserve the natural partial order of R n .Such a homeomorphism is called a dihomeomorphism (see e.g. [5]). Proposition 2.1.1.
Consider an element x ∈ [ a , b ] × · · · × [ a n , b n ] . Then x and φ ( x ) have the same number of minimal coordinates and the same number ofmaximal coordinates. WEAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA
Proof.
Consider the sets A = { u ∈ [ a , b ] × · · · × [ a n , b n ] : u ≤ x } = [ a , x ] × · · · × [ a n , x n ]and B = { v ∈ [ c , d ] × · · · × [ c n , d n ] : v ≤ φ ( x ) } = [ c , φ ( x )] × · · · × [ c n , φ n ( x )] . Then φ restricts to a homeomorphism from A to B . It follows that A and B arespaces of the same dimension. Hence x and φ ( x ) have the same number of minimalcoordinates. In the same way one shows that x and φ ( x ) have the same number ofmaximal coordinates. (cid:3) Proposition 2.1.2.
Consider an index k ∈ { , . . . , n } . Then there exists an index l ∈ { , . . . , n } such that φ ([ a , b ] × · · · × [ a k − , b k − ] × { a k } × [ a k +1 , b k +1 ] × · · · × [ a n , b n ])= [ c , d ] × · · · × [ c l − , d l − ] × { c l } × [ c l +1 , d l +1 ] × · · · × [ c n , d n ] and φ ([ a , b ] × · · · × [ a k − , b k − ] × { b k } × [ a k +1 , b k +1 ] × · · · × [ a n , b n ])= [ c , d ] × · · · × [ c l − , d l − ] × { d l } × [ c l +1 , d l +1 ] × · · · × [ c n , d n ] . Proof.
Consider the point ( b , . . . , b k − , a k , b k +1 , . . . , b n ) ∈ [ a , b ] × · · · × [ a n , b n ].By 2.1.1, there exists an index l ∈ { , . . . , n } such that φ ( b , . . . , b k − , a k , b k +1 , . . . , b n ) = ( d , . . . , d l − , c l , d l +1 , . . . , d n ) . It follows that φ ([ a , b ] × · · · × [ a k − , b k − ] × { a k } × [ a k +1 , b k +1 ] × · · · × [ a n , b n ])= φ ( { x ∈ [ a , b ] × · · · × [ a n , b n ] : x ≤ ( b , . . . , b k − , a k , b k +1 , . . . , b n ) } )= { y ∈ [ c , d ] × · · · × [ c n , d n ] : y ≤ ( d , . . . , d l − , c l , d l +1 , . . . , d n ) } = [ c , d ] × · · · × [ c l − , d l − ] × { c l } × [ c l +1 , d l +1 ] × · · · × [ c n , d n ]In the same way, one shows that there exists an index p ∈ { , . . . , n } such that φ ([ a , b ] × · · · × [ a k − , b k − ] × { b k } × [ a k +1 , b k +1 ] × · · · × [ a n , b n ])= [ c , d ] × · · · × [ c p − , d p − ] × { d p } × [ c p +1 , d p +1 ] × · · · × [ c n , d n ] . It remains to show that l = p . Suppose that this is not the case. Consider the points x θ = ( a + b , . . . , a k − + b k − , a k + θ ( b k − a k ) , a k +1 + b k +1 , . . . , a n + b n ), θ ∈ [0 , φ ( x ) = ( s , . . . , s l − , c l , s l +1 , . . . , s n ) and φ ( x ) = ( t , . . . , t p − , d p , t p +1 , . . . , t n ).Since x ≤ x , we have s j ≤ t j for all j ∈ { , . . . , n } \ { l, p } . Consider the element v of [ c , d ] × · · · × [ c n , d n ] given by v j = c l , j = l,d p , j = p,s j , j ∈ { , . . . n } \ { l, p } . Then φ ( x ) ≤ v ≤ φ ( x ) and hence x ≤ φ − ( v ) ≤ x . It follows that there exists θ ∈ [0 ,
1] such that φ − ( v ) = x θ , i.e. φ ( x θ ) = v . By 2.1.1, x θ has at least oneminimal and one maximal coordinate. This is impossible, and therefore l = p . (cid:3) EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 7
Weak morphisms of precubical sets. A weak morphism from a precubicalset P to a precubical set P ′ is a continuous map f : | P | → | P ′ | such that thefollowing two conditions hold:(1) for every vertex v ∈ P there exists a (necessarily unique) vertex f ( v ) ∈ P ′ such that f ([ v, ()]) = [ f ( v ) , ()];(2) for all integers n, k , . . . , k n ≥ χ : J , k K ⊗· · ·⊗ J , k n K → P there exist integers k ′ , . . . , k ′ n ≥
1, a morphismof precubical sets χ ′ : J , k ′ K ⊗ · · · ⊗ J , k ′ n K → P ′ and a dihomeomorphism φ : | J , k K ⊗ · · · ⊗ J , k n K | = [0 , k ] × · · · × [0 , k n ] → | J , k ′ K ⊗ · · · ⊗ J , k ′ n K | = [0 , k ′ ] × · · · × [0 , k ′ n ]such that f ◦ | χ | = | χ ′ | ◦ φ . Examples. (i) The geometric realisation of a morphism of precubical sets is a weakmorphism.(ii) The map | J , K ⊗ | = [0 , → | J , K ⊗ | = [0 , , ( s, t ) ( t, s ) is a weakmorphism that is not the geometric realization of a morphism of precubical sets J , K ⊗ → J , K ⊗ .(iii) The map | J , K | = [0 , → | J , K | = [0 , t t is a weak morphism. Thisweak morphism is the homeomorphism of a subdivision in the sense of 2.4.It is important to note that weak morphisms are stable under composition. Thisfact will enable us in sections 2.6, 3.9 and 4.1 to use weak morphisms to definepreorder relations for higher dimensional automata. Proposition 2.2.1.
Let f : | P | → | Q | and g : | Q | → | R | be weak morphisms ofprecubical sets. Then g ◦ f : | P | → | R | is a weak morphism of precubical sets, and ( g ◦ f ) = g ◦ f . (cid:3) The maps χ ′ and φ . Our purpose in this subsection is to show that themorphism of precubical sets χ ′ and the dihomeomorphism φ in condition (2) of thedefinition of weak morphisms in section 2.2 are unique and that φ is itself a weakmorphism. We need four lemmas. Lemma 2.3.1.
Let f : P → R and g : Q → R be morphisms of precubical sets and α : | P | → | Q | be a continuous map such that | g | ◦ α = | f | . Then there exists amorphism of precubical sets h : P → Q such that g ◦ h = f and | h | = α .Proof. Let x ∈ P be an element of degree r and u ∈ ]0 , r . Consider the uniquelydetermined elements y ∈ Q and v ∈ ]0 , deg( y ) such that α ([ x, u ]) = [ y, v ]. Wehave [ g ( y ) , v ] = | g | ([ y, v ]) = | g | ◦ α ([ x, u ]) = | f | ([ x, u ]) = [ f ( x ) , u ] and thereforedeg( y ) = r , g ( y ) = f ( x ) and v = u . Consider u ′ ∈ ]0 , r . For each t ∈ [0 , y t ∈ Q r such that α ([ x, (1 − t ) u + tu ′ ]) = [ y t , (1 − t ) u + tu ′ ]. Sinceevery continuous path in the subspace of open r -cells of | Q | must stay in one ofthe cells, we have y t = y for all t ∈ [0 , h ( x ) = y . We have shown that g ( h ( x )) = f ( x ) and that α ([ x, u ]) = [ h ( x ) , u ] for all u ∈ ]0 , r . It remains to showthat h is a morphism of precubical sets. Consider an element x ∈ P of degree r > i ∈ { , . . . , r } and k ∈ { , } . Consider the path [0 , → | P | , t [ x, u t ] where theelement u t ∈ [0 , r is defined by( u t ) j = ( , j = i, − t + tk, j = i. WEAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA
For t ∈ [0 , u t ∈ ]0 , r and therefore α ([ x, u t ]) = [ h ( x ) , u t ]. Hence also α ([ x, u ]) =[ h ( x ) , u ]. Thus, [ h ( d ki x ) , ( , . . . , )] = α ([ d ki x, ( , . . . , )]) = α ([ x, δ ki ( , . . . , )]) = α ([ x, u ]) = [ h ( x ) , u ] = [ h ( x ) , δ ki ( , . . . , )] = [ d ki h ( x ) , ( , . . . , )]. It follows that h ( d ki x ) = d ki h ( x ) and hence that h is a morphism of precubical sets. (cid:3) Lemma 2.3.2.
Consider integers n, k , . . . , k n ≥ and a morphism of precubicalsets f : J , k K ⊗· · ·⊗ J , k n K → J , l K ⊗· · ·⊗ J , l n K such that f (0 , . . . ,
0) = (0 , . . . , .Then k i ≤ l i for all i ∈ { , . . . , n } and f is the inclusion.Proof. Note first that necessarily l , . . . l n ≥
1. Consider the set K = { , . . . , k − } × · · · × { , . . . , k n − } . For ( i , . . . , i n ) ∈ K set c i ,...,i n = ([ i , i + 1] , . . . , [ i n , i n + 1]) . The c i ,...,i n are the n -cubes of J , k K ⊗ · · · ⊗ J , k n K . In the same way, we use the notation c ′ i ,...,i n forthe n -cubes of J , l K ⊗ · · · ⊗ J , l n K . Consider the total order on K defined by( j , . . . , j n ) < ( i , . . . , i n ) ⇔ ∃ r ∈ { , . . . , n } : j = i , . . . , j r − = i r − , j r < i r . We show by induction that for each ( i , . . . , i n ) ∈ K , i < l , . . . , i n < l n and f ( c i ,...,i n ) = c ′ i ,...,i n . The only n -cube of J , l K ⊗ · · · ⊗ J , l n K having (0 , . . . , c ′ ,..., . Therefore f ( c ,..., ) = c ′ ,..., . Suppose that ( i , . . . , i n ) > (0 , . . . ,
0) and that the assertion holds all ( j , . . . , j n ) < ( i , . . . , i n ). Let r be anindex such that i r >
0. By the inductive hypothesis, f ( c i ,...,i r − ,i r − ,i r +1 ,...,i n ) = c ′ i ,...,i r − ,i r − ,i r +1 ,...,i n . It follows that d r f ( c i ,...,i n )= f ( d r c i ,...,i n )= f ( d r c i ,...,i r − ,i r − ,i r +1 ,...,i n )= d r f ( c i ,...,i r − ,i r − ,i r +1 ,...,i n )= d r c ′ i ,...,i r − ,i r − ,i r +1 ,...,i n = ([ i , i + 1] , . . . , [ i r − , i r − + 1] , i r , [ i r +1 , i r +1 + 1] , . . . , [ i n , i n + 1])This can only happen if i < l , . . . , i n < l n and f ( c i ,...,i n ) = c ′ i ,...,i n . The resultfollows. (cid:3) Lemma 2.3.3.
Consider integers n, k , . . . , k n , l , . . . , l n ≥ , morphisms of pre-cubical sets ξ : J , k K ⊗ · · · ⊗ J , k n K → P and ζ : J , l K ⊗ · · · ⊗ J , l n K → P anda homeomorphism α : | J , k K ⊗ · · · ⊗ J , k n K | → | J , l K ⊗ · · · ⊗ J , l n K | such that | ζ | ◦ α = | ξ | . Then k i = l i for all i ∈ { , . . . , n } , α = id and ξ = ζ .Proof. By 2.3.1, there exist morphisms of precubical sets h : J , k K ⊗ · · · ⊗ J , k n K → J , l K ⊗ · · · ⊗ J , l n K and h ′ : J , l K ⊗ · · · ⊗ J , l n K → J , k K ⊗ · · · ⊗ J , k n K such that | h | = α , | h ′ | = α − , ζ ◦ h = ξ and ξ ◦ h ′ = ζ . We show that h (0 , . . . ,
0) =(0 , . . . , h (0 , . . . ,
0) = d x for some x ∈ ( J , l K ⊗ · · · ⊗ J , l n K ) . Therefore h ′ ◦ h (0 , . . . ,
0) = h ′ ( d x ) = d h ′ ( x ). On theother hand, [ h ′ ◦ h (0 , . . . , , ()] = | h ′ ◦ h | ([(0 , . . . , , ()]) = α − ◦ α ([(0 , . . . , , ()]) =[(0 , . . . , , ()] and hence h ′ ◦ h (0 , . . . ,
0) = (0 , . . . , , . . . ,
0) = d h ′ ( x ).This is impossible. It follows that h (0 , . . . ,
0) = (0 , . . . , h ′ (0 , . . . ,
0) =(0 , . . . , k i = l i for all i ∈ { , . . . , n } and h = h ′ = id . This implies that α = id and ξ = ζ . (cid:3) EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 9
Lemma 2.3.4.
Consider a weak morphism of precubical sets f : | P | → | P ′ | , twomorphisms of precubical sets χ : Q → P and χ ′ : Q ′ → P ′ and a continuous map g : | Q | → | Q ′ | such that f ◦ | χ | = | χ ′ | ◦ g . Then g is a weak morphism of precubicalsets.Proof. Consider a vertex v ∈ Q . Let x ∈ Q ′ and u ∈ ]0 , deg( x ) be the uniquelydetermined elements such that g ([ v, ()]) = [ x, u ]. We have [ χ ′ ( x ) , u ] = | χ ′ | ([ x, u ]) = | χ ′ | ◦ g ([ v, ()]) = f ◦ | χ | ([ v, ()]) = f ([ χ ( v ) , ()]) = [ f ( χ ( v )) , ()] and hence deg( x ) = 0and u = ().Let n, k , . . . , k n ≥ ξ : J , k K ⊗ · · ·⊗ J , k n K → Q be a morphismof precubical sets. Since f is a weak morphism, there exist integers k ′ , . . . , k ′ n ≥
1, amorphism of precubical sets θ : J , k ′ K ⊗ · · · ⊗ J , k ′ n K → P ′ and a dihomeomorphism φ : | J , k K ⊗ · · · ⊗ J , k n K | → | J , k ′ K ⊗ · · · ⊗ J , k ′ n K | such that f ◦ | χ ◦ ξ | = | θ | ◦ φ .Consider the continuous map α = g ◦| ξ |◦ φ − : | J , k ′ K ⊗· · ·⊗ J , k ′ n K | → | Q ′ . We have α ◦ φ = g ◦| ξ | and | χ ′ |◦ α = | χ ′ |◦ g ◦| ξ |◦ φ − = f ◦| χ |◦| ξ |◦ φ − = | θ |◦ φ ◦ φ − = | θ | . Bylemma 2.3.1, there exists a morphism of precubical sets ξ ′ : J , k ′ K ⊗· · ·⊗ J , k ′ n K → Q ′ such that χ ′ ◦ ξ ′ = θ and | ξ ′ | = α . The result follows. (cid:3) We are finally ready to establish the uniqueness of the maps χ ′ and φ in thedefinition of weak morphisms and the fact that φ is itself a weak morphism: Proposition 2.3.5.
Let f : | P | → | P ′ | be a weak morphism of precubical sets, n, k , . . . , k n ≥ be integers and χ : J , k K ⊗ · · · ⊗ J , k n K → P be a morphism ofprecubical sets. Then there exist unique integers k ′ , . . . , k ′ n ≥ , a unique morphismof precubical sets χ ′ : J , k ′ K ⊗ · · · ⊗ J , k ′ n K → P ′ and a unique dihomeomorphism φ : | J , k K ⊗ · · · ⊗ J , k n K | → | J , k ′ K ⊗ · · · ⊗ J , k ′ n K | such that f ◦ | χ | = | χ ′ | ◦ φ .Moreover, φ is a weak morphism.Proof. The existence of k ′ , . . . , k ′ n , χ ′ and φ is guaranteed by the definition of weakmorphisms. The fact that φ is a weak morphism follows from 2.3.4. We have to showthe uniqueness of k ′ , . . . , k ′ n , χ ′ and φ . Suppose that the integers l , . . . , l n ≥
1, themorphism of precubical sets ζ : J , l K ⊗ · · · ⊗ J , l n K → Q and the dihomeomorphism ψ : | J , k K ⊗ · · · ⊗ J , k n K | → | J , l K ⊗ · · · ⊗ J , l n K | satisfy f ◦ | χ | = | ζ | ◦ ψ . Considerthe homeomorphism α = ψ ◦ φ − : | J , k ′ K ⊗ · · · ⊗ J , k ′ n K | → | J , l K ⊗ · · · ⊗ J , l n K | .Then | ζ | ◦ α = | ζ | ◦ ψ ◦ φ − = f ◦ | χ | ◦ φ − = | χ ′ | ◦ φ ◦ φ − = | χ ′ | . By lemma 2.3.3,it follows that k ′ i = l i for all i ∈ { , . . . , n } , φ = ψ and χ ′ = ζ . (cid:3) Subdivisions. A subdivision of a precubical set P is a precubical set Q to-gether with a homeomorphism f : | P | → | Q | satisfying the following two conditions:(1) for every vertex v ∈ P there exists a vertex w ∈ Q such that f ([ v, ()]) =[ w, ()];(2) for every n ≥ x ∈ P n there exist integers k . . . , k n ≥ ξ : J , k K ⊗ · · · ⊗ J , k n K → Q and dihomeo-morphisms φ i : | J , K | = [0 , → | J , k i K | = [0 , k i ] ( i ∈ { , . . . , n } ) such that f ◦ | x ♯ | = | ξ | ◦ ( φ × · · · × φ n ). Example.
For all n, k , . . . , k n ≥
1, the precubical set J , k K ⊗ · · · ⊗ J , k n K togetherwith the homeomorphism f : | J , K ⊗ n | = [0 , n → | J , k K ⊗ · · · ⊗ J , k n K | = [0 , k ] × · · · × [0 , k n ] , ( t , . . . , t n ) ( k t , . . . , k n t n ) is a subdivision of the precubical n -cube J , K ⊗ n . It is clear that condition (1) of thedefinition of subdivisions is satisfied. For condition (2) consider an integer m ≥ x = ( x , . . . , x n ) ∈ ( J , K ⊗ n ) m . Since deg( x ) = m , there existindices 1 ≤ i < . . . < i m ≤ n such that x i = . . . = x i m = [0 ,
1] and x i ∈ { , } for i / ∈ { i , . . . , i m } . Consider the dihomeomorphisms φ j : [0 , → [0 , k i j ], t k i j t ( j ∈ { , . . . , m } ) and the morphism of precubical sets ξ : J , k i K ⊗ · · · ⊗ J , k i m K → J , k K ⊗ · · · ⊗ J , k n K defined by ξ ( a , . . . a m ) = ( b , . . . , b n ) with b i j = a j for j ∈{ , . . . , m } and b i = k i x i for i / ∈ { i , . . . , i m } . Then f ◦ | x ♯ | = | ξ | ◦ ( φ × · · · × φ m ). Proposition 2.4.1.
Let P be a precubical set and ( Q, f ) be a subdivision of P .Then f is a weak morphism.Proof. We only have to show that condition (2) of the definition of weak morphisms(see 2.2) holds. Let n ≥ k , . . . , k n ≥ χ : J , k K ⊗ · · ·⊗ J , k n K → P . We show that there existintegers p , . . . , p n ≥
1, dihomeomorphisms φ j : [0 , k j ] → [0 , p j ] ( j ∈ { , . . . , n } ) anda morphism of precubical sets χ ′ : J , p K ⊗ · · · ⊗ J , p n K → Q such that f ◦ | χ | = | χ ′ | ◦ ( φ × · · · × φ n ). If all k i = 1, then this holds by condition (2) of the definitionof subdivisions. Suppose inductively that P ni =1 k i > n and that the claim holdsfor all morphisms of precubical sets J , l K ⊗ · · · ⊗ J , l n K → P such that P ni =1 l i < P ni =1 k i . Let s ∈ { , . . . , n } be an index such that k s >
1. The precubical set J , k K ⊗ · · · ⊗ J , k n K is the union of the precubical subsets A = J , k K ⊗ · · · ⊗ J , k s − K ⊗ J , K ⊗ J , k s +1 K ⊗ · · · ⊗ J , k n K and B = J , k K ⊗ · · · ⊗ J , k s − K ⊗ J , k s K ⊗ J , k s +1 K ⊗ · · · ⊗ J , k n . K Denote the restrictions of χ to A and B by α and β respectively. By the inductivehypothesis, there exist integers q , . . . , q n ≥
1, dihomeomorphisms σ s : [0 , → [0 , q s ] and σ j : [0 , k j ] → [0 , q j ] ( j ∈ { , . . . , n } \ { s } ) and a morphism of precubicalsets α ′ : J , q K ⊗ · · · ⊗ J , q n K → Q such that f ◦ | α | = | α ′ | ◦ ( σ × · · · × σ n ).Consider the isomorphism of precubical sets η : J , k K ⊗ · · · ⊗ J , k s − K ⊗ J , k s − K ⊗ J , k s +1 K ⊗ · · · ⊗ J , k n K → B given by ( y , . . . , y s − , i, y s +1 , . . . , y n ) ( y , . . . , y s − , i + 1 , y s +1 , . . . , y n )for i ∈ { , . . . , k s − } and by( y , . . . , y s − , [ i − , i ] , y s +1 , . . . , y n ) ( y , . . . , y s − , [ i, i + 1] , y s +1 , . . . , y n )for i ∈ { , . . . , k s − } . By the inductive hypothesis, there exist integers r , . . . , r n ≥
1, dihomeomorphisms ψ j : [0 , k j ] → [0 , r j ] ( j ∈ { , . . . , m }\{ s } ) and ψ s : [0 , k s − → [0 , r s ] and a morphism of precubical sets γ : J , r K ⊗ · · · ⊗ J , r n K → Q such that f ◦ | β ◦ η | = | γ | ◦ ( ψ × · · · × ψ n ).Consider the precubical set B ′ = J , r K ⊗ · · · ⊗ J , r s − K ⊗ J q s , q s + r s K ⊗ J , r s +1 K ⊗ · · · ⊗ J , r n K and the isomorphism of precubical sets µ : J , r K ⊗ · · · ⊗ J , r n K → B ′ given by( y , . . . , y s − , i, y s +1 , . . . , y n ) ( y , . . . , y s − , q s + i, y s +1 , . . . , y n )for i ∈ { , . . . , r s } and by( y , . . . , y s − , [ i − , i ] , y s +1 , . . . , y n ) ( y , . . . , y s − , [ q s + i − , q s + i ] , y s +1 , . . . , y n ) EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 11 for i ∈ { , . . . , r s } . Consider the morphism of precubical sets β ′ : B ′ → Q definedby β ′ = γ ◦ µ − and the dihomeomorphism θ s : [1 , k s ] → [ q s , q s + r s ] given by θ s ( t ) = q s + ψ s ( t − j ∈ { , . . . , n }\{ s } set θ j = ψ j . Then f ◦| β | = | β ′ |◦ ( θ ×· · ·× θ n ).Set A ′ = J , q K ⊗ · · · ⊗ J , q n K . We show that q j = r j and σ j = θ j for all j ∈ { , . . . , n } \ { s } and that α ′ and β ′ coincide on A ′ ∩ B ′ . If n = 1, we only haveto show that α ′ and β ′ coincide on A ′ ∩ B ′ = { q s } . We have | α ′ | ( q s ) = | α ′ | ◦ σ s (1) = f ◦ | α | (1) = f ◦ | χ | (1) = f ◦ | β | (1) = | β ′ | ◦ θ s (1) = | β ′ | ( q s ). Thus, α ′ ( q s ) = β ′ ( q s ).Suppose now that n >
1. Consider the composite morphisms of precubical sets ξ : J , k K ⊗ · · · ⊗ J , k s − K ⊗ J , k s +1 K ⊗ · · · ⊗ J , k n K ∼ = → A ∩ B ֒ → A and ξ ′ : J , q K ⊗ · · · ⊗ J , q s − K ⊗ J , q s +1 K ⊗ · · · ⊗ J , q n K ∼ = → J , q K ⊗ · · · ⊗ J , q s − K ⊗ { q s } ⊗ J , q s +1 K ⊗ · · · ⊗ J , q n K ֒ → A ′ . We have ( σ × · · · × σ n ) ◦ | ξ | = | ξ ′ | ◦ ( σ × · · · × σ s − × σ s +1 × · · · × σ n ) and hence f ◦ | α ◦ ξ | = | α ′ ◦ ξ ′ | ◦ ( σ × · · · × σ s − × σ s +1 × · · · × σ n ). Consider the compositemorphisms of precubical sets ζ : J , k K ⊗ · · · ⊗ J , k s − K ⊗ J , k s +1 K ⊗ · · · ⊗ J , k n K ∼ = → A ∩ B ֒ → B and ζ ′ : J , r K ⊗ · · · ⊗ J , r s − K ⊗ J , r s +1 K ⊗ · · · ⊗ J , r n K ∼ = → J , r K ⊗ · · · ⊗ J , r s − K ⊗ { q s } ⊗ J , r s +1 K ⊗ · · · ⊗ J , r n K ֒ → B ′ . We have ( θ × · · · × θ n ) ◦ | ζ | = | ζ ′ | ◦ ( θ × · · · × θ s − × θ s +1 × · · · × θ n ) and hence f ◦ | β ◦ ζ | = | β ′ ◦ ζ ′ | ◦ ( θ × · · · × θ s − × θ s +1 × · · · × θ n ). Since α ◦ ξ = β ◦ ζ , we have | β ′ ◦ ζ ′ |◦ ( θ ×· · ·× θ s − × θ s +1 ×· · ·× θ n ) ◦ ( σ ×· · ·× σ s − × σ s +1 ×· · ·× σ n ) − = | α ′ ◦ ξ ′ | .It follows by 2.3.3 that for all j ∈ { , . . . , n } \ { s } , q j = r j and σ j = θ j . Moreover, α ′ ◦ ξ ′ = β ′ ◦ ζ ′ . Since α ′ ◦ ξ ′ = β ′ ◦ ζ ′ , α ′ and β ′ coincide on A ′ ∩ B ′ .Set p s = q s + r s and p j = q j = r j ( j ∈ { , . . . , n } \ { s } ). Then we have J , p K ⊗ · · · ⊗ J , p n K = A ′ ∪ B ′ . Let χ ′ : J , p K ⊗ · · · ⊗ J , p n K → Q be the uniquemorphism of precubical sets such that χ ′ | A ′ = α ′ and χ ′ | B ′ = β ′ . Consider thedihomeomorphism φ s : [0 , k s ] → [0 , p s ] defined by φ s ( t ) = (cid:26) σ s ( t ) , ≤ t ≤ ,θ s ( t ) , ≤ t ≤ k s . For j ∈ { , . . . , n } \ { s } set φ j = σ j = θ j . We have f ◦ | χ | = | χ ′ | ◦ ( φ × · · · × φ n ).This terminates the induction and the proof. (cid:3) Weak morphisms and paths.
Let f : | P | → | P ′ | be a weak morphism ofprecubical sets and ω : J , k K → P ( k ≥
0) be a path. If k >
0, we denote by f I ( ω ) the unique path ω ′ : J , k ′ K → P ′ for which there exists a dihomeomorphism φ : | J , k K | → | J , k ′ K | such that f ◦ | ω | = | ω ′ | ◦ φ . If k = 0, f I ( ω ) is defined to bethe path in P ′ of length 0 given by f I ( ω )(0) = f ( ω (0)). Note that if g : P → P ′ is a morphism of precubical sets, then | g | I ( ω ) = g ◦ ω . The next three propositionscontain the basic properties of the maps f I : P I → P ′ I . Proposition 2.5.1.
Let f : | P | → | P ′ | be a weak morphism of precubical sets and ω : J , k K → P be a path. Then f I ( ω ) is a path from f ( ω (0)) to f ( ω ( k )) .Proof. This is clear for k = 0. Suppose that k > k ′ = length( f I ( ω )).Let φ : | J , k K | = [0 , k ] → | J , k ′ K | = [0 , k ′ ] be the dihomeomorphism such that f ◦ | ω | = | f I ( ω ) | ◦ φ . Since φ is a dihomeomorphism, φ (0) = 0 and φ ( k ) = k ′ .We therefore have [ f I ( ω )(0) , ()] = | f I ( ω ) | (0) = | f I ( ω ) | ◦ φ (0) = f ◦ | ω | (0) = f ([ ω (0) , ()]) = [ f ( ω (0)) , ()] and [ f I ( ω )( k ′ ) , ()] = | f I ( ω ) | ( k ′ ) = | f I ( ω ) | ◦ φ ( k ) = f ◦ | ω | ( k ) = f ([ ω ( k ) , ()]) = [ f ( ω ( k )) , ()]. It follows that f I ( ω )(0) = f ( ω (0)) and f I ( ω )( k ′ ) = f ( ω ( k )). (cid:3) Proposition 2.5.2.
Let f : | P | → | Q | and g : | Q | → | R | be weak morphisms ofprecubical sets. Then ( g ◦ f ) I = g I ◦ f I .Proof. Let ω : J , k K → P be a path. Suppose first that k = 0. Then f I ( ω ), g I ◦ f I ( ω )and ( g ◦ f ) I ( ω ) are paths of length 0. We have g I ◦ f I ( ω )(0) = g ◦ f ( ω (0)) = ( g ◦ f ) ( ω (0)) = ( g ◦ f ) I ( ω )(0)and therefore ( g ◦ f ) I ( ω ) = g I ◦ f I ( ω ). Suppose that k > k ′ = length( f I ( ω ))and k ′′ = length( g I ( f I ( ω ))). Let φ : | J , k K | → | J , k ′ K | and ψ : | J , k ′ K | → | J , k ′′ K | bedihomeomorphisms such that f ◦ | ω | = | f I ( ω ) | ◦ φ and g ◦ | f I ( ω ) | = | g I ( f I ( ω )) | ◦ ψ .Then g ◦ f ◦ | ω | = | g I ( f I ( ω )) | ◦ ψ ◦ φ . Since ψ ◦ φ is a dihomeomorphism, this impliesthat ( g ◦ f ) I ( ω ) = g I ( f I ( ω )). (cid:3) Proposition 2.5.3.
Let f : | P | → | Q | be a weak morphism of precubical sets and ω : J , k K → P and ν : J , l K → P be paths in P such that ω ( k ) = ν (0) . Then f I ( ω · ν ) = f I ( ω ) · f I ( ν ) .Proof. We may suppose that k, l >
0. Suppose that f I ( ω ) and f I ( ν ) are of length k ′ and l ′ respectively. Note that by 2.5.1, f I ( ω )( k ′ ) = f ( ω ( k )) = f ( ν (0)) = f I ( ν )(0).Let φ : | J , k K | → | J , k ′ K | and ψ : | J , l K | → | J , l ′ K | be dihomeomorphisms such that | f I ( ω ) | ◦ φ = f ◦ | ω | and | f I ( ν ) | ◦ ψ = f ◦ | ν | . Consider the dihomeomorphism α : | J , k + l K | = [0 , k + l ] → | J , k ′ + l ′ K | = [0 , k ′ + l ′ ] defined by α ( t ) = (cid:26) φ ( t ) , ≤ t ≤ k,k ′ + ψ ( t − k ) , k ≤ t ≤ k + l. For 0 ≤ t ≤ k , | f I ( ω ) · f I ( ν ) | ◦ α ( t ) = | f I ( ω ) · f I ( ν ) | ( φ ( t )) = | f I ( ω ) | ( φ ( t )) = f ◦ | ω | ( t ) = f ◦ | ω · ν | ( t ). For k ≤ t ≤ k + l , | f I ( ω ) · f I ( ν ) | ◦ α ( t ) = | f I ( ω ) · f I ( ν ) | ( k ′ + ψ ( t − k )) = | f I ( ν ) | ( ψ ( t − k )) = f ◦ | ν | ( t − k ) = f ◦ | ω · ν | ( t ). Wetherefore have | f I ( ω ) · f I ( ν ) | ◦ α = f ◦ | ω · ν | and hence f I ( ω · ν ) = f I ( ω ) · f I ( ν ). (cid:3) Weak morphisms of HDAs. A weak morphism from an M -HDA A =( P, I, F, λ ) to an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) is a weak morphism f : | P | → | P ′ | such that f ( I ) ⊆ I ′ , f ( F ) ⊆ F ′ and λ ′ ◦ f I = λ . We say that an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) is a subdivision of an M -HDA A = ( P, I, F, λ ) if there exists aweak morphism f from A to B such that f ( I ) = I ′ , f ( F ) = F ′ and ( P ′ , f ) is asubdivision of P .We remark that a morphism from an M -HDA A = ( P, I, F, λ ) to an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) is a weak morphism. More preceisely, if g : P → P ′ is a morphismof precubical sets such that g ( I ) ⊆ I ′ , g ( F ) ⊆ F ′ and λ ′ ◦ g = λ , then | g | is a weakmorphism of M -HDAs. EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 13
Definition 2.6.1.
Let A and B be two M -HDAs. We write A → B if there existsa weak morphism from A to B . We write A ↔ B if A → B and
B → A .It follows from 2.2.1 and 2.5.2 that → and ↔ are, respectively, a preorder andan equivalence relation on the class of M -HDAs. Example.
Consider the 1-dimensional precubical set P given by P = { u, v, w } , P = { x, y, z } , d x = d z = u , d x = d y = v and d y = d z = w . Considerthe set of labels Σ = { a, b } and the Σ ∗ -HDAs A = ( P, { u } , { w } , λ ) and B =( P \ { z } , { u } , { w } , λ | { x,y } ) where the labelling function λ is given by λ ( x ) = a , λ ( y ) = b and λ ( z ) = ab . Then A ↔ B . Note that there is no morphism ofΣ ∗ -HDAs from A to B . Proposition 2.6.2.
A → B ⇒ L ( A ) ⊆ L ( B ) and A ↔ B ⇒ L ( A ) = L ( B ) .Proof. Let A = ( P, I, F, λ ) and B = ( P ′ , I ′ , F ′ , λ ′ ) be M -HDAs such that A → B .Let f : | P | → | P ′ | be a weak morphism from A to B . Consider a path ω : J , k K → P such that ω (0) ∈ I and ω ( k ) ∈ F . Then f I ( ω )(0) = f ( ω (0)) ∈ I ′ and f I ( ω )(length( f I ( ω ))) = f ( ω ( k )) ∈ F ′ . We have λ ( ω ) = λ ′ ( f I ( ω )) ∈ L ( B ). Thisshows that L ( A ) ⊆ L ( B ). The result follows. (cid:3) Traces
We define the trace language and the trace category of an HDA and study howthese objects behave under weak morphisms. The fundamental concept for bothconstructions is dihomotopy of paths. We also introduce trace equivalences ofHDAs and the preorder relation of trace equivalent abstraction. The trace categoryis invariant under trace equivalent abstraction by definition, and we show that thisis often also the case for the trace language.3.1.
Dihomotopy.
Two paths ω and ν in a precubical set P are said to be ele-mentarily dihomotopic (or contiguous or adjacent ) if there exist paths α, β ∈ P I and an element z ∈ P such that d d z = α (length( α )), d d z = β (0) and { ω, ν } = { α · ( d z ) ♯ · ( d z ) ♯ · β, α · ( d z ) ♯ · ( d z ) ♯ · β } (see [5, 9]). The dihomo-topy relation, denoted by ∼ , is the equivalence relation generated by elementarydihomotopy. Remarks. (i) For a notion of homotopy for more general paths (essentially the cubepaths we consider in section 4) the reader is referred to [8].(ii) Dihomotopic paths have the same end points and the same length.(iii) Dihomotopy is a congruence. More precisely, if α, β, ω and ν are paths ina precubical set such that β (0) = α (length( α )), ν (0) = ω (length( ω )), α ∼ ω and β ∼ ν , then α · β ∼ ω · ν .(iv) If α and ω are (elementarily) dihomotopic paths in a precubical set P and f : P → Q is a morphism of precubical sets, then f ◦ α and f ◦ ω are (elementarily)dihomotopic paths in Q .3.2. The trace language of an HDA.
Let A = ( P, I, F, λ ) be an M -HDA. Wesay that a, b ∈ M are independent in A if(1) a, b ∈ λ ( P );(2) a = b ; (3) for all paths ω ∈ P I and elements u, v ∈ M with λ ( ω ) ∈ { uabv, ubav } thereexists a path ν ∈ P I such that ω and ν are dihomotopic and { λ ( ω ) , λ ( ν ) } = { uabv, ubav } .If two elements of M are not independent in A , we say that they are dependent in A . We denote by ≡ A the smallest congruence relation in M such that ab and ba arecongruent for all independent elements a, b ∈ M . The quotient monoid M/ ≡ A iscalled the trace monoid of A , and the canonical projection M → M/ ≡ A is denotedby tr A . The trace language of A is the set T L ( A ) = tr A ( L ( A )) ⊆ M/ ≡ A . Remarks. (i) We have x ≡ A y if and only if there exist elements x , . . . , x n ∈ M ( n ≥
1) such that x = x , y = x n and for each i ∈ { , . . . , n − } , x i = u i a i b i v i and x i +1 = u i b i a i v i for certain elements a i , b i , u i , v i ∈ M with a i and b i independent.(ii) If M = Σ ∗ for some finite set Σ and λ ( P ) ⊆ Σ, then the set D = { ( a, b ) ∈ Σ × Σ : a and b are dependent in A} is a dependency in Σ and M/ ≡ A is the classical trace monoid M (Σ , D ) (see e.g.[2, 12]). Moreover, in this situation, condition (3) of the definition of independencecan be replaced by the following condition:(3’) for any path ω = x ♯ x ♯ : J , K → P with { λ ( x ) , λ ( x ) } = { a, b } thereexists an element z ∈ P such that either d z = x and d z = x or d z = x and d z = x . Lemma 3.2.1.
Let A = ( P, I, F, λ ) be an M -HDA, ω ∈ P I and m ∈ M such that λ ( ω ) ≡ A m . Then there exists a path ν ∈ P I such that ω ∼ ν and λ ( ν ) = m .Proof. Since λ ( ω ) ≡ A m , there exist x , . . . , x n ∈ M such that λ ( ω ) = x , m = x n and for all i ∈ { , . . . , n − } , x i = u i a i b i v i and x i +1 = u i b i a i v i for certain elements a i , b i , u i , v i ∈ M with a i and b i independent. A simple induction using condition(3) of the definition of independence establishes the result. (cid:3) Lemma 3.2.2.
Let A = ( P, I, F, λ ) be an M -HDA. Then L ( A ) = tr − A ( T L ( A )) .Proof. It is enough to show that tr − A ( T L ( A )) ⊆ L ( A ). Consider an element m ∈ tr − A ( T L ( A )). Then there exists a path ω ∈ P I beginning in I and ending in F suchthat tr A ( λ ( ω )) = tr A ( m ), i.e. λ ( ω ) ≡ A m . By the preceding lemma, there exists apath ν ∈ P I with the same end points as ω and λ ( ν ) = m . Thus, m ∈ L ( A ). (cid:3) Theorem 3.2.3. If A = ( P, I, F, λ ) is an M -HDA such that P and P are finite,then T L ( A ) is a rational subset of the monoid M/ ≡ A . If, moreover, M is a finitelygenerated free monoid, then T L ( A ) is a recognisable subset of M/ ≡ A .Proof. It is well-known that the image of a rational set under a morphism of monoidsis rational and that a subset of a monoid is recognisable if its preimage under asurjective morphism of monoids is recognisable [13, prop. II.1.7, cor. II.2.12]. Theresult now follows from 1.9 and 3.2.2. (cid:3)
Stable HDAs.
In a general M -HDA it is possible that dependent actions arelocally independent. In stable M -HDAs, which we define next, this is impossible. Definition 3.3.1.
We say that an M -HDA A = ( P, I, F, λ ) is stable if for notwo dependent elements a, b ∈ M there exists an element z ∈ P such that { λ ( d z ) , λ ( d z ) } = { a, b } . EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 15
Proposition 3.3.2.
Let A = ( P, I, F, λ ) be a stable M -HDA. Then any two diho-motopic paths have congruent labels.Proof. Consider elementarily dihomotopic paths ω = α · ( d z ) ♯ · ( d z ) ♯ · β and ν = α · ( d z ) ♯ · ( d z ) ♯ · β . Since A is stable, the elements λ ( d z ) = λ ( d z ) and λ ( d z ) = λ ( d z ) are independent. Therefore λ ( d z ) λ ( d z ) ≡ A λ ( d z ) λ ( d z ). Hence λ ( ω ) = λ ( α ) λ ( d z ) λ ( d z ) λ ( β ) ≡ A λ ( α ) λ ( d z ) λ ( d z ) λ ( β ) = λ ( ν ). (cid:3) Deterministic HDAs.
A converse of 3.3.2 holds in deterministic HDAs,which are defined as follows:
Definition 3.4.1.
We say that an M -HDA is deterministic if it has exactly oneinitial state and if any two paths with the same starting point and the same labelare equal. Proposition 3.4.2.
Let A = ( P, I, F, λ ) be a deterministic M -HDA. Then any twopaths beginning at the same vertex and with congruent labels are dihomotopic.Proof. Let ω, ν ∈ P I be two paths such that ω (0) = ν (0) and λ ( ω ) ≡ A λ ( ν ).By 3.2.1, there exists a path α ∈ P I such that ω ∼ α and λ ( α ) = λ ( ν ). Since α (0) = ω (0) = ν (0), α = ν . (cid:3) Corollary 3.4.3.
Let A = ( P, I, F, λ ) be a deterministic and stable M -HDA. Thentwo paths with the same starting point are dihomotopic if and only if they havecongruent labels. Subdivided cubes.
Any subdivided cube J , k K ⊗· · ·⊗ J , k n K ( n, k , . . . , k n ≥
1) is the underlying precubical set of a stable and deterministic HDA A over thefree monoid on the set Σ = { l , . . . , l n } . The labelling function of A is given by λ ( j , . . . , j p − , [ j p , j p + 1] , j p +1 , . . . , j n ) = l p , and the sets of initial and final states are I = { (0 , . . . , } and F = { ( k , . . . , k n ) } .Any two distinct elements of Σ are independent in A , and for any element z ofdegree 2, λ ( d z ) = λ ( d z ). Consequently, A is stable. It follows from the factthat any vertex of J , k K ⊗ · · · ⊗ J , k n K is the starting point of at most one edgewith a given label that A is deterministic. One easily sees that given a path ω : J , r K → J , k K ⊗ · · · ⊗ J , k n K from ( a , . . . , a n ) to ( b , . . . , b n ), one has b i ≥ a i and λ ( ω ) ≡ A l b − a · · · l b n − a n n . Consequently, any two paths in A with the sameend points have congruent labels and are dihomotopic.3.6. Trace languages and weak morphisms.
The existence of a weak morphismbetween two M -HDAs permits us to compare their trace languages. The main pointis that weak morphisms preserve dihomotopy: Proposition 3.6.1.
Let f : P → Q be a weak morphism of precubical sets and ω and ν be dihomotopic paths in P . Then f I ( ω ) and f I ( ν ) are dihomotopic paths in Q .Proof. We may suppose that ω and ν are elementarily dihomotopic and that thereexist an element z ∈ P and paths α and β such that ω = α · ( d z ) ♯ · ( d z ) ♯ · β and ν = α · ( d z ) ♯ · ( d z ) ♯ · β . By 2.5.3, f I ( ω ) = f I ( α ) · f I (( d z ) ♯ · ( d z ) ♯ ) · f I ( β )and f I ( ν ) = f I ( α ) · f I (( d z ) ♯ · ( d z ) ♯ ) · f I ( β ). It follows that f I ( ω ) ∼ f I ( ν ) if f I (( d z ) ♯ · ( d z ) ♯ ) ∼ f I (( d z ) ♯ · ( d z ) ♯ ). We may thus suppose that ω = ( d z ) ♯ · ( d z ) ♯ and ν = ( d z ) ♯ · ( d z ) ♯ . Consider the morphism of precubical sets z ♯ : J , K ⊗ → P . Since f is a weak morphism, there exist integers k, l ≥
1, a morphism of precubicalsets ζ : J , k K ⊗ J , l K → Q and a dihomeomorphism φ : | J , K ⊗ | = [0 , → | J , k K ⊗ J , l K | = [0 , k ] × [0 , l ]such that f ◦ | z ♯ | = | ζ | ◦ φ . Consider the paths σ = ( d ι ) ♯ · ( d ι ) ♯ and τ =( d ι ) ♯ · ( d ι ) ♯ in J , K ⊗ . By 2.5.1, φ I ( σ ) and φ I ( τ ) are paths in J , k K ⊗ J , l K from φ (0 ,
0) = (0 ,
0) to φ (1 ,
1) = ( k, l ). Thus, φ I ( σ ) ∼ φ I ( τ ) (see 3.5). It follows that ζ ◦ φ I ( σ ) ∼ ζ ◦ φ I ( τ ). We have z ♯ ◦ σ = ω and z ♯ ◦ τ = ν . Hence f I ( ω ) = f I ( z ♯ ◦ σ ) = f I ( | z ♯ | I ( σ )) = ( f ◦ | z ♯ | ) I ( σ ) = ( | ζ | ◦ φ ) I ( σ ) = | ζ | I ( φ I ( σ )) = ζ ◦ φ I ( σ ) and similarly f I ( ν ) = ζ ◦ φ I ( τ ). The result follows. (cid:3) Theorem 3.6.2.
Let A = ( P, I, F, λ ) and B = ( P ′ , I ′ , F ′ , λ ′ ) be two M -HDAs suchthat B is stable and there exists a weak morphism f : A → B . Then for any twolabels l, m ∈ L ( A ) , l ≡ A m ⇒ l ≡ B m and a map T L ( A ) → T L ( B ) is given by tr A ( l ) tr B ( l ) . Consequently, if A and B are stable M -HDAs and A ↔ B , then
T L ( A ) = T L ( B ) .Proof. Let l, m ∈ L ( A ) such that l ≡ A m . By 3.2.1, there exist dihomotopic paths ω, ν ∈ P I beginning in I and ending in F such that λ ( ω ) = l and λ ( ν ) = m .By 3.6.1, f I ( ω ) ∼ f I ( ν ). Since B is stable, λ ′ ( f I ( ω )) ≡ B λ ′ ( f I ( ν )). The resultfollows. (cid:3) The trace category of an HDA.
The fundamental category of a precubicalset P is the category ~π ( P ) whose objects are the vertices of P and whose morphismsare the dihomotopy classes of paths in P (cp. [9, 11]). Given a weak morphismof precubical sets f : | P | → | Q | , the functor f ∗ : ~π ( P ) → ~π ( Q ) is defined by f ∗ ( v ) = f ( v ) ( v ∈ P ) and f ∗ ([ ω ]) = [ f I ( ω )] ( ω ∈ P I ). It is clear that thefundamental category construction is functorial.A vertex v of a precubical set P is said to be maximal (minimal) if there is noelement x ∈ P such that d x = v ( d x = v ). The sets of maximal and minimalelements of P are denoted by M ( P ) and m ( P ) respectively. The trace category ofan M -HDA A = ( P, I, F, λ ), T C ( A ), is the full subcategory of ~π ( P ) generated by I ∪ F ∪ m ( P ) ∪ M ( P ). Our definition of the trace category of an HDA is a variantof Bubenik’s definition of the fundamental bipartite graph of a d-space [1].Note that if f is a weak morphism from an M -HDA A = ( P, I, F, λ ) to an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) such that f ( m ( P ) ∪ M ( P )) ⊆ I ′ ∪ F ′ ∪ m ( P ′ ) ∪ M ( P ′ ),then the functor f ∗ : ~π ( P ) → ~π ( Q ) restricts to a functor f ∗ : T C ( A ) → T C ( B ).3.8. Accessible HDAs. An M -HDA A = ( P, I, F, λ ) is said to be accessible if forevery vertex v ∈ P there exists a path in P from an initial state to v (cp. [13, def.I.1.10]). If A and B are accessible, deterministic and stable M -HDAs and A ↔ B ,then L ( A ) = L ( B ), T L ( A ) = T L ( B ) and T C ( A ) ∼ = T C ( B ). This follows from 2.6.2,3.6.2 and the following proposition, which shows that in the case of accessible anddeterministic HDAs, A ↔ B is a very strong condition:
Proposition 3.8.1.
Let A = ( P, I, F, λ ) and B = ( P ′ , I ′ , F ′ , λ ′ ) be two accessibleand deterministic M -HDAs and f : A → B and g : B → A be weak morphisms.Then(i) f : P → P ′ and g : P ′ → P are inverse bijections;(ii) f I : P I → P ′ I and g I : P ′ I → P I are inverse bijections;(iii) f and g preserve maximal and minimal elements; EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 17 (iv) f ∗ : ~π ( P ) → ~π ( P ′ ) and g ∗ : ~π ( P ′ ) → ~π ( P ) are inverse isomorphisms;(v) f ∗ : T C ( A ) → T C ( B ) and g ∗ : T C ( B ) → T C ( A ) are inverse isomorphisms.Proof. It is enough to show (i), (ii) and (iii).(i) Since A and B are deterministic, there exist a ∈ P and a ′ ∈ P ′ such that I = { a } and I ′ = { a ′ } . We have f ( a ) = a ′ and g ( a ′ ) = a . Consider a path ω ∈ P I suchthat ω (0) = a . Then g I ( f I ( ω ))(0) = g ( f I ( ω )(0)) = g ( f ( ω (0))) = g ( f ( a )) = a = ω (0). Since λ ( g I ( f I ( ω ))) = λ ( ω ) and A is deterministic, g I ( f I ( ω )) = ω . Analo-gously, f I ( g I ( ω ′ )) = ω ′ for any path ω ′ ∈ P ′ I with ω ′ (0) = a ′ .Let v ∈ P be any vertex. Since A is accessible, there exists a path ω : J , k K → P such that ω (0) = a and ω ( k ) = v . As we have seen, g I ( f I ( ω )) = ω . Supposethat length( f I ( ω )) = k ′ . We have g ( f ( v )) = g ( f ( ω ( k ))) = g ( f I ( ω )( k ′ )) = g I ( f I ( ω ))( k ) = ω ( k ) = v . Similarly, f ( g ( v ′ )) = v ′ for all vertices v ′ ∈ P ′ . Thisshows that f and g are inverse bijections.(ii) Let ω ∈ P I be any path. Then g I ( f I ( ω ))(0) = g ( f ( ω (0)) = ω (0). Since λ ( g I ( f I ( ω ))) = λ ( ω ) and A is deterministic, g I ( f I ( ω )) = ω . Analogously, f I ( g I ( ω ′ )) = ω ′ for any path ω ′ ∈ P ′ I . This shows that f I and g I are inversebijections.(iii) It is enough to show that f preserves maximal and minimal elements..Let v ∈ P be an element such that f ( v ) is not maximal. Then there exists anelement x ′ ∈ P ′ such that x ′ ♯ (0) = d x ′ = f ( v ). Then g I ( x ′ ♯ )(0) = g ( x ′ ♯ (0)) = g ( f ( v )) = v . Since length( g I ( x ′ ♯ )) > v is not maximal. This shows that f preserves maximal elements. Similarly, f preserves minimal elements. (cid:3) Trace equivalences. A trace equivalence from an M -HDA A = ( P, I, F, λ )to an M -HDA B = ( P ′ , I ′ , F ′ , λ ′ ) is a weak morphism f from A to B such that f ( I ) = I ′ , f ( F ) = F ′ , f ( m ( P )) = m ( P ′ ), f ( M ( P )) = M ( P ′ ) and the functor f ∗ : T C ( A ) → T C ( B ) is an isomorphism. We write A ∼ t → B and say that A is a traceequivalent abstraction of B , or that B is a trace equivalent refinement of A , if thereexists a trace equivalence from A to B . It is clear that ∼ t → is a preorder on the classof M -HDAs. We have seen in 3.8 that for accessible and deterministic M -HDAs A and B , A ↔ B ⇒ A ∼ t → B . In the next section, we give another condition underwhich one has A ∼ t → B . Here, we show that in the stable and deterministic case, thetrace language of an M -HDA is invariant under trace equivalent abstraction: Theorem 3.9.1.
Let A be a stable M -HDA and B be a stable and deterministic M -HDA such that A ∼ t → B . Then a bijection T L ( A ) → T L ( B ) is given by tr A ( l ) tr B ( l ) .Proof. Write A = ( P, I, F, λ ) and B = ( P ′ , I ′ , F ′ , λ ′ ), and let f be a trace equiva-lence from A to B . By theorem 3.6.2, the map Ψ : T L ( A ) → T L ( B ), tr A ( l ) tr B ( l )is well-defined.We show that Ψ is surjective. Consider l ′ ∈ L ( B ). Let ω ′ be a path in P ′ beginning in I ′ and ending in F ′ such that λ ′ ( ω ′ ) = l ′ . Since f ∗ : T C ( A ) → T C ( B )is an isomorphism, there exists a path ω in P with end points in I ∪ F ∪ m ( P ) ∪ M ( P )such that f ∗ ([ ω ]) = [ ω ′ ]. It follows that f I ( ω ) ∼ ω ′ and that f I ( ω ) begins in I ′ and ends in F ′ . Since the functor f ∗ : T C ( A ) → T C ( B ) is an isomorphism,the map f : I ∪ F ∪ m ( P ) ∪ M ( P ) → I ′ ∪ F ′ ∪ m ( P ′ ) ∪ M ( P ′ ) is a bijection.Since f ( I ) = I ′ and f ( F ) = F ′ , it follows that ω begins in I and ends in F . Therefore λ ( ω ) ∈ L ( A ). By 3.3.2, l ′ = λ ′ ( ω ′ ) ≡ B λ ′ ( f I ( ω )) = λ ( ω ). Thus, tr B ( l ′ ) = tr B ( λ ( ω )) = Ψ( tr A ( λ ( ω ))).It remains to show that Ψ is injective. Consider l, m ∈ L ( A ) such that tr B ( l ) = tr B ( m ), i.e. l ≡ B m . Let ω and ν be paths in P beginning in I and ending in F such that λ ( ω ) = l and λ ( ν ) = m . Then f I ( ω ) and f I ( ν ) start in the only initialstate of B and λ ′ ( f I ( ω )) = l ≡ B m = λ ′ ( f I ( ν )). By 3.4.2, f I ( ω ) ∼ f I ( ν ). Thus, f ∗ ([ ω ]) = f ∗ ([ ν ]). Since f ∗ is an isomorphism, ω ∼ ν . By 3.3.2, l ≡ A m , i.e. tr A ( l ) = tr A ( m ). (cid:3) Homeomorphic abstraction
In this last section, we introduce the preorder relation of homeomorphic abstrac-tion and show that, under a mild condition, it is stronger than trace equivalentabstraction. The main point is to construct an inverse “up to dihomotopy” of themap induced on paths sets by a weak morphism that is a homeomorphism. Theconstruction of this inverse relies on cube paths and carrier sequences.4.1.
The preorder relation of homeomorphic abstraction.
Consider two M -HDAs A = ( P, I, F, λ ) and B = ( P ′ , I ′ , F ′ , λ ′ ). We say that A is a homeo-morphic abstraction of B , or that B is a homeomorphic refinement of A , if thereexists a weak morphism f from A to B that is a homeomorphism and satisfies f ( I ) = I ′ and f ( F ) = F ′ . In particular, if B is a subdivision of A , then A is ahomeomorphic abstraction of B . We use the notation A ≈ → B to indicate that A isa homeomorphic abstraction of B . It is clear that the relation ≈ → is a preorder onthe class of M -HDAs.4.2. Cube paths.
The cube paths we define next are essentially the same as thefull cube paths considered in [3] and the paths considered in [8]. A cube path ina precubical set P is a sequence of elements c = ( c , . . . , c m ) such that for all i ∈ { , . . . , m − } one of the following conditions holds:(i) c i = c i +1 ;(ii) ∃ r ∈ { , . . . , deg( c i ) } : c i +1 = d r c i ;(iii) ∃ r ∈ { , . . . , deg( c i +1 ) } : c i = d r c i +1 .The concatenation of two cube paths c = ( c , . . . , c m ) and d = ( d , . . . , d n ) suchthat d = c m is the cube path c · d = ( c , . . . , c m = d , . . . , d n ). If χ : Q → P isa morphism of precubical sets and c = ( c , . . . , c m ) is a cube path in Q , then wedenote by χ ( c ) the cube path ( χ ( c ) , . . . , χ ( c m )) in P .With every cube path we can associate an ordinary path. In the definition ofthis path we use the following notation: Let b ∈ P be an element of degree n > r ∈ { , . . . , n } . We define the element ˆ d r b ∈ P byˆ d r b = (cid:26) b, n = 1 ,d · · · d r − d r +1 · · · d n b, n > . Note that the element ˆ d r b is an edge of b leading from the initial vertex of b to theinitial vertex of the face d r b , i.e. we have d ˆ d r b = d · · · d b and d ˆ d r b = d · · · d d r b . Definition 4.2.1.
Let c = ( c , . . . , c m ) be a cube path in P . The path associ-ated with c is the path γ ( c ) ∈ P I defined recursively as follows: If either m = 1or m = 2 and deg( c ) ≤ deg( c ), then γ ( c ) is the path of length 0 defined by γ ( c )(0) = c ♯ (0 , . . . , m = 2 and deg( c ) > deg( c ), then consider the least EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 19 r ∈ { , . . . , deg( c ) } such that c = d r c and set γ ( c ) = ( ˆ d r c ) ♯ . For m > γ ( c ) = γ ( c , c ) · γ ( c , c ) · · · γ ( c m − , c m ). Remarks. (i) We remark that γ ( c ) is a path in P from c ♯ (0 , . . . ,
0) to c m♯ (0 , . . . , γ ( c · d ) = γ ( c ) · γ ( d ).4.3. Carrier sequences.
Let f : | P | → | P ′ | be a weak morphism of precubicalsets that is a homeomorphism. We shall show that with every path in P ′ we canassociate a cube path, which we call its carrier sequence. The concept of carriersequence is an adaptation of the one considered in [4]. Definition 4.3.1.
The carrier of an element b ∈ P ′ with respect to f is theunique element c ( b ) ∈ P for which there exists an element u ∈ ]0 , deg( c ( b )) suchthat f ([ c ( b ) , u ]) = [ b, ( , . . . , )]. The carrier sequence of a path ν = y ♯ · · · y n♯ in P ′ ( n ≥
1) with respect to f is the sequence c ( ν ) = (cid:0) c ( d y ) , c ( y ) , c ( d y ) , c ( y ) , c ( d y ) , . . . , c ( y n ) , c ( d y n ) (cid:1) . For a path ν ∈ P ′ I of length 0 we set c ( ν ) = ( c ( ν (0))). Remark.
For all a ∈ P , c ( f ( a )) = a . Indeed, f ([ a, ()]) = [ f ( a ) , ()]. Proposition 4.3.2.
The carrier sequence of a path in P ′ is a cube path in P .Proof. The proposition is an immediate consequence of lemma 4.3.3 below. (cid:3)
Remarks. (i) Since the carrier sequence of a path ω ∈ P ′ I is a cube path in P , wemay consider the path γ ( c ( ω )) in P . If v and w are vertices in P and ω is a pathin P ′ from f ( v ) to f ( w ), then γ ( c ( ω )) is a path in P from v to w .(ii) If ω and ν are paths in P ′ such that ω (length( ω )) = ν (0), then c ( ω · ν ) = c ( ω ) · c ( ν ). Lemma 4.3.3.
Consider elements b ∈ P ′ and k ∈ { , } . Then either c ( d k b ) = c ( b ) or there exists an r ∈ { , . . . , deg( c ( b )) } such that c ( d k b ) = d kr c ( b ) .Proof. Set n = deg( c ( b )). Note that n ≥ f ( c ( b )) , ()] = [ b, ]. Let u ∈ ]0 , n be the element such that f ([ c ( b ) , u ]) = [ b, ].Since f is a weak morphism, there exist integers k , . . . , k n ≥
1, a morphism ofprecubical sets β : J , k K ⊗ · · · ⊗ J , k n K → P ′ and a dihomeomorphism φ : | J , K ⊗ n | = [0 , n → | J , k K ⊗ · · · ⊗ J , k n K | = [0 , k ] × · · · × [0 , k n ]such that f ◦ | c ( b ) ♯ | = | β | ◦ φ . Since[ b, ] ∈ f ( | c ( b ) ♯ | ([0 , n )) = | β ( J , k K ⊗ · · · ⊗ J , k n K ) | , we have b ∈ β ( J , k K ⊗ · · · ⊗ J , k n K ). Write b = β ( i , . . . , i j − , [ i j , i j + 1] , i j +1 , . . . , i n ) . Since f ([ c ( b ) , φ − ( i , . . . , i j − , i j + , i j +1 , . . . , i n )])= | β | ( i , . . . , i j − , i j + , i j +1 , . . . , i n )= | β | ([( i , . . . , i j − , [ i j , i j + 1] , i j +1 , . . . , i n ) , ])= [ b, ]= f ([ c ( b ) , u ]) , we have [ c ( b ) , φ − ( i , . . . , i j − , i j + , i j +1 , . . . , i n )] = [ c ( b ) , u ]. Since u ∈ ]0 , n , itfollows that φ − ( i , . . . , i j − , i j + , i j +1 , . . . , i n ) = u and that i r / ∈ { , k r } for r = j .We have d b = β ( i , . . . , i n ) and f ([ c ( b ) , φ − ( i , . . . , i n )]) = | β | ( i , . . . , i n ) =[ d b, ()]. If i j >
0, then φ − ( i , . . . , i n ) ∈ ]0 , n and therefore c ( d b ) = c ( b ). If i j = 0, then, by proposition 2.1.1, exactly one coordinate of φ − ( i , . . . , i n ) isequal to 0 and none is equal to 1. This implies that there exist elements r ∈{ , . . . , n } and v ∈ ]0 , n − such that φ − ( i , . . . , i n ) = δ r ( v ). Therefore [ d b, ()] = f ([ c ( b ) , δ r ( v )]) = f ([ d r c ( b ) , v ]) and hence c ( d b ) = d r c ( b ). An analogous argumentshows that either c ( d b ) = c ( b ) or there exists an index r ∈ { , . . . , n } such that c ( d b ) = d r c ( b ). (cid:3) The map γ ◦ c . Consider a weak morphism of precubical sets f : | P | → | P ′ | that is a homeomorphism. We show that the map γ ◦ c : P ′ I → P I is a retractionof f I : Proposition 4.4.1.
For every path ω ∈ P I , γ ◦ c ( f I ( ω )) = ω .Proof. Suppose first that length( ω ) = 0. Then length( f I ( ω )) = 0 and f I ( ω )(0) = f ( ω (0)). Therefore c ( f I ( ω )) = ( c ( f ( ω (0))) = ( ω (0)). Thus, γ ( c ( f I ( ω ))) = γ ( ω (0)). Since γ ( ω (0))(0) = ω (0) ♯ (0) = ω (0), we have γ ( c ( f I ( ω ))) = ω .Suppose now that ω = x ♯ for some x ∈ P . Suppose that length( f I ( ω )) = k and that φ : | J , K | = [0 , → | J , k K | = [0 , k ] is a dihomeomorphism such that | f I ( ω ) | ◦ φ = f ◦ | ω | . Write y i = f I ( ω )([ i − , i ]). Then f I ( ω ) = y ♯ · · · y k♯ and hence c ( f I ( ω )) = c ( y ♯ ) · · · c ( y k♯ ). For i ∈ { , . . . , k } , φ − ( i − ) ∈ ]0 ,
1[ and f ([ x, φ − ( i − )]) = f ◦ | ω | ( φ − ( i − )) = | f I ( ω ) | ( i − ) = [ f I ( ω )([ i − , i ]) , ]) =[ y i , ]. Thus, c ( y i ) = x . For i ∈ { , . . . , k − } , φ − ( i ) ∈ ]0 ,
1[ and f ([ x, φ − ( i )]) = f ◦ | ω | ( φ − ( i )) = | f I ( ω ) | ( i ) = [ f I ( ω )( i ) , ()] = [ d y i , ()]. Thus, c ( d y i ) = x . We have c ( d y ) = c ( f I ( ω )(0)) = c ( f ( ω (0))) = ω (0) = d x and c ( d y k ) = c ( f I ( ω )( k )) = c ( f ( ω (1))) = ω (1) = d x . It follows that c ( f I ( ω )) = ( d x, x, . . . , x, d x ) and hencethat γ ( c ( f I ( ω )) = γ ( d x, x ) · γ ( x, x ) · · · γ ( x, x ) · γ ( x, d x ) = ( d x ) ♯ · · · ( d x ) ♯ · x ♯ = x ♯ = ω .Suppose finally that ω = x ♯ · · · x k♯ where ( x , . . . , x k ) is a sequence of ele-ments of P such that d x j +1 = d x j for all 1 ≤ j < k . Then γ ( c ( f I ( ω ))) = γ ( c ( f I ( x ♯ · · · x k♯ ))) = γ ( c ( f I ( x ♯ ))) · · · γ ( c ( f I ( x k♯ ))) = x ♯ · · · x k♯ = ω . (cid:3) Regular and weakly regular elements.
We say that an element x of aprecubical set is regular if x ♯ is injective. We say that an element x of degree n is weakly regular if the restrictions of x ♯ to the graded subsets ( J , K \ { } ) ⊗ n and ( J , K \ { } ) ⊗ n of J , K ⊗ n are injective. A precubical set or an M -HDA issaid to be (weakly) regular if all of its elements are (weakly) regular. Every 0- or1-dimensional precubical set is weakly regular. The directed circle , which is theprecubical set consisting of exactly one vertex and exactly one 1-cube, is weaklyregular but not regular. It can be shown that a precubical set is weakly regular ifand only if all elements of degree 2 are weakly regular. Since we do not need thisresult in this paper, we do not include a proof.4.6. Compatibility of γ ◦ c with dihomotopy. The purpose of this subsectionis to show that the map γ ◦ c sends dihomotopic paths to dihomotopic paths, atleast if we restrict ourselves to weakly regular precubical sets. EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 21
Lemma 4.6.1.
Consider two weak morphisms of precubical sets f : | P | → | P ′ | and g : | Q | → | Q ′ | and two morphisms of precubical sets χ : Q → P and χ ′ : Q ′ → P ′ such that f and g are homeomorphisms and f ◦ | χ | = | χ ′ | ◦ g . Then for all a ∈ Q ′ , c ( χ ′ ( a )) = χ ( c ( a )) .Proof. Consider u ∈ ]0 , deg( c ( a )) such that g ([ c ( a ) , u ]) = [ a, ( , . . . , )]. We have f ([ χ ( c ( a )) , u ]) = f ◦ | χ | ([ c ( a ) , u ]) = | χ ′ | ◦ g ([ c ( a ) , u ]) = | χ ′ | ([ a, ( , . . . , )]) =[ χ ′ ( a ) , ( , . . . , )] and hence c ( χ ′ ( a )) = χ ( c ( a )). (cid:3) Lemma 4.6.2.
Let χ : Q → P be a morphism of precubical sets such that P isweakly regular. Then for any cube path c in Q , γ ( χ ( c )) = χ ◦ γ ( c ) .Proof. Let c = ( c , . . . , c m ) be a cube path in Q . Recall that χ ( c ) is the cube path( χ ( c ) , . . . , χ ( c m )) in P . In order to proof the lemma, it is enough to consider thecase m ≤
2. If either m = 1 or m = 2 and deg( c ) ≤ deg( c ), then γ ( c ) is the pathin Q of length 0 defined by γ ( c )(0) = c ♯ (0 , . . . , χ ◦ γ ( c ) is the path in P of length 0 given by ( χ ◦ γ ( c ))(0) = χ ( c ♯ (0 , . . . , χ ( c ) ♯ (0 , . . . , χ ◦ γ ( c ) = γ ( χ ( c )). Suppose that m = 2 and deg( c ) > deg( c ). Consider theleast r ∈ { , . . . , deg( c ) } such that c = d r c . We have γ ( c ) = ( ˆ d r c ) ♯ and hence χ ◦ γ ( c ) = χ ◦ ( ˆ d r c ) ♯ = χ ( ˆ d r c ) ♯ . Moreover, χ ( c ) = χ ( d r c ) = d r ( χ ( c )). Since P is weakly regular, this is the only such r and γ ( χ ( c )) = ( ˆ d r χ ( c )) ♯ = χ ( ˆ d r c ) ♯ = χ ◦ γ ( c ). (cid:3) Proposition 4.6.3.
Let P be a weakly regular precubical set and f : | P | → | P ′ | bea weak morphism that is a homeomorphism. Then for any two paths ω, ν ∈ P ′ I , ω ∼ ν ⇒ γ ◦ c ( ω ) ∼ γ ◦ c ( ν ) .Proof. We may suppose that ω and ν are elementarily dihomotopic paths of length2. We may further suppose that there exists an element y ∈ P ′ such that ω =( d y ) ♯ · ( d y ) ♯ and ν = ( d y ) ♯ · ( d y ) ♯ . Set x = c ( y ) and n = deg( x ). Since f is a weakmorphism, there exist a morphism of precubical sets β : J , k K ⊗ · · · ⊗ J , k n K → P ′ and a dihomeomorphism φ : | J , K ⊗ n | = [0 , n → | J , k K ⊗ · · · ⊗ J , k n K | = [0 , k ] × · · · × [0 , k n ]such that f ◦ | x ♯ | = | β | ◦ φ . Since[ y, ( , )] ∈ f ( | x ♯ | ([0 , n )) = | β ( J , k K ⊗ · · · ⊗ J , k n K ) | , we have y ∈ β ( J , k K ⊗ · · · ⊗ J , k n K ). Consider z ∈ J , k K ⊗ · · · ⊗ J , k n K suchthat β ( z ) = y . Consider the paths ξ = ( d z ) ♯ · ( d z ) ♯ and θ = ( d z ) ♯ · ( d z ) ♯ .Since ξ and θ have the same end points, so do γ ( c ( ξ )) and γ ( c ( θ )). Since anytwo paths in J , K ⊗ n with the same end points are dihomotopic (see 3.5), we have γ ( c ( ξ )) ∼ γ ( c ( θ )) and hence x ♯ ◦ γ ( c ( ξ )) ∼ x ♯ ◦ γ ( c ( θ )). Since P is weakly regular,lemma 4.6.2 implies that γ ( x ♯ ( c ( ξ ))) ∼ γ ( x ♯ ( c ( θ ))). By lemma 4.6.1, we have x ♯ ( c ( ξ )) = x ♯ ( c (( d z ) ♯ · ( d z ) ♯ ))= x ♯ ( c ( d d z ) , c ( d z ) , c ( d d z ) , c ( d z ) , c ( d d z ))= ( x ♯ ( c ( d d z )) , x ♯ ( c ( d z )) , x ♯ ( c ( d d z )) , x ♯ ( c ( d z )) , x ♯ ( c ( d d z )))= ( c ( β ( d d z )) , c ( β ( d z )) , c ( β ( d d z )) , c ( β ( d z )) , c ( β ( d d z )))= ( c ( d d y ) , c ( d y ) , c ( d d y ) , c ( d y ) , c ( d d y ))= c ( ω ) . Similarly, x ♯ ( c ( θ )) = c ( ν ). Thus, γ ( c ( ω )) ∼ γ ( c ( ν )). (cid:3) The composite f I ◦ γ ◦ c . Let f : | P | → | P ′ | be a weak morphism of precubicalsets that is a homeomorphism. We have seen in 4.4.1 that the composite γ ◦ c ◦ f I is the identity. Here, we show that “up to dihomotopy”, the composite f I ◦ γ ◦ c isthe identity for paths with end points in f ( P ). We need three lemmas. Lemma 4.7.1.
Consider elements b ∈ J , k K ⊗ · · · ⊗ J , k m K ( m, k , . . . , k m ≥ and u ∈ ]0 , deg( b ) . Then b ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ) if and only if [ b, u ] ∈ [0 , k [ × · · · × [0 , k m [ .Proof. Write b = ( b , . . . , b m ) and suppose deg( b ) = p . Then there exist indices 1 ≤ i < · · · < i p ≤ m such that deg( b i q ) = 1 for q ∈ { , . . . , p } and b i ∈ { , . . . , k i } for i / ∈ { i , . . . , i p } . For each q ∈ { , . . . , p } there exists an element j q ∈ { , . . . , k q − } such that b i q = [ j q , j q + 1]. We have [ b, u ] = [ b, ( u , . . . , u p )] = ( t , . . . , t m ) ∈ [0 , k ] × · · · × [0 , k m ] where t i = (cid:26) b i , i / ∈ { i , . . . , i p } ,j q + u q , q ∈ { , . . . , p } . The result follows. (cid:3)
Lemma 4.7.2.
Let φ : | J , l K ⊗ · · · ⊗ J , l m K | → | J , k K ⊗ · · · ⊗ J , k m K | ( k i , l i ≥ be a weak morphism of precubical sets that is a dihomeomorphism. Then for anyelement b ∈ J , k K ⊗ · · · ⊗ J , k m K , b ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ) if andonly if c ( b ) ∈ ( J , l K \ { l } ) ⊗ · · · ⊗ ( J , l m K \ { l m } ) .Proof. Consider an element b ∈ J , k K ⊗· · ·⊗ J , k m K , and let u ∈ ]0 , deg( c ( b )) be theelement such that φ ([ c ( b ) , u ]) = [ b, ( , . . . , )]. By proposition 2.1.1, [ b, ( , . . . , )] ∈ [0 , k [ × · · ·× [0 , k m [ if and only if [ c ( b ) , u ] ∈ [0 , l [ × · · ·× [0 , l m [. The preceding lemmaimplies the result. (cid:3) Lemma 4.7.3.
Suppose that P is weakly regular. Let x ∈ P n ( n ≥ be anelement, β : J , k K ⊗ · · · ⊗ J , k n K → P ′ ( k , . . . , k n ≥ be a morphism of precubicalsets and φ : | J , K ⊗ n | → | J , k K ⊗ · · · ⊗ J , k n K | be a dihomeomorphism such that f ◦ | x ♯ | = | β | ◦ φ . Then the restriction of β to ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k n K \ { k n } ) is injective.Proof. Let a, b ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k n K \ { k n } ) be distinct elements ofthe same degree. Then [ a, ( , . . . , )] = [ b, ( , . . . , )] ∈ [0 , k [ × · · · × [0 , k n [ andtherefore, by 2.1.1, φ − ([ a, ( , . . . , )]) = φ − ([ b, ( , . . . , )]) ∈ [0 , n . Let u ∈ ]0 , deg( c ( a )) and v ∈ ]0 , deg( c ( b )) be the uniquely determined elements such that φ ([ c ( a ) , u ]) = [ a, ( , . . . , )] and φ ([ c ( b ) , v ]) = [ b, ( , . . . , )]. By 4.7.2, c ( a ) , c ( b ) ∈ ( J , K \ { } ) ⊗ n . If c ( a ) = c ( b ), then u = v and hence | x ♯ | ([ c ( a ) , u ]) = | x ♯ | ([ c ( b ) , v ]).If c ( a ) = c ( b ), then this holds because x is weakly regular. Since f is injective, itfollows that | β | ([ a, ( , . . . , )]) = | β | ([ b, ( , . . . , )]) and hence that β ( a ) = β ( b ). (cid:3) Proposition 4.7.4.
Suppose that P is weakly regular. Let v, w ∈ P be verticesand ω be a path in P ′ from f ( v ) to f ( w ) . Then f I ( γ ◦ c ( ω )) ∼ ω .Proof. It is enough to show that there exists a path α ∈ P I such that f I ( α ) ∼ ω .Indeed, given such a path α , we have, by 4.4.1 and 4.6.3, α = γ ◦ c ( f I ( α )) ∼ γ ◦ c ( ω )and hence, by 3.6.1, f I ( γ ◦ c ( ω )) ∼ f I ( α ) ∼ ω . EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 23
In order to construct α , we proceed by induction on the length of ω . Supposefirst that length( ω ) = 0. Consider the path α : J , K → P defined by α (0) = v .Then f I ( α ) is the path of length 0 given by f I ( α )(0) = f ( α (0)) = f ( v ) = ω (0).Hence f I ( α ) = ω . It follows that the assertion holds for paths of length 0.Suppose that length( ω ) = n > P ′ that has length < n and end points in f ( P ). Write ω = y ♯ · · · y n♯ . Since d y n = ω ( n ) = f ( w ), we have c ( d y n ) = w . It follows that deg( c ( d y n )) = 0 < c ( y n )). Let l be the lowest index such that deg( c ( d y l )) < deg( c ( y l )).Consider the path ν = y ♯ · · · y l♯ , and let σ be the unique path in P ′ such that ω = ν · σ . Set x = c ( y l ) and m = deg( x ). We shall show that there exist an index r ∈ { , . . . , m } and a path ξ in P ′ such that ν ∼ f I (( ˆ d r x ) ♯ ) · ξ . This permits usto terminate the induction as follows: Since ξ · σ is a path of length < n from f ( d · · · d d r x ) to f ( w ), the inductive hypothesis implies that there exists a path τ ∈ P I such that f I ( τ ) ∼ ξ · σ . Since f ( τ (0)) = f I ( τ )(0) = f ( d · · · d d r x ) and f is a homeomorphism, τ (0) = d · · · d d r x and we may set α = ( ˆ d r x ) ♯ · τ . We thenhave f I ( α ) = f I (( ˆ d r x ) ♯ ) · f I ( τ ) ∼ f I (( ˆ d r x ) ♯ ) · ξ · σ ∼ ν · σ = ω .It remains to determine r and ξ . Let β : J , k K ⊗ · · · ⊗ J , k m K → P ′ be theunique morphism of precubical sets and φ : | J , K ⊗ m | → | J , k K ⊗ · · · ⊗ J , k m K | be the unique dihomeomorphism such that f ◦ | x ♯ | = | β | ◦ φ . We construct apath θ : J , l K → J , k K ⊗ · · · ⊗ J , k m K such that θ (0) = (0 , . . . ,
0) and | β | I ( θ ) = β ◦ θ = ν . For i ∈ { , . . . , l } , by 4.3.3, we have either c ( d y i ) = c ( y i ) or c ( d y i ) = d q c ( y i ) for some q ∈ { , . . . , deg( c ( y i )) } . For i < l , by the definition of l , c ( y i ) = c ( d y i ) = c ( d y i +1 ). Hence for i < l , c ( y i ) = c ( y i +1 ) or c ( y i ) = d q c ( y i +1 ) forsome q ∈ { , . . . , deg( c ( y i +1 )) } . It follows that c ( y i ) ∈ x ♯ (( J , K \ { } ) ⊗ m ) for all i ∈ { , . . . , l } . For i ∈ { , . . . , l } let u i ∈ ]0 , deg( c ( y i )) be the unique element suchthat [ y i , ] = f ([ c ( y i ) , u i ]). Since c ( y i ) ∈ x ♯ (( J , K \{ } ) ⊗ m ), there exists an element v i ∈ [0 , m such that [ c ( y i ) , u i ] = [ x, v i ] and hence [ y i , ] = f ([ x, v i ]) = f ◦| x ♯ | ( v i ) = | β |◦ φ ( v i ). By 2.1.1, φ ( v i ) ∈ [0 , k [ × · · ·× [0 , k m [. By 4.7.1, it follows that there existelements z i ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ) and w i ∈ ]0 , deg( z i ) such that φ ( v i ) = [ z i , w i ]. We have [ β ( z i ) , w i ] = | β | [ z i , w i ] = | β | ◦ φ ( v i ) = [ y i , ] and hencedeg( z i ) = 1, β ( z i ) = y i and w i = . Since z i ∈ ( J , k K \{ k } ) ⊗· · ·⊗ ( J , k m K \{ k m } ),also d z i ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ). For i < l , c ( d z i ) = c ( z i )because otherwise there would exist an index q such that c ( d z i ) = d q c ( z i ). Lemma4.6.1 would then imply that c ( d y i ) = c ( β ( d z i )) = x ♯ ( c ( d z i )) = x ♯ ( d q c ( z i )) = d q x ♯ ( c ( z i )) = d q c ( β ( z i )) = d q c ( y i ), but c ( d y i ) = c ( y i ). By lemma 4.7.2, since z i ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ), c ( z i ) ∈ ( J , K \ { } ) ⊗ m . Hencefor i < l , c ( d z i ) ∈ ( J , K \ { } ) ⊗ m . This implies, again by lemma 4.7.2, that,for i < l , d z i ∈ ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ). By lemma 4.7.3, therestriction of β to the set ( J , k K \ { k } ) ⊗ · · · ⊗ ( J , k m K \ { k m } ) is injective. For i < l , β ( d z i ) = d y i = d y i +1 = β ( d z i +1 ) and therefore d z i = d z i +1 . Thus, θ = z ♯ · · · z l♯ is a well-defined path in J , k K ⊗ · · · ⊗ J , k m K . By construction, | β | I ( θ ) = β ◦ θ = ν . We have d y = ν (0) = f ( v ) and hence c ( d y ) = v . It followsthat v ∈ x ♯ (( J , K \ { } ) ⊗ m ) and hence that v = d · · · d x . We have β (0 , . . . ,
0) = | β | ◦ φ (0 , . . . ,
0) = f ◦ | x ♯ | (0 , . . . ,
0) = f ( d · · · d x ) = f ( v ) = ν (0) = β ( θ (0))and therefore θ (0) = (0 , . . . , x ♯ ( c ( z l )) = c ( β ( z l )) = c ( y l ) = x . Hence c ( z l ) = ι m . We have c ( d z l ) = d r c ( z l ) = d r ι m for some r ∈ { , . . . , m } . Otherwise we would have c ( d z l ) = c ( z l ) = ι m and hence c ( d y l ) = c ( β ( d z l )) = x ♯ ( c ( d z l )) = x ♯ ( ι m ) = x , which isimpossible because deg( c ( d y l )) < deg( x ). By proposition 2.1.2, there exists anindex s ∈ { , . . . , m } such that φ ( δ r ([0 , m − )) = [0 , k ] × · · · × [0 , k s − ] × { k s } × [0 , k s +1 ] × · · · × [0 , k m ] . Since φ − ([ d z l , ()]) ∈ δ r ([0 , m − ), we have d z l ∈ J , k K ⊗ · · · ⊗ J , k s − K ⊗ { k s } ⊗ J , k s +1 K ⊗ · · · ⊗ J , k m K . Consider the path φ I (( ˆ d r ι m ) ♯ ) in J , k K ⊗· · ·⊗ J , k m K . Set p = length( φ I (( ˆ d r ι m ) ♯ )).We have φ I (( ˆ d r ι m ) ♯ )(0) = φ (( ˆ d r ι m ) ♯ (0)) = φ (0 , . . . ,
0) = (0 , . . . , φ I (( ˆ d r ι m ) ♯ )( p ) = φ (( ˆ d r ι m ) ♯ (1)) = φ ( d · · · d d r ι m ) = (0 , . . . , , k s , , . . . , ρ be any path in J , k K ⊗ · · · ⊗ J , k m K such that ρ (0) = (0 , . . . , , k s , , . . . ,
0) and ρ (length( ρ )) = d z l . By 3.5, θ ∼ φ I (( ˆ d r ι m ) ♯ ) · ρ . Set ξ = | β | I ( ρ ) = β ◦ ρ . We have ν = | β | I ( θ ) ∼ | β | I ( φ I (( ˆ d r ι m ) ♯ ) · ρ ) = ( | β | ◦ φ ) I (( ˆ d r ι m ) ♯ ) · ξ = ( f ◦ | x ♯ | ) I (( ˆ d r ι m ) ♯ ) · ξ = f I ( | x ♯ | I (( ˆ d r ι m ) ♯ )) · ξ = f I (( ˆ d r x ) ♯ ) · ξ . (cid:3) Maximal and minimal elements.
Consider again a weak morphism of pre-cubical sets f : | P | → | P ′ | that is a homeomorphism. Proposition 4.8.1.
We have f ( M ( P )) = M ( P ′ ) and f ( m ( P )) = m ( P ′ ) .Proof. Consider an element v ∈ P such that f ( v ) is not maximal in P ′ . Thenthere exists an element y ∈ P ′ such that d y = f ( v ). This implies that c ( y ) ∈ P and that d c ( y ) = c ( d y ) = v . Thus, v is not maximal in P . It follows that f ( M ( P )) ⊆ M ( P ′ ). In order to establish the reverse inclusion, consider a maximalelement v ′ ∈ P ′ . We show first that c ( v ′ ) is a vertex in P . Suppose that this isnot the case and that deg( c ( v ′ )) = m >
0. Let β : J , k K ⊗ · · · ⊗ J , k m K → P ′ bethe unique morphism of precubical sets and φ : | J , K ⊗ m | → | J , k K ⊗ · · · ⊗ J , k m K | be the unique dihomeomorphism such that f ◦ | c ( v ′ ) ♯ | = | β | ◦ φ . Let u ∈ ]0 , m be the uniquely determined element such that f ([ c ( v ′ ) , u ]) = [ v ′ , ()]. Considerelements z ∈ J , k K ⊗ · · · ⊗ J , k m K and w ∈ ]0 , deg( z ) such that φ ( u ) = [ z, w ]. Since[ β ( z ) , w ] = | β | ◦ φ ( u ) = f ◦ | c ( v ′ ) ♯ | ( u ) = f ([ c ( v ′ ) , u ]) = [ v ′ , ()], we have deg( z ) =0, w = () and β ( z ) = v ′ . Since u ∈ ]0 , m , [ z, ()] = φ ( u ) ∈ ]0 , k [ × · · · × ]0 , k m [.Therefore z = ( k , . . . , k m ) and there exists a 1-cube y ∈ J , k K ⊗ · · · ⊗ J , k m K suchthat d y = z . It follows that d β ( y ) = v ′ , which is impossible. Therefore c ( v ′ ) is avertex in P . We have f ( c ( v ′ )) = v ′ . We show that c ( v ′ ) ∈ M ( P ). Suppose that thisis not the case. Then there exists an element x ∈ P such that c ( v ′ ) = d x . Since f I ( x ♯ )(0) = f ( c ( v ′ )) = v ′ , there exists a path in P ′ of non-zero length beginning in v ′ . This contradicts the hypothesis that v ′ is maximal in P ′ . Thus, c ( v ′ ) ∈ M ( P ).It follows that M ( P ′ ) ⊆ f ( M ( P )) and hence that f ( M ( P )) = M ( P ′ ).An analogous argument shows that f ( m ( P )) = m ( P ′ ). (cid:3) Homeomorphic abstraction and trace equivalence.
Let A = ( P, I, F, λ )and B = ( P ′ , I ′ , F ′ , λ ′ ) be two M -HDAs. Theorem 4.9.1. If A is weakly regular and A ≈ → B , then A ∼ t → B .Proof. Let f : | P | → | P ′ | be a weak morphism from A to B that is a homeomor-phism and that satisfies f ( I ) = I ′ and f ( F ) = F ′ . By 4.8.1, f ( M ( P )) = M ( P ′ )and f ( m ( P )) = m ( P ′ ). It follows that the functor f ∗ : ~π ( P ) → ~π ( P ′ ) re-stricts to a functor f ∗ : T C ( A ) → T C ( B ) that is a bijection on objects. Con-sider vertices v, w ∈ I ∪ F ∪ m ( P ) ∪ M ( P ). By 4.4.1, 4.6.3 and 4.7.4, the map EAK MORPHISMS OF HIGHER DIMENSIONAL AUTOMATA 25