generalising conduché’s theorem
TRANSCRIPT
Appl Categor StructDOI 10.1007/s10485-009-9200-9
Generalising Conduché’s Theorem
Stefano Kasangian · Anna Labella · Andrea Montoli
Received: 6 September 2008 / Accepted: 7 April 2009© Springer Science + Business Media B.V. 2009
Abstract In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009)we proved a form of Conduché’s theorem for LSymcat-categories, where L was ameet-semilattice monoid. The original theorem was proved in Conduché (CR AcadSci Paris 275:A891–A894, 1972) for ordinary categories. We showed also that the“lifting factorisation condition” used to prove the theorem is strictly related to thenotion of state for processes whose semantics is modeled by LSymcat-categories. Inthis note we resume the content of Kasangian and Labella (J Pure Appl Algebra,2009) in order to generalise the theorem to other situations, mainly arising fromcomputer science. We will consider PSymcat-categories, where P is slightly moregeneral than a meet-semilattice monoid, in which the lifting factorisation conditionfor a PSymcat-functor still implies the existence of a right adjoint to its correspondinginverse image functor.
Keywords Enriched categories · 2-categories · Computing processes
Mathematics Subject Classifications (2000) 18D20 · 68Q85
As a further output of the “Australian-Italian axis”, this paper is dedicated to Max Kelly whoseinvitation to the first author commenced this international collaboration,which continued alsowith the second author.
S. Kasangian (B) · A. MontoliDipartimento di Matematica, Università di Milano, Milan, Italye-mail: [email protected]
A. Montolie-mail: [email protected]
A. LabellaDipartimento di Informatica, “Sapienza” Università di Roma, Rome, Italye-mail: [email protected]
S. Kasangian et al.
1 Introduction
According to Lawvere [13] (Lawvere, unpublished manuscript), when modelingphysical processes, a control functor strictly reflecting morphism factorisation in-duces a notion of state on its domain. This intuition works also in the case of com-puting processes [1, 4], if we consider enriched categories and functors [10] instead oftheir ordinary counterparts. The factorisation condition in its set-theoretical form isa necessary and sufficient condition for a functor to prove Conduché’s theorem [2],i.e. the existence of a right adjoint to the corresponding inverse image functor. Wewill call it the Conduché condition.
In some companion works [7] we considered the case of Conduché’s theoremfor models for computational processes [6] and we proved that the analogue ofConduché condition for an enriched functor F is still sufficient to guarantee theexistence of a right adjoint to the corresponding inverse image functor. This factconsists in a generalisation of Conduché’s theorem from (Set-based) categories tocategories based on LSymcat, where L is the lp 2-category [6] generated by acomplete meet-semilattice monoid, in particular by a free monoid (see Section 5).It is immediate that Set is a particular case of LSymcat, when L is the trivial freemonoid generated by the empty set. Characterisation of models for interleavingconcurrent processes [14] and bisimulation relations [15] can be achieved in termsof this property (De Nicola et al., submitted for publication) and the related notionof state and its preservation by the given equivalences.
The category LSymcat has sums, a (non-commutative) tensor product which isright distributive w.r.t. sums, and it is left closed. In fact, when L is a free monoid,the finite part of LSymcat is the free category on L with such properties [11].
Keeping this intuition in mind, we can think about LSymcat as the category ofL-labeled trees, because, in the case L is a free monoid, a LSymcat-category is a setof paths labeled by words in L and glued together by a common prefix.
Example 1 Consider the trees
� �
��
��
��
��a a
c b
y x
��
���
��
��
a
c b
y x
They both have two paths, x and y, that are labeled with ab and ac respectively; inthe one on the left, x and y are not glued at all (i.e., they are glued by the empty wordε); in the one on the right, x and y are glued by the initial a).
Since a form of Conduché’s theorem has been proved both in the cases of ordinarycategories and LSymcat-categories, the next problem has been looking for otherinstances in which we can still prove the same kind of result. A first idea wasto consider a concrete category T with properties which do actually characteriseLSymcat, that is a monoidal category with sums, a tensor product which is rightdistributive w.r.t. sums, and left closed. Under these conditions we immediately
Generalising Conduché’s theorem
found counterexamples supplied by algebraic categories like Ab (the category ofabelian groups).
On the other hand, we are able to prove a form of Conduché’s theorem for acategory PSymcat, where P is a sufficiently general lp 2-category, instead of a meet-semilattice monoid (see Section 3). The difference between L and P relies on thefact that the lp 2-category structure in L is induced by its multiplication as a monoid,which actually provides also the tensor product, whereas the lp 2-category P isslightly more general than a meet-semilattice and a tensor product on it is definedindependently.
We will describe some examples where instances of the category P have a precisemeaning in the theory of computing processes.
2 The Classical Case and the Conduché Condition in the Enriched Case
It is well known that, given a functor between two (Set)-categories F : B → C, thereexists an inverse image functor F∗ : Cat|C → Cat|B with a left adjoint �F.
In 1972 Conduché proved (see [2]) the following:
Theorem 1 Given a functor between two (Set)-categories F : B → C, the inverseimage functor F∗ has a right adjoint �F : Cat|B → Cat|C iff F satisfies the followingreflecting factorisation condition:
if a morphism F(α) = β2β1 in C, then there is at least a factorisation α = α2α1
in B such that F(α1) = β1 and F(α2) = β2. If two such factorisations do exist for thesame α, then they are connected by a zig-zag of morphisms between the intermediateobjects.1 These morphisms must make commutative the resulting diagrams, and bemapped by F into identities.
We recall that Cat, i.e. the category of (Set)-categories with their functors isdefinable because Set is monoidal, being cartesian closed [3]. On the other hand,the proof of the existence of an inverse image functor, as well as the proof of thetheorem, exploits the fact that Cat has pullbacks defined from pullbacks in Set.
Let us now replace Set with a monoidal category T. From now on, we will supposethat T is concrete and has pullbacks.
Definition 1 A concrete category T is an ordinary category equipped with a faithfulfunctor Supp : T → Set.
Proposition 1 Let T be a monoidal category (with pullbacks). If F is a T-functorbetween two T-categories B and C, then there is an inverse image functor F∗ fromT-Cat|C to T-Cat|B.
1This means that there exist morphisms γ1, . . . , γk and objects k0, . . . , kn such that k0 = k, kn = k′,where k and k′ are the intermediate objects, and γi belongs to either C[ki, ki+1] or C[ki+1, ki].
S. Kasangian et al.
Proof Let ψ : Y → C be a T-functor, F−1(Y) is the T-category with pairs (b , y)
s.t. F(b) = ψ(y) as objects and enriched on T as follows: F−1(Y)[(b , y), (b ′, y′)] isthe pullback B[b , b ′] × Y[y, y′] over C[F(b) = ψ(y), F(b ′) = ψ(y′)]. Now, F∗ψ isthe T-functor from F−1(Y) to B corresponding to the first projection. Given ψ ′ :Z → C and a T-functor α : Y → Z s.t. ψ ′α = ψ , then F∗(α) is defined from F−1(Y)
to F−1(Z) as follows: F∗(α)(b , y) = (b , α(y)) and F∗(α) induces a morphism fromF−1(Y)[(b , y), (b ′, y′)] to F−1(Z)[(b , α(y)), (b ′, α(y′))], exploiting the morphism go-ing from Y[y, y′] to Z[α(y), α(y′)] and the pullback property. ��
F∗ has always a left adjoint �F, given by the composition. Let us now reformulatethe factorisation condition for a T-functor over a monoidal category T and try toprove the analogue of Conduché’s theorem, at least to state a condition sufficient toguarantee the existence of a right adjoint �F to F∗.
Definition 2 Given a T-category C, we say that (k, f1, f2) is a factorisation of C[c′, c]for (t1 ⊗ t2, f ), where t1 is a singleton supported object and f : t1 ⊗ t2 → C[c′, c] is amorphism in T, if there is an object k and two arrows f1 : t1 → C[k, c] and f2 : t2 →C[c′, k] such that the following diagram commutes (m is the multiplication morphismin C):
t1 ⊗ t2 �f1 ⊗ f2 C[k, c] ⊗ C[c′, k]
�
mc′,k,c
���������f
C[c′, c]
Definition 3 Given a T-functor F : B → C, F is Conduché iff
– given a path π in C[F(b ′), F(b)] and an inverse image of it, say p ∈ (Fb ′,b )−1(π)
in B[b ′, b ], for every factorisation (k, f1, f2) of C[F(b ′), F(b)] for (t1 ⊗ t2, f ),with π ∈ Im( f ), and f ′ s. t. Fb ′,b f ′ = f and p ∈ Im( f ′), there is a factorisation(h, f ′
1, f ′2) of B[b ′, b ], such that F(h) = k and the following diagram commutes
�Fh,b ⊗ Fb ′,h
�Fb ′,b
B[h, b ] ⊗ B[b ′, h]
B[b ′, b ]
������ f ′1 ⊗ f ′
2
������� f ′�
mb ′,h,b
C[k, F(b)] ⊗ C[F(b ′), k]
t1 ⊗ t2
C[F(b ′), F(b)]
�������� f1 ⊗ f2
��������f �
mF(b ′),k,F(b)
Generalising Conduché’s theorem
– If two such factorisations (h, f ′1, f ′
2) and (h′, f ′′1 , f ′′
2 ) exist for p, then theyare connected, i.e. there exist a family of factorisations and a zig-zag of pathsconnecting the intermediate objects in B,2 making the resulting diagrams com-mute and becoming trivial when mapped by F.
Remark 1 In some points of the main proof we will use a weaker form of factorisa-tion, namely the one closer to the set-theoretical formulation, where both t1 and t2 aresupposed to be singleton supported objects. We will refer to it as light factorisationand, to the corresponding Conduché condition, as light Conduché condition.
3 The General Case of PSymcat
Let us consider a locally posetal symmetric 2-category P [16], where
– hom-posets are complete subsets of a unique meet-semilattice S and– composition of 1-cells is given by the intersection in S– a monoidal structure is defined on P by a tensor product •.
Then the corresponding category T = PSymcat, if it is provided with a tensorproduct defined from •, satisfying some conditions, will allow us to prove in the nextsection the existence of a right adjoint for the inverse image functor of any ConduchéT-functor, as in Conduché’s theorem.
For this purpose, we first describe the structure of T = PSymcat.
Definition 4 A symmetric P-category X is a triple (X, eX , aX), where X is the set ofpaths or support, eX : X → Ob j(P) is the extent map and aX : X × X → Arr(P) isthe agreement between paths, such that, for every x, y, z ∈ X, it holds that:1. aX(x, x) = eX(x)
2. aX(x, y) : eX(x) → eX(y)
3. aX(x, y) ∧ aX(y, z) ≤ aX(x, z)
4. aX(x, y) = aX(y, x).
A P-functor f : X → Y is a function mapping paths into paths, strictly preservinglabeling and non decreasing agreement between them, i.e.:
1. eX(x) = eY( f (x))
2. aX(x, x′) ≤ aY( f (x), f (x′)).
Example 2 In particular, if A∗ is the free monoid generated by the alphabet A, anA∗-category X will consist in an A∗-labeled tree, as we will see in Section 5.
PSymcat is the category of symmetric P-categories and P-functors betweenthem. Notice that PSymcat is always a concrete category with sums and pullbacks.Pullbacks are obtained as in Cat by considering pairs of objects mapped on the same
2This means that there exist a family of factorisations (hi, f i1, f i
2), 0 ≤ i ≤ k + 1, with (h, f ′1, f ′
2) =(h0, f 0
1 , f 02 ) and (h′, f ′′
1 , f ′′2 ) = (hk+1, f k+1
1 , f k+12 ), and paths p1, . . . , pk, where pi belongs to either
B[hi, hi+1] or B[hi+1, hi].
S. Kasangian et al.
object by the two functors, but here they must have also the same extent. Agreementbetween two such pairs is the maximal one compatible with the projections, i.e. theintersection of the two original agreements.
Let us now suppose that we can provide a tensor product ⊗ on PSymcat withidentity I, defined from • satisfying the following properties:
1. The set of paths of X ⊗ Y is given by X × Y2. eX⊗Y(x, y) = eX(x) • eY(y)
3. aX⊗Y((x, y), (x′, y′)) ≤ aX(x, x′) • aY(y, y′)
This can be done in several ways, as it can be seen in the examples taken fromComputer Science (see Section 5). In the sequel, we will denote the path (x, y) inX ⊗ Y by x;y.
To simplify the notation and in analogy with the category of trees, we will denotethis category by T. T-categories can be also thought of as ordinary categories withextra structure. In fact the object of T between two given objects of a T-category ismade out of a “set of paths” that can be considered as a set of morphisms with extrainformation about extent and agreement. Similarly, T-functors are ordinary functorspreserving extent and agreement of paths, i.e. their structure. This can be formallyseen by considering the forgetful monoidal functor Supp : T → Set mapping everytree to the set of its paths. The corresponding change of base makes a T-categoryinto an ordinary category. In analogy with the case of the category of trees, whichis our leading example, we will call elements in Supp(C[c′, c]) paths from c to c′, forevery T-category C.
Notice that it is quite reasonable to think of the set of T- morphisms from asingleton-supported object J in T to Supp(C[c′, c]) as “elements” of C[c′, c], becausewhat is called “ an element” in general category theory, i.e. a morphism from theterminal object, or in enriched category theory, i.e. a morphism from the unit of thetensor product, would be of very little interest in this case.
Example 3 (Pseudo)ultrametric spaces can be viewed as symmetric P-categories,where P is the poset R+ of non-negative real numbers, as in [12], ordered w.r.t. ≥,now completed with a greatest element ∞. For our purposes R+ can be thought ofas a locally posetal 2-category with only one object 0, numbers as 1-cells (composedvia the function max), the order as 2-cells. As for the tensor product on P we couldconsider, for example, max itself.
Example 4 Let us now consider the poset ([0, 1],≤) of real numbers between 0 and1. It can be seen as a locally posetal 2-category P with numbers between 0 and 1as objects, and, given two such numbers, numbers smaller (or equal) than them as1-cells composed by min, the order as 2-cells. As for the tensor product on P we couldconsider, for example, min itself both on objects and 1-cells. Probability spaces canbe thought of as particular symmetric P-categories. A triple (X, eX , aX) should beinterpreted in the following way: the set X is the σ -algebra of events, the extent mapeX associates to every event its probability, and the agreement map aX associatesto every pair of events their joint probability. Probability spaces are then symmetricP-categories for which the map eX is countably additive.
Generalising Conduché’s theorem
4 The Generalisation of Conduché’s Theorem
Let us now take advantage of the category T = PSymcat defined in the previoussection, in order to extend Conduché’s theorem to T-Cat. The proof of the mainresult will proceed as follows: we will make our constructions in Set and thenguarantee that there are objects in T supported by those sets, i.e. that we can properlydefine extent and agreement for them, enjoying the required properties.
Lemma 1 Given a Conduché T-functor F : B → C, given π, π ′ ∈ Fb ′,b (B[b ′, b ]),for every p ∈ F−1
b ′,b (π) there exists a p′ ∈ F−1b ′,b (π ′) such that aB[b ′,b ](p, p′) =
aC[Fb ′,Fb ](π, π ′).
Proof Suppose aC[Fb ′,Fb ](π, π ′) = σ and let us take the subobject d of C[Fb ′, Fb ]made out of the set {π, π ′} with their extent and agreement. Then I ⊗ d producesa trivial factorisation for π (that always does exist) and since F is a Conduché T-functor, there is a corresponding factorisation of p in B[b ′, b ]. Hence, there is a T-morphism from d to B[b ′, b ] sending π to p. The image of π ′ in this morphism willbe the required to be p′. ��
Theorem 2 Let F : B → C be a Conduché T-functor between two T-categories, thenthe inverse image functor F∗ has a right adjoint �F : T-Cat|B → T-Cat|C.
Proof
– Let us give first a definition of �F in analogy with the non enriched case, takinginto account that it has to be the candidate for the right adjoint to F∗. Given theT-functor φ : X → B, we have to produce �F(φ) : �F(X) → C.If it does exist, the T-category �F(X) must have T-functors in X from inverseimages of objects of C as objects, T-functors in X from inverse images of pathsbetween objects of C as morphisms, taking care that there should be an objectin T with this support; concatenations of paths must also be obtained by pullingback along F those in C and mapping the result on X.
1. More formally, an object c in C is a T-functor from the trivial one-objectT-category 1 to C. 1 has only one object ∗ and 1[∗, ∗] is the unit object I.F∗(c) is a T-functor from F−1(1) to B. Similarly, an object in �F(X) will bea T-functor from 1 to �F(X). By the adjunction property, objects in �F(X)
must be T-functors θ from F−1(1) to X s.t. φθ = F∗(c) for some c.2. An arrow π ∈ C[c′, c] is a T-functor π from the T-category 2π with two
objects ∗ and ∗∗ and only one non-trivial path in 2π [∗, ∗∗], whose extent isthe same of π ’s, into C.F∗(π) is a T-functor from F−1(2π ) to B. Similarly, an arrow in �F(X) will bea T-functor from 2π to �F(X). By the adjunction property, the support ofarrow-objects Supp(�F(X)[θ ′, θ ]) must be the union, for any π ∈ C[c′, c],of the sets of T-functors κπ from F−1(2π ) to X s.t. φκπ = F∗(π) if θ =F−1(dom)κπ and θ ′ = F−1(cod)κπ . dom and cod are the obvious functorsfrom 1 to 2π sending the unique object in 1 respectively in the first and thesecond object in 2π . The set of such functors κπ , π ∈ C[c′, c], is the supportof an object in T candidate to be �F(X)[θ ′, θ ].
S. Kasangian et al.
The structure of �F(X)[θ ′, θ ] as a T-object is defined as follows:e(κπ ) = e(π) and a(κπ , κπ ′) = ∧
a(κπ (π, pi), κπ ′(π ′, p′j)) for pi ∈ F−1(π),
p′j ∈ F−1(π ′) and a(pi, p′
j) = a(π, π ′). This definition makes sense becauseof Lemma 1 and the completeness condition.
3. A concatenation of paths π; ρ, labeled by s • t, in C is a T-functor from athree-object T-category 3π,ρ to C. 3π,ρ has three objects ∗, ∗∗ and ∗ ∗ ∗ and3π,ρ[∗, ∗∗], 3π,ρ[∗∗, ∗ ∗ ∗], 3π,ρ[∗, ∗ ∗ ∗] are the one path objects labeled s, tand s • t, respectively.If �F is supposed to be a right adjoint to F∗, compositions in Supp(�F(X))
must be defined as the T-functors λ from F−1(3π,ρ) to X s.t. φλ = F∗(π; ρ)
for some π and ρ. Now, such a condition defines composition, i.e. just the“ commutative triangles of paths” in Supp(�F(X)) making it a category, iffthe “resulting” path cannot be present without its components, and thoseare uniquely determined up to trivially labeled subpaths.This means that every path in F∗(π; ρ) must factorise as a path in F∗(π)
concatenated with a path in F∗(ρ); if two such factorisations do exist, theintermediate objects must be connected through trivially labeled paths.This is not always the case, but this is equivalent to the fact that F islight Conduché (a factorisation of a path in F∗(π; ρ) is nothing but a lightfactorisation, indeed). Hence, this is implied by the fact that F is Conduché.The fact that these compositions are morphisms in T, and not only set-theoretical functions, comes easily from the properties of the tensor producton T.
�F(φ) will be defined by mapping every T-functor above in its original object(resp. arrow) in C.
– We will now check the adjointness condition. As in the non-enriched case, thevery definition of �F(X) implies the assertion. It is only left to prove that pairsof maps corresponding to each other in the adjunction do preserve agreementif one of them does (the same claim about extent is immediate). The reader hascertainly noticed that, by Lemma 1, given ψ : Y → C and a pair of paths η, η′in Y[y′, y], there are always pairs of inverse images in F∗(Y, ψ) such that theiragreement will reach the agreement in (Y, ψ) between them.Let us suppose α : F∗(Y, ψ) → (X, φ) is given. By definition of pullback, we havethat a((pi, η), (p′
i, η′)) = a(η, η′) ∧ a(pi, p′
i).If a(pi, p′
i) = a(π, π ′), where π is the common image of pi and η and π ′ is thecommon image of p′
i and η′, then a((pi, η), (p′i, η
′)) = a(η, η′).Being α a T-Cat|B-functor, a((pi, η), (p′
i, η′)) ≤ a(α(pi, η), α(p′
i, η′)).
We must show that its transposed α′ : (Y, ψ) → �F(X, φ) preserves the agree-ment. In fact,
α′(η) : F−1 (2π ) → X
α′ (η′) : F−1 (2π ′) → X
where ψ(η) = π and ψ(η′) = π ′. As usual, we define
α′(η)(pi) = α(pi, η)
α′ (η′)(
p′j
)= α
(p′
j, η′)
.
Generalising Conduché’s theorem
We have to prove that
a(η, η′) ≤ a
(α′(η), α′ (η′)) .
By definition,
a(α′(η), α′ (η′)) =
∧a
(α′(η)(pi), α
′ (η′)(
p′j
))
for all pairs (pi, p′j) such that a(pi, p′
j) = a(π, π ′). But, for all such pairs, we knowthat
a(η, η′) = a
((pi, η),
(p′
j, η′))
≤ a(α (pi, η) , α
(p′
j, η′))
Given β : (Y, ψ) → �F(X, φ) (making commutative the suitable diagram), bythe functoriality of T-Cat|C, we have that a(η, η′) ≤ a(β(η), β(η′)) where
β(η) : F−1(2π ) → X
β(η′) : F−1(2π ′) → X
Its transposed β ′ : F∗(Y, ψ) → (X, φ) is defined as follows
β ′(pi, η) = β(η)(pi)
β ′(
p′j, η
′)
= β(η′)
(p′
j
)
and preserves the agreement of arrow-objects in F∗Y. In fact, if a((pi, η), (p′j, η
′))does make sense, i.e. if pi and p′
j have the same domain and the same codomain,we can consider two cases:
– a(pi, p′j) = a(π, π ′), then
a((pi, η),
(p′
j, η′))
= a(η, η′) ≤ a
(β(η), β
(η′)) ≤ a
(β ′(pi, η), β ′
(p′
j, η′))
by definition of agreement between elements of �F(X, φ).– a(pi, p′
j) = σ ≤ a(π, π ′), then
a((pi, η),
(p′
j, η′))
= a(η, η′) ∧ a
(pi, p′
j
)≤ a
(β(η), β
(η′)) ∧ a
(pi, p′
j
).
We are left to prove that
a(β(η), β
(η′)) ∧ a
(pi, p′
j
)≤ a
(β ′(pi, η), β ′
(p′
j, η′))
.
From Lemma 1, there exist p′i ∈F−1(π ′) and pj ∈F−1(π) such that a(pi, p′
i)=a(pj, p′
j) = a(π, π ′). By the definition of objects of T, it holds a(p′i, p′
j) ≥ σ ,hence, being β(η′) a T-Cat|B-functor,
a(β
(η′) (
p′i
), β
(η′)
(p′
j
))≥ σ
Now,
a(β(η), β
(η′)) ∧ a
(pi, p′
j
)≤ a
(β(η)(pi), β
(η′) (
p′i
)) ∧ a(
pi, p′j
).
S. Kasangian et al.
By what we have just proved, a(pi, p′j) = σ ≤ a(β(η′)(p′
i), β(η′)(p′j)), hence
a(β(η)(pi), β
(η′) (
p′i
)) ∧ a(
pi, p′j
)≤ a
(β(η)(pi), β
(η′) (
p′i
))
∧ a(β
(η′) (
p′i
), β
(η′)
(p′
j
))
and, by definition of tree, using point 3 in the definition of P-category,
a(β(η)(pi), β
(η′) (
p′i
))∧a(β
(η′) (
p′i
), β
(η′)
(p′
j
))≤a
(β ′(pi, η), β ′
(p′
j, η′))
.
��
As a corollary, we have that the theorem above applies to categories enriched inpseudoultrametric and probability spaces (see Examples 3 and 4), provided that asuitable tensor product is defined.
5 Examples from Computer Science
5.1 Recalling the Case of LSymcat
In a previous work [7] we considered the case of L-labeled trees, which representedbehaviors of concurrent processes. There we were able to prove a form of Con-duché’s theorem for the category TreeL = LSymcat, which is a particular case ofthe one proved in the present paper. L was a meet semilattice monoid (with leftcancellation property), i.e. a monoid where composition defines a meet-semilatticestructure by using the notion of prefix, e.g. the free monoid generated by an alphabetA. Composition defines also the tensor product. TreeL can be considered as acategorical model for concurrent processes: a class of processes can be enriched overTreeL if we choose to consider a tree between two objects as the local behaviorperformed by a process (state) to reach another process (state). Paths representindividual computations, agreement represents non-determinism, while time flow isrepresented by the order in the structure of objects of L (natural numbers in the caseof a free monoid: see later).
Definition 5 A complete meet-semilattice monoid is a monoid (L, •, 1) such that theprefix relation between its elements (as usual defined as s ≤ t iff there exists u ∈ Lsuch that s • u = t) induces a complete meet-semilattice structure. A complete meet-semilattice monoid enjoys the left-cancellation property if, for every s, t, u ∈ L suchthat s • t = s • u, it holds that t = u.
Given a meet-semilattice monoid L, it can be considered as a 2-category asfollows:
– objects are elements of L;– 1-cells between two given objects are elements smaller than both. Composition
of 1-cells is given by intersection.– 2-cells are given by the order.
Generalising Conduché’s theorem
Proposition 2 L is a monoidal 2-category if the tensor product ;© is defined as follows:
– w1 ;© w2 = w1 • w2;– If s1 ∈ hom[w1, w
′1] and s2 ∈ hom[w2, w
′2], then s1 ;© s2 ∈ hom[w1 ;© w2, w
′1 ;©
w′2], will be s1 • s2, if w1 = w′
1 = s1, otherwise it is s1. The empty word is the unit ofthe tensor product.
A L-tree X = (X, eX , aX) is a L-category as in Definition 4.Given a meet-semilattice monoid L and two L-trees X and Y , we can form the
sequential composition of X and Y , X ⊗ Y = (Z , eZ , aZ ), as follows:
– Z = X × Y– eZ (x, y) = eX(x) • eY(y)
– aZ ((x, y), (x′, y′)) is aX(x, x′), if x �= x′, and eX(x) • aY(y, y′), otherwise.
T = TreeL = LSymcat, where L is a complete meet-semilattice monoid viewedas a locally posetal 2-category, yields a category with sums and pullbacks, and thetensor product just defined is right distributive w.r.t. sums. TreeL is left-closed w.r.t.the tensor product, hence it is also a TreeL-category in a canonical way. Let us callit Tree. A TreeL-subcategory of it, namely Beh (see [6]) provides a useful model forthe behavior of concurrent computing processes.
Now, it is possible to prove (see [7]) that both TreeL-categories, Tree and Beh,enjoy a very nice property: they are equipped with a Conduché TreeL-functor into Lviewed as a TreeL-category. L is the terminal TreeL-category, with only one objectand the tree (L, 1L,∧) as hom-object, actually. This last object is a tree, namely theterminal object in TreeL, with elements of L as paths, the identity function as extentsand agreement given by the intersection. Hence, there is a unique TreeL-functorfrom every TreeL-category to L; in particular, this TreeL-functor is Conduché forboth Tree and Beh.
As we anticipated in the Introduction, by considering L as a control category, sucha Conduché property for TreeL-categories means the possibility of defining a goodnotion of state for the processes modelled thereby. Conduché property for TreeL-functors between such TreeL-categories means preservation of this notion of state.
This kind of approach allowed us to prove (see De Nicola et al., submitted forpublication) that the most common notions of bisimulations [15] between concurrentprocesses can be classified and characterised as TreeL-endofuntors on Beh, accordingto their characteristic of satisfying Conduché property or, equivalently, of preservingdeterminacy [14], i.e. the notion of state.
5.2 Some Cases Where Agreement Is Interpreted as (Non-)Determinism
In the previous subsection we illustrated the result in the case of a meet-semilatticemonoid L. Let us make it more explicit in some particular cases, which get a specificmeaning in process theory.
Definition 6 Given an alphabet A, a word is a partial function N+ → A, defined onan initial segment, i.e. defined for n iff n ≤ m for a fixed m ∈ N. The empty functionwill correspond to the trivial word ε.
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The set of words on A is the free monoid (A∗, •, ε). A∗ is a meet-semilattice (A∗,≤,∧, ε), where s ≤ t if and only if s is a prefix of t and s ∧ t is the maximal commonprefix of s and t. The empty word ε is the bottom element of the semilattice, as wellas its identity.
TreeA∗ provides models for processes whose behavior is supposed to evolve in adiscrete time (see [6]), due to the fact that A∗ can be thought of as a set of partialfunctions from natural numbers to A.
Every path of a tree is interpreted as a possible evolution of a process, while agree-ment between two paths means how far two possible evolutions can be consideredequal. Hence agreement is the measure of (non-)determinism in the behavior. Forinstance, in Example 1 the leftmost behavior is less deterministic then the other one.
Analogously, we can provide models for processes whose behavior is supposed toevolve in a continuous time [5].
Definition 7 Given an alphabet A of atomic actions, a partial piece-wise constantfunction (pc-function, for short) f is a function from R+ (the non-negative realnumbers) to A defined on a bounded interval [0, t) ⊆ R+, such that for any α∈A,f −1(α) is the union of finitely many intervals of the form [r, s) , with r < s, whereboth r and s are non-negative real numbers.
Let us denote by Ac the set of pc-functions. (Ac, •,≤,∧, �) can be viewed as ameet-semilattice monoid where:
i) f ≤ g iff if x∈dom( f ) then x∈dom(g) and f (x) = g(x),ii) f ∧ g is defined as follows:
a) dom( f ∧ g) = [0, t) where t is s.t. for all x < t both f (x) and g(x) aredefined, f (x) = g(x), and, if f (t) and g(t) are both defined, then f (t) �= g(t),
b) ( f ∧ g)(x) = f (x) for every x ∈ dom( f ∧ g).
iii) the bottom element is the empty function �.iv) composition of two pc-functions f : [0, s)→A, g : [0, t)→A, f • g : [0, s +
t)→A, is defined as follows:( f • g)(x) = f (x) if x < s, ( f • g)(x) = g(x − s) if s ≤ x < s + t.
The identity of • is the empty function �. One can easily see that ≤ is the prefix orderw.r.t. this composition.
The corresponding 2-category Ac allows us to describe trees representing behav-iors evolving in the continuous time case. By replacing R+ by N we may recover thediscrete time case.
As a further example of this situation we can consider, for future work, modelsfor hybrid systems, namely processes involving two different notions of time, e.g. adiscrete and a continuous one. One can think of a computer sampling at discreteintervals through a webcam human gestures in order to modify its computation. Ofcourse, the computing time is discrete, while the human being’s time can be assumedto be continuous. There, a model of behavior is provided by a two-sided enrichedcategory from A∗ to Ac according to [9].
Intuitively, in that case a path is the coordination of some data/time flows,because it assigns “rendez-vous” between them. We do not develop here our theoryfor Hybrid = A∗ AcSymcat, but we do not see any difficulty a priori in proving
Generalising Conduché’s theorem
Conduché’s theorem also in this case, following the lines of the proof for meet-semilattice monoids.
5.3 A Case Where Agreement Is Interpreted as Synchronisation
Alternatively, we can describe a single thread in the behavior of a “family of concur-rent processes”, as a set of paths glued together at some moments, not necessarily atthe beginning (cfr. [8]). We will think of such a structure as a generalisation of a treeand call it a dendroid.
Given an alphabet A, we can define a locally posetal 2-category AN as follows:
– objects are words on the alphabet A;– if w1 and w2 are words, hom[w1, w2] is made out of all possible subsets W of the
subset of N+ × A defined by: {(n, a)|w1(n) = w2(n) = a}. Composition of 1-cellsis given by intersection.
– 2-cells are given by inclusion.
Proposition 3 AN is a monoidal 2-category if the tensor product ;© is defined asfollows:
– w1 ;© w2 is their concatenation as words;– If S1 ∈ hom[w1, w
′1] and W2 ∈ hom[w2, w
′2], then W1 ;© W2 ∈ hom[w1 ;© w2, w
′1
;© w′2], will be W1 ∪ W ′
2, where W ′2 is W2 when first components are shifted by the
length of w1, if w1 = w′1 = W1, otherwise it is W1. The empty word is the unit of
the tensor product.
AN is a symmetric lp 2-category where hom-posets are complete subsets of a uniquemeet-semilattice A∗, so Theorem 2 holds for symmetric AN-categories.
Definition 8 A dendroid (X, e, a) is a symmetric AN-category. Dendr is the categoryof dendroids with AN-functors as morphisms, i.e. AN SymCat.
Example 5 Consider the dendroid
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It has two paths, x and y, that are labeled with abac, i.e. {(1, a), (2, b), (3, a), (4, c)},and cbabb , i.e. {(1, c), (2, b), (3, a), (4, b), (5, b)}, respectively; x and y agree on theset {{(3, a)}}.
Proposition 4 Dendr has
– an initial object, given by the empty dendroid, 0 = (∅,∅,∅);– finite coproducts (i.e., sums), given by considering the juxtaposition of two den-
droids (formally, given (X, eX , aX) and (Y, eY , aY), their sum is the dendroid(X � Y, eX � eY , aX � aY), where ‘�’ denotes disjoint union);
– pullbacks, defined by taking pairs of paths with the same extent and mapped to thesame path. Agreement between two such pairs is given by the intersection of thetwo original agreements.
We can define concatenation of dendroids as well, making Dendr a monoidalcategory.
Given two A-dendroids X and Y , we can form the sequential composition of Xand Y , X ⊗ Y = (Z , eZ , aZ ), as follows:
– Z = X × Y– eZ (x, y) = eX(x) ;©eY(y)
– aZ ((x, y), (x′, y′)) is aX(x, x′), if x �= x′, otherwise eX(x) ;©aY(y, y′).
The unit element for the tensor product is the dendroid I with only one element, theempty word as extent and the empty set as agreement.
Notice that in this case we can produce an easy example where a Dendr-functor islight Conduché, but not Conduché.
Example 6 Consider a Dendr-functor F locally operating as follows, sending the leftdendroid to the right dendroid, mapping x to x′ and y to y′:
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We can factorise the dendroid on the right using a one-path dendroid d1 labeled{(1, a), (2, b)} concatenated with a dendroid d2 with two paths labeled {(1, a), (2, c)}and {(1, a), (2, b), (3, b)} respectively and with agreement {{(1, a)}}. We cannot find acorresponding factorisation of the left dendroid, because there is no morphism from
Generalising Conduché’s theorem
d1 ⊗ d2 to it. On the other hand, every “light” factorisation, i. e. if d2 is also a one-path dendroid, can be reflected. This is due to the fact that, also in this case, a “state”must have a “unique past”.
6 Conclusions and Future Work
In this work we made a further step in order to extend Conduché’s result beyond thecase of Set-based categories and of LSymcat-based categories. For us, Conduché’sresult amounts to finding a condition sufficient to guarantee, for an enriched functorenjoying the condition itself, a right adjoint to the inverse image functor. Applica-tions to computer science gave us further inputs to our research. For this purpose,we had to investigate the world of categories enriched over PSymcat, where we putdifferent kinds of 2-categories in the place of P. PSymcat is still concrete, henceit represents a quite natural generalisation of Set. On the other hand, Set can beconsidered a category of “collapsed” trees, where all paths are trivially labeled by thefree monoid generated by the empty set. In common PSymcat-categories, modelingthe behavior of suitable computational processes, the factorisation condition strictlyrelated to the result, means also the possibility of recovering a “good” notion ofstate. This notion of state is preserved by bisimulations, as we have already pointedout in previous works (De Nicola et al., submitted for publication) [7]. In fact, asit is claimed by Lawvere in the case of physical processes, objects in the involvedcategories can be thought of as processes in a given “state”, and the property ofreflecting factorisations is strictly related to the possibility of singling out a “statereached after performing a part of the evolution of the process”. As one can seefrom the two series of examples, different kinds of processes can be encompassed inthis treatment.
Here we generalised the result given in [7], which concerned meet semilatticemonoids only. Further work can be done in order to cover other cases coming fromcomputer science, e.g. a model for hybrid processes [4]. We can model this last caseas a “category enriched on two sides”, according to the bright intuition of Max Kellyin the paper [9].
Examining the properties common to all the examples for which we alreadyproved that Conduché’s theorem holds, we can try to establish a more general state-ment, involving other well known mathematical structures, described by concretecategories, as hinted by Examples 3 and 4. We can notice, for instance, that inboth cases of LSymcat and PSymcat the support functor over Set has a left adjointpreserving singletons: this property is fundamental in order to generalize the originalConduché’s proof to these new cases, and it is satisfied by many other interestingcategories, like Top (the category of topological spaces), but not by more algebraiccategories, like Gp or Ab . Hence this property could be a starting point in order toinvestigate further generalisations of the result exposed in the present paper.
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